the klimontovich description of complex plasma systems;510526/fulltext01.pdflist of papers this...

The Klimontovich Description of Complex PlasmaSystems;

Low Frequency Electrostatic Modes, Spectral Densities of Fluctuations and CollisionIntegrals

PANAGIOTIS TOLIAS

Doctoral Thesis in Physical ElectrotechnologyStockholm, Sweden 2012

TRITA-EE 2012:008ISSN 1653-5146ISRN KTH/EE--12/008--SEISBN 978-91-7501-284-1

Rymd- och PlasmafysikSkolan för Elektro- och systemteknik, KTH

SE-100 44 StockholmSWEDEN

Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framläggestill offentlig granskning för avläggande av teknologie doktorsexamen i fysikaliskelectroteknik fredagen den 23 mars 2012 klockan 13.15 i H1, Teknikringen 33, KTH,Stockholm.

© Panagiotis Tolias, 2012

Tryck: Universitetsservice US-AB

Abstract

Plasmas seeded with solid particulates of nanometer to micron sizes (complexplasma systems) are a ubiquitous feature of intergalactic, interstellar and plane-tary environments but also of plasma processing applications or even fusion devices.Their novel aspects compared with ideal multi-component plasmas stem from (i) thelarge number of elementary charges residing on the grain surface, (ii) the variabilityof the charge over mass ratio of the dust component, (iii) the inherent opennessand dissipative nature of such systems.

Their statistical description presents a major challenge; On one hand by treatingdust grains as point particles new phase space variables must be introduced aug-menting the classical Hamiltonian phase space, while the microphysics of interactionbetween the plasma and the grains will introduce additional coupling between thekinetic equations of each species, apart from the usual fine-grained electromagneticfield coupling. On the other hand complex plasma systems do not always exist ina gaseous state but can also condensate, i.e. form liquid, solid or crystalline states.

In this thesis we study gaseous partially ionized complex plasma systems fromthe perspective of the Klimontovich technique of second quantization in phase-space, initially, in regimes typical of dust dynamics. Starting from the Klimon-tovich equations for the exact phase space densities, theory deliverables such asthe permittivity, the spectral densities of fluctuations and the collision integrals areimplemented either for concrete predictions related to low frequency electrostaticwaves or for diagnostic purposes related to the enhancement of the ion density andelectrostatic potential fluctuation spectra due to the presence of dust grains. Par-ticular emphasis is put to the comparison of the self-consistent kinetic model withmulti-component kinetic models (treating dust as an additional massive chargedspecies) as well as to the importance of the nature of the plasma particle source.

Finally, a new kinetic model of complex plasmas (for both constant and fluctu-ating sources) is formulated. It is valid in regimes typical of ion dynamics, whereplasma discreteness can no longer be neglected, and, in contrast to earlier models,does not require relatively large dust densities to be valid.

iii

Acknowledgements

I would like to express my gratitude to My Tsarina, Svetlana Ratynskaia; heruncompromising attitude, her enthusiasm and dedication to science and her vol-canic (bi)polar temperament have made my two year long Ph.D experience a realpleasure. It has been a blessing having a supervisor with whom I share commonviewpoints about life and science, that has always been available for scientific dis-cussions regardless of time or workload, that has been supportive in both scientificand personal mishaps, and most of all has managed to tame (partly) my overam-bitious nature. Always caring not about the amount of papers we can produce butabout my overall scientific awareness and ethos, you had been a model supervisorfor me, even though you were asphyxiating in those pedagogical courses enforcedon you ...

It has been an honor to work with The Professore, Umberto de Angelis, whosework on the kinetic theory of complex plasmas has been a rare source of inspiration,but also frustration since such a diamond work going unnoticed makes you wonder...

I am thankful to Lars Blomberg and Nickolay Ivchenko for being co-operativeand encouraging and thus making my Department life easier, and Johanna Bergmanfor helping me to struggle through every day formalities.

Last but certainly not least, I would like to thank my family, friends and allthose who stood by me in the tragedy that befell me in the year 2010. Not tomention music, whiskey and silence for keeping me company in those late nightworking hours that I always cherish.

v

Στην μνημη του πατερα μου

Σπυριδωνα Γεωργιου Τολια

Τα παντα ρει και ουδεν μενει

vii

List of Papers

This thesis is based on the work incorporated in the following papers:

I: P. Tolias, S. Ratynskaia and U. de Angelis, Regimes for experimental tests ofkinetic effects in dust acoustic waves, Phys. Plasmas 17, 103707 (2010).

II: P. Tolias, S. Ratynskaia and U. de Angelis, Kinetic models of partially ionizedcomplex plasmas in the low frequency regime, Phys. Plasmas 18, 073705 (2011).

III: P. Tolias, Low-frequency electrostatic modes in partially ionized complex plas-mas: a kinetic approach, New J. Phys. 14, 013002 (2012).

IV: S. A. Khrapak, P. Tolias, S. Ratynskaia, M. Chaudhuri, A. Zobnin, A. Us-achev, C. Rau, M. H. Thoma, O. F. Petrov, V. E. Fortov and G. E. Morfill, Graincharging in an intermediately collisional plasma, Europhys. Lett. 97, 35001 (2012).

V: P. Tolias, S. Ratynskaia and U. de Angelis, Spectra of ion density and potentialfluctuations in weakly ionized plasmas in presence of dust grains, Phys. Rev. E 85,026408 (2012).

VI: U. de Angelis, P. Tolias and S. Ratynskaia, Effects of dust particles in plasmakinetics; ion dynamics time scales, Phys. Plasmas 19, POP37972 (2012).

ix

Contents

Abstract iii

Acknowledgements v

vii

List of Papers ix

Contents x

1 Introduction 1

2 The Klimontovich Description of Ideal Un-magnetized Plasmas 52.1 Statistical description of plasmas . . . . . . . . . . . . . . . . . . . . 52.2 Microscopic phase-space densities and small parameters . . . . . . . 72.3 Systems of non-interacting particles and the natural statistical cor-

relator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 The Klimontovich equation and its decomposition . . . . . . . . . . 122.5 The fluctuation equation and the permittivity . . . . . . . . . . . . . 142.6 The plasma kinetic equation and the collision integral . . . . . . . . 162.7 The spectral densities of fluctuations . . . . . . . . . . . . . . . . . . 22

3 The Klimontovich Description of Complex Plasmas 273.1 The effect of charge variability in the structure of the Klimontovich

kinetic scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 Basic complex plasma parameters . . . . . . . . . . . . . . . . . . . . 293.3 Basic kinetic assumptions and their critical assessment . . . . . . . . 303.4 The Klimontovich equations for the dust/plasma components . . . . 353.5 The decomposition of the Klimontovich equations . . . . . . . . . . . 363.6 The assumption of small deviations from the dust equilibrium charge 383.7 The permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.8 The collision integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 533.9 The spectral densities of fluctuations . . . . . . . . . . . . . . . . . . 57

x

CONTENTS xi

3.10 Kinetic phenomena unique in complex plasmas . . . . . . . . . . . . 58

4 The Klimontovich Description of Partially Ionized Complex Plas-mas 614.1 The Bhatnagar-Gross-Krook collision integral . . . . . . . . . . . . . 614.2 The effect of neutrals in the structure of the Klimontovich equations 634.3 The Klimontovich equations for the dust/plasma components . . . . 654.4 Decomposition in regular and fluctuating parts . . . . . . . . . . . . 664.5 The permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5 Charging of Non-emitting Grains in Presence of Neutrals 815.1 Charging in the collisionless regime . . . . . . . . . . . . . . . . . . . 825.2 Charging in the weakly collisional regime . . . . . . . . . . . . . . . 835.3 Charging in the intermediate collisional regime . . . . . . . . . . . . 855.4 Charging in the strongly collisional regime . . . . . . . . . . . . . . . 855.5 Charging in the fully collisional regime . . . . . . . . . . . . . . . . . 875.6 Maximum of the charge as a function of pressure . . . . . . . . . . . 87

6 Dust Charging Experiments in PK-4 916.1 The discharge tube and the microgravity experiments . . . . . . . . 916.2 Dust charging experiments . . . . . . . . . . . . . . . . . . . . . . . . 926.3 Comparison with charging models . . . . . . . . . . . . . . . . . . . 93

7 Summary 957.1 Paper I: Regimes for experimental tests of kinetic effects in dust

acoustic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957.2 Paper II: Kinetic models of partially ionized complex plasmas in the

low frequency regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 967.3 Paper III: Low frequency electrostatic modes in partially ionized

complex plasmas; a kinetic approach . . . . . . . . . . . . . . . . . . 987.4 Paper IV: Grain charging in an intermediately collisional plasma . . 1007.5 Paper V: Spectra of ion density and potential fluctuations in weakly

ionized plasmas in presence of dust grains . . . . . . . . . . . . . . . 1017.6 Paper VI: Effects of dust particles in plasma kinetics; ion dynamics

time scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

8 Discussion and Outlook 1058.1 Self-consistent treatment of electrostatic waves in complex plasmas . 1058.2 Development of an in situ dust diagnostic based on the fluctuation

spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1068.3 Applications of the new kinetic model . . . . . . . . . . . . . . . . . 107

Appendices 109

Appendices 111

xii CONTENTS

A Generalized Approach in the Computation of the Integral Re-sponses 111A.1 Calculation of the ion responses . . . . . . . . . . . . . . . . . . . . . 111A.2 Calculation of the dust responses . . . . . . . . . . . . . . . . . . . . 136

B Fluid Description of Dust Acoustic Waves 147

Bibliography 153

Chapter 1

Introduction

Complex (dusty) plasmas are non-ideal plasma systems seeded with solid partic-ulates. Such systems are ubiquitous in astrophysical environments [Goertz, 1989;Whittet, 2002; Draine, 2003], laboratory applications [Bouchoule, 1999; Boufendiand Bouchoule, 2002] or even fusion devices [Krasheninnikov et al., 2010]. Theirnovel aspects compared with ideal multi-component plasmas stem from the fact thatthe charge over mass ratio of the dust species is not constant. In particular, thedust charge "breathes" with the local plasma parameters and its quasi-stationaryequilibrium value is set up by the balance of the plasma fluxes absorbed and emittedby the grain. Moreover, complex plasmas are open systems, they exchange particlesand energy with the ambient environment in order for the grains to maintain theircharges. Through the continuous charging process grains serve as a sink of plasmaparticles and/or radiation which should be replenished by an external source, whosenature plays an important role in the kinetics and thermodynamics of the system[Tsytovich et al., 2008].

Complex plasma systems are most commonly met in nature in a gaseous state,where the species kinetic energy greatly exceeds the interaction energy. The Klimon-tovich technique of second quantization in phase-space [Klimontovich, 1958] offers aself-consistent approach for the description of such systems [Tsytovich and de Ange-lis, 1999] and also a unifying framework for the study and interpretation of diversetopics such as the spectral densities of fluctuations and the scattering of radiation,wave dispersion and stochastic acceleration, hydrodynamic equations and transportcoefficients, collective dust interaction inducing transitions from disordered weaklycoupled to ordered strongly coupled states.

The main motivation behind this thesis is twofold. Initially, the thesis is fo-cused on the comprehensive study of the previously "unexplored" kinetic models ofpartially ionized complex plasmas [Tsytovich et al., 2005]. More specifically, differ-ent kinetic models are formulated in presence of neutrals and compared to identifyregimes where the self-consistent description is necessary, low frequency electro-static waves are investigated self-consistently and a new long-wavelength mode is

1

2 CHAPTER 1. INTRODUCTION

discovered in presence of strong electron impact ionization of neutrals, the spectraldensities of fluctuations are derived and numerically studied as a new diagnostictool for sub-micron dust detection and composition. In addition, the thesis is de-voted to the formulation and development of a kinetic model of complex plasmasin frequency ranges typical of ion dynamics. Previous kinetic models have laid thetheoretical foundations for the extension of the Klimontovich approach for complexplasma systems. However, they endure severe mathematical complexities and havea limited parameter range of validity. They are not only restricted to low frequencyregimes typical of dust dynamics, but also they impose a strong restriction on thedust densities, that should be high enough for binary plasma collisions to be ne-glected in comparison to collisions with dust and simultaneously low enough for thedust species to be in its gaseous phase.

The basic scientific results of this thesis are gathered in six papers.Paper I employs a self-consistent kinetic model of fully ionized complex plasmas

for the study of dust acoustic waves and the identification of experimentally acces-sible parameter regimes where the deviations from the commonly used collisionlesshydrodynamic description are most pronounced.

Paper II formulates the "standard" and multi-component kinetic models inpresence of neutrals and compares with a self-consistent kinetic model of partiallyionized complex plasmas in terms of the static and dynamic permittivities. More-over, the low frequency responses of the dust and plasma species are computedtaking into account the effect of neutrals in the ion capture cross-sections in theweakly collisional regime. A criterion for the omission of induced dust densityfluctuations in frequencies larger than the dust plasma frequency is numericallyinvestigated.

Paper III is focused on the kinetic study of low frequency electrostatic waves.The effect of ionization and neutral pressure in the dust acoustic waves is investi-gated. A novel long-wavelength mode is discovered and attributed to the interplaybetween plasma absorption on dust and electron impact ionization of neutrals, aphysical mechanism is proposed and its properties are numerically investigated.

Paper IV is devoted to the experimental / theoretical study of dust charging inthe intermediate collisional regime of ions with neutrals, such a regime lies betweenthe pair-collision and hydrodynamic regimes and analytical expressions do not existfor the ion fluxes to the dust grain. The force balance method is used for thedetermination of the grain charge in experiments run in the Plasmakristall-4 facility,an interpolation formula is proposed for the ion flux that is in remarkable agreementwith the experiments.

Paper V deals with the spectral densities of ion density and electrostatic po-tential fluctuations, that are derived in presence of neutrals. Numerical results arepresented investigating the dependence on pressure and electron temperature forthe first time, the parameters used refer to realistic quiescent plasma laboratoryconfigurations. The results support the feasibility of the use of the spectral den-sities as a new diagnostic tool for the determination of the dust composition. A

3

condition is derived for the omission of plasma discreteness that properly definesthe low frequency regime of dust dynamics.

Paper VI formulates a kinetic model of complex plasmas in time scales relevantfor ion dynamics (for both constant and fluctuating plasma sources). The permit-tivity, the collision integrals and the spectral densities are derived in their generalforms. The structure of the ion kinetic equation is analyzed and applications forboth astrophysical and laboratory environments are discussed.

Chapter 2

The Klimontovich Description ofIdeal Un-magnetized Plasmas

This chapter can be regarded as a compendium summarizing results from the ki-netic description of ideal un-magnetized plasmas. The problem is viewed from theperspective of the Klimontovich approach as implemented by Tsytovich [Tsytovich,1989; Tsytovich, 1995], this method despite being mathematically elegant is rarelyencountered even in specialized textbooks.

We start by reviewing different methods of kinetic description of ideal un-magnetized plasmas and continue by presenting the Klimontovich approach in ageneralized fashion that will facilitate the transition to complex plasma systems.The system is assumed to consist of an arbitrary number of negatively and posi-tively charged species, which implies that the results also correspond to simplifiedmulti-component models of complex plasmas.

2.1 Statistical description of plasmas

The first attempt for a statistical description of plasmas originates back to Lan-dau and his derivation of the Landau kinetic equation [Landau, 1937]. The startingpoint in Landau’s approach is the Boltzmann kinetic equation. The latter, originallyderived for rarified gases, incorporates only pair-wise interactions in the collision in-tegral, justified due to the short-range nature of the molecular forces. However, dueto the long-range nature of Coulomb forces, such a treatment is not self-consistentfor plasmas. This inconsistency manifests itself as a number of divergent integralsarising in the treatment of the collision integral.

1. The collision integral diverges for Coulomb cross-sections and largedistances between charged particles. This implies that collisions betweencharged particles are important at large distances, where the change in mo-mentum and the scattering angle are small. This led Landau to a first orderexpansion of the collision integral with respect to the momentum transfer

5

6CHAPTER 2. THE KLIMONTOVICH DESCRIPTION OF IDEAL

UN-MAGNETIZED PLASMAS

and is similar to the argument underlying the Fokker-Planck description ofplasmas; "The cumulative effect of many small angle deflections in the rateof change of a particle’s kinetic energy turns out to be much larger than theeffect of the infrequent large angle deflections".

2. The collision integral diverges logarithmically for both small andlarge distances. Heuristically, we can say that the collision integral con-tains integrals of the form

∫kl

k2 dk, which result from the Fourier transformof the bare Coulomb potential, ϕ(k) ∝ e

k2 . In order to encounter this prob-lem Landau added upper and lower integration limits to the wavenumber k.The low integration limit is set on kmin = 1

λD, where λ−2

D = λ−2De + λ−2

Di isthe plasma Debye length, it expresses the fact that the Debye sphere definesthe sphere of interaction of many charged particles. It can also stem fromsubstituting the Coulomb potential with the more appropriate Yukawa po-tential, with a Fourier transform ϕ(k) ∝ e

k2+λ−2D

. The upper integration limitis set on kmax = 1

Rc, where Rc is the Coulomb radius defined by equating

the Coulomb potential energy with the particle thermal energy, Rc = e2

T . Itexpresses the weak coupling approximation, for the first order expansion tobe valid the particle kinetic energy must be weakly perturbed by the fieldsof the colliding particles, i.e r > Rc. Overall, the wavenumber integral willnow give

∫kl

k2 dk ∼∫ kmax

kmin

1k dk ∼ ln kmax

kmin∼ ln λD

Rc, where the argument of

the natural logarithm is proportional to the number of particles in the Debyesphere, nλ3

D ≫ 1. Even though the limits of integration are approximate,the low sensitivity of the logarithm in case of large arguments, ensures thecorrectness of results.

Landau’s treatment is rather a result of his brilliant physical intuition than of math-ematical rigor. However, his treatment essentially includes the physics behind morestrict statistical approaches, namely, (i) small momentum transfer in collisions, (ii)static screening of the fields, (iii) omission of non-linear effects. We should also notethat in the more precise collision integrals to be derived later, the Landau collisionintegral can be recovered by using ϵk,k·v′ ≃ 1. This demonstrates that a subtleeffect not taken into account is the dynamic plasma polarization during collisions,that can be important especially for systems far from equilibrium.

The strict theoretical foundations of a kinetic theory of gases/plasmas werelaid by Bogoliubov [Bogoliubov, 1946]. To illustrate the main ideas, let us as-sume a system of N particles, where each can be fully described by a set ofX = x1 , x2 , ... xX dynamic variables. Every particle is allocated X coordi-nates in a X × N dimensional phase space, where the system is represented by asingle point at any given time instant. Then the exact density of the system willbe a product of X × N δ-functions centered at the solutions of the deterministicequations describing the evolution of the system with given initial conditions.

Statistics can then be made by assuming an ensemble of such systems, eachprepared with different initial conditions. Then one can define distribution func-

2.2. MICROSCOPIC PHASE-SPACE DENSITIES AND SMALLPARAMETERS 7

tions and also fN (X1 , X2 , ... Xk , ...XN , t)dX1 dX2 ... dXk ... dXN to be thejoint probability that X1(t) lies within X1 , X1 + dX1 and X2(t) lies withinX2 , X2 + dX2 ... and Xk(t) lies within Xk , Xk + dXk ... and XN (t)lies within XN , XN + dXN , with the normalization condition that the integralof fN over all phase space coordinates is unity. Similarly, one can also considerreduced probability distributions by fk(X1 , X2 , ... Xk , t)dX1 dX2 ... dXk beingthe joint probability that X1(t) lies within X1 , X1 +dX1 and X2(t) lies withinX2 , X2 + dX2 ... and Xk(t) lies within Xk , Xk + dXk, irrespective of thecoordinates of the particles k +1 , k +2 , ... N . The reduced probabilities will resultfrom an integration of fN over dXk , dXk+1 , ...dXN with the appropriate normal-ization. The evolution equation for fk will be coupled to the fk+1 reduced distri-bution function, giving rise to an infinite and still exact hierarchy of equations, theBogoliubov-Born-Green-Kirkwood-Yvon hierarchy [Born and Green, 1949; Balescu,1975; Nicholson, 1983]. Truncation of the hierarchy is dictated by physical argu-ments. For example, due to the existence of small parameters in weakly coupledplasmas, truncation is justified already for k = 2.

On the other hand, the Klimontovich approach utilizes a compact X-dimensionalphase space, where the system is represented by a set of points. Separation insmooth and fluctuating parts will still produce a hierarchy in the order of fluc-tuations, that is equivalent to the BBGKY hierarchy. The approach is knownas method of second-quantization in phase-space or method of microscopic phase-space densities and is widely used for classical, quantum and relativistic plasmas[Klimontovich, 1958; Klimontovich, 1959; Klimontovich and Ebeling, 1972].

2.2 Microscopic phase-space densities and small parameters

In the Klimontovich kinetic scheme the starting point is the microscopic single par-ticle distribution function of each species fα(X, t), that in the case of point particlesis simply a sum of δ-functions positioned at exact phase-space trajectories. HereX denotes the set of phase-space variables, these are independent variables thatfully characterize any possible state of the particle and typically refer to the po-sition and momentum. However, for complex systems they can be complementedby the angular momentum and the internal energy - e.g plasma-molecular systems[Klimontovich et al., 1989]- or the charge, mass, surface temperature - e.g com-plex plasma systems [Schram et al., 2003; Tsytovich et al., 2004]-. Moreover, incase of open systems, fα(X, t) can become more complicated in order to allow forbirth/death processes and chemical transformations -e.g partially ionized systems,chemically reactive systems, complex plasmas [Klimontovich et al., 1987] -.

The phase space variables X follow sets of deterministic dynamic equations,that, when complemented by the sets of equations describing the fields that areproduced by the particles themselves, can fully characterize any future state of themany-body system provided initial conditions are known. For example for idealmagnetized plasmas, the deterministic equations consist of the Hamilton equations



of motion and the Maxwell equations for the electromagnetic fields.Integro-differential equations for fα(X, t), the Klimontovich equations, can be

constructed (i) either by their differentiating their known solution with respectto time and using the dynamic equations (ii) or by repeated integration of theLiouville equation over the phase-space variables of all particles except one, andthey represent an exact description of the system [Klimontovich, 1967].

Departure from such a deterministic description is achieved by decomposing thedistribution functions and the self-consistent fields in regular and fluctuating parts[Tsytovich, 1995],

fα(X, t) = Φα(X, t) + δfα(X, t) (2.1)

where Φα(X, t) = ⟨fα(X, t)⟩ is the smooth regular part with the brackets denot-ing the average over the Gibbs ensemble 1 of all discrete species and δfα(X, t)with ⟨δfα(X, t)⟩ = 0 is the spiky fluctuating part. The latter can be furtherseparated into the natural fluctuations δfα,(0)(X, t), connected directly to the dis-crete nature of the system, and fluctuations induced by the self-consistent fieldsδfα,(ind)(X, t). This decomposition has both physical and mathematical grounds,since δfα,(0)(X, t) is the homogeneous solution of the fluctuation equation in ab-sence of fields (free streaming particles) and δfα,(ind)(X, t) is the particular solu-tion.

The scheme proceeds with applying the decomposition in the Klimontovichand the field equations and ensemble averaging. This will yield a partial integro-differential equation for the regular part Φα(X, t), the kinetic equation, that willalso contain ensemble averages of the product of two fluctuating quantities, i.e thecollision integral. Subtraction of the averaged part from the decomposed equation,will result to an equation for the fluctuating part which is then Fourier transformedin space and time. The above system of equations is closed and can be easily treatedmathematically after invoking the following basic assumptions;

The first basic assumption of the Klimontovich kinetic scheme is thatproducts of fluctuating quantities and their ensemble averages can beomitted in the equations for the fluctuating parts. The omission of high or-der terms in fluctuations is equivalent to the omission of triple correlation functions.It is essential for the formation of a closed system of equations and its justificationis connected with the existence of small dimensionless parameters that characterizethe nature of the system [Klimontovich, 1997]. Nevertheless, such an assumptiondefinitely restricts the systems for which the description is applicable, i.e molecularsystems should be rarefied and fully ionized systems should be in the gaseous state.

In the case of rarefied gases: the radius r0 of action of the molecular forces issmall compared to the mean intermolecular distance rav. Thus, the simultaneous

1In statistic descriptions instead of focusing on a unique system, we focus on an ensembleconsisting of a very large number of identical realizations of our original system, all preparedsubject to whatever conditions are specified. In general, the systems in the ensemble will be indifferent states and characterized by different macroscopic parameters. By definition, the conceptof ensemble is directly linked to the large number of experiments that can be conducted on amacroscopic system and the concept of probability space in mathematics.

2.3. SYSTEMS OF NON-INTERACTING PARTICLES AND THE NATURALSTATISTICAL CORRELATOR 9

approach of three molecules within the sphere of action r0 is a rare event due tothe smallness of the parameter ϵg = r3

0r3

av. In that sense triple correlation functions

can be ignored being second order on ϵg and a closed system of equations for singleparticle regular distributions and double correlation functions or equivalently singleparticle regular distributions and their fluctuations can be constructed.

In the case of plasmas: the charged particles are interacting via long-rangeCoulomb forces. The effective radius of action of the forces in a plasma is the

Debye radius λD defined for a multi-component plasma via 1λ2

D

=∑

α

4π e2αnα

Tα,

while the small parameter is ϵp = r3av

λ3D

, which implies that there are many particlesin the Debye sphere (also notice that the ratio of interaction to kinetic energy fora particle is approximately proportional to ϵ

2/3p which means that the system is

in the gaseous phase). However, the ratio of collisional to collective effects is alsoapproximately proportional to ϵp. Therefore, the pair correlation function will beof the order of ϵp and the triple correlation function of the order of ϵ2

p and thusnegligible.

The second basic assumption of the Klimontovich kinetic schemerefers to the Bogoliubov hypothesis of a hierarchical structure in thecharacteristic temporal and spatial scales of variations. We begin with therelaxation time τ0 of a hydrodynamic quantity, it can be approximated by the ratioof the system’s scale L to the sound velocity cs, τ0 = L/cs. The characteristictime τ1 for a single particle distribution function to relax to its local equilibriumvalues is approximated by the ratio of the mean free path in Coulomb collisionsto the thermal velocity of the particles, which is the inverse of the collision fre-quency, τ1 = lmfp/uth = 1/νc. Finally, the characteristic time τ2 for the relaxationof a pair correlation function is estimated by the average time for a particle totravel over a correlation distance, hence it is approximately the ratio between theDebye length and the thermal velocity, which is the inverse of plasma frequencyτ2 = λD/uth = 1/ωp.

The above lead to the inequalities L ≫ lmfp ≫ λD, τ0 ≫ τ1 ≫ τ2, that arealways satisfied in a plasma due to the smallness of the parameter ϵp = νcol

ωp. This

space-time scale separation is of paramount importance in the treatment of theequations for the fluctuating parts, enabling us to treat ensemble averaged quanti-ties as constant in the fluctuations space-time scales [Ichimaru, 1992].

2.3 Systems of non-interacting particles and the naturalstatistical correlator

Before proceeding in the Klimontovich equation for ideal un-magnetized plasmas,we take a small de-tour to analyze systems of non-interacting particles. The ap-plication of fluctuation theory for such systems results in a relation for the natural



statistical correlator, a key relation to the Klimontovich description of any system[Tsytovich, 1995].

We assume an unbounded system of non-interacting particles of different species.In lack of particle collisions and in absence of external fields, the system’s Hamil-

tonian will be H(p1, p2, ..., pN ) =∑

1≤i≤N

p2i

2mi, whereas the momentum of each

particle will be independent of time. Therefore, the phase-space variables will beX = r , p and the microscopic phase-space densities for each species can be writ-ten as fα

p (r, t) =∑

1≤i≤Nα

δ(r − ri(t))δ(p − pi) and differentiation with respect to

time, together with ∂ri

∂t = vi(t) will yield the Klimontovich equation

(∂

∂t+ v · ∂

∂r

)fα

p (r, t) = 0 . (2.2)

In such a system the exact distribution function can be decomposed into regularand fluctuating parts, fα

p (r, t) = Φαp(r, t) + δf

α,(0)p (r, t), where the latter appear

solely as a consequence of the discrete nature of matter. The regular part of theKlimontovich equation is

(∂

∂t+ v · ∂

∂r

)Φα

p(r, t) = 0 , (2.3)

while the fluctuating part is

(∂

∂t+ v · ∂

∂r

)δfα,(0)

p (r, t) = 0 . (2.4)

Due to the random character of the natural fluctuations in a gas, one cannotacquire deterministic relations and only a stochastic description is viable. Knowingthat their mean value is zero, it suffices to obtain a relation for the second moment,i.e the natural statistical correlator ⟨δf

α,(0)p (r, t)δf

β,(0)p′ (r′, t′)⟩ or more conveniently

in Fourier space ⟨δfα,(0)p,k,ωδf

β,(0)p′,k′,ω′⟩ .

In absence of an interaction potential, the fluctuating part of the distributionfunction is stationary in space and time, hence all correlations between fluctuationsare functions of the time difference and the difference between the spatial coor-dinates only, ⟨δf

α,(0)p (r, t)δf

β,(0)p′ (r′, t′)⟩ = ⟨δf

α,(0)p δf

β,(0)p′ ⟩(r − r′, t − t′). Fourier

2.3. SYSTEMS OF NON-INTERACTING PARTICLES AND THE NATURALSTATISTICAL CORRELATOR 11

transforming in space and time we get

⟨δfα,(0)p,k,ω

δfβ,(0)p′,k′,ω′ ⟩ =

1(2π)8 ⟨

∫ +∞

−∞

δfα,(0)p (r, t)e

ı(ωt−k·r)dtd

3r ×∫ +∞

−∞

δfβ,(0)p′ (r

′, t

′)eı(ω′t′−k′·r′)

dt′d

3r

′ ⟩

=1

(2π)8

∫ +∞

−∞

⟨δf

α,(0)p (r, t)δf

β,(0)p′ (r

′, t

′)⟩

eı(ωt+ω′t′)

e−ı(k·r+k′·r′)

dtdt′d

3rd

3r

′

=1

(2π)8

∫ +∞

−∞

⟨δf

α,(0)p δf

β,(0)p′

⟩(r − r

′, t − t

′) eı(ωt+ω′t′)

e−ı(k·r+k′·r′)

dtdt′d

3rd

3r

′

=1

(2π)8

∫ +∞

−∞

⟨δf

α,(0)p δf

β,(0)p′

⟩(R, τ)e

ıωτe

−ık·Rd

3Rdτ ×∫ +∞

−∞

eı(ω+ω′)t′

e−ı(k+k′)·r′

dt′d

3r

′

=1

(2π)4

∫ +∞

−∞

⟨δf

α,(0)p δf

β,(0)p′

⟩(R, τ)e

ıωτe

−ık·Rd

3Rdτ δ(ω + ω

′)δ(k + k′)

=⟨

δfα,(0)p δf

β,(0)p′

⟩(ω, k) δ(ω + ω

′)δ(k + k′) .

By Fourier transforming the fluctuating part of the Klimontovich equation inspace and time we get

−ıωδfα,(0)p,k,ω + ı(k · v)δf

α,(0)p,k,ω = 0

(ω − k · v) δfα,(0)p,k,ω = 0

δfα,(0)p,k,ω ∝ δ(ω − k · v) ,

which is the only non-trivial solution of the algebraic equation. The δ−functionsimply states that non-interacting particles stream freely with a constant phasespace velocity v.

Combining we get ⟨δfα,(0)p,k,ωδf

β,(0)p′,k′,ω′⟩ ∝

⟨δf

α,(0)p δf

β,(0)p′

⟩(ω, k)δαβδ(p − p′)δ(ω +

ω′)δ(k + k′)δ(ω − k · v) with Kronecker’s delta stating that there is no correlationbetween particles of different species and δ(p − p′) stating that due to the absenceof collisions there can be no correlations for different momenta p = p′.

Another relation can be found by employing the relation for independent eventsin any statistics. Let A be a large number of statistically independent elements,then the mean square of fluctuations is equal to the mean value, i.e ⟨(δA)2⟩ = ⟨A⟩with δA = A − ⟨A⟩. By choosing A to correspond to the number of particles of onespecies that have their momenta in the range p, p+dp, A will simply be the exactdistribution function fα

p and ⟨(δfα,(0)p )2⟩ = Φα

p , which results to ⟨δfα,(0)p,k,ωδf

β,(0)p′,k′,ω′⟩ ∝

Φαpδαβδ(p − p′)δ(ω + ω′)δ(k + k′)δ(ω − k · v).

The only undetermined part is a numerical factor, that stems from the solu-tion of the fluctuation equation. It is found by using the normalization condition



⟨(δN)2⟩ = ⟨N⟩, where N is the number of particles in a volume V. Since the naturalcorrelator uses Fourier components, which refer to an infinite gas volume, whereasthe normalization condition is valid for a finite volume, we should use a Fourierseries expansion in a finite cubic volume of edge L and then consider the limitL → ∞. The procedure is trivial and yields unity for the numerical factor.

Thus, the natural statistical correlator is given by

⟨δfα,(0)p,k,ωδf

β,(0)p′,k′,ω′⟩ = δαβΦα

pδ(p − p′)δ(ω + ω′)δ(k + k′)δ(ω − k · v) . (2.5)

It is important to notice that Φαp is an arbitrary solution of the regular Klimontovich

equation and does not have to be thermal; the system of non-interacting particlesdoes not have to be in equilibrium for the above relation to be valid.

2.4 The Klimontovich equation and its decomposition

In ideal un-magnetized plasmas the particles interact through Coulomb potentials.In absence of external fields the Hamiltonian is H(r1, r2, ..., rN , p1, p2, ..., pN ) =∑1≤i≤N

p2i

2mi+ 1

2

i =j∑1≤i,j≤N

U(|ri −rj |) , where U(|ri −rj |) = eiej

|ri−rj | is the unscreened

Coulomb potential energy. The phase-space variables will be X = r , p and theexact distribution function for a species α will fα

p (r, t) =∑

1≤i≤Nα

δ(r − ri(t))δ(p −

pi(t)), where the dynamics of the particles are governed by the classical equationsof motion for the electric force, ∂ri(t)

∂t = vi(t) and ∂pi(t)∂t = eαE(ri(t), t), with

E denoting the fine-grained electrostatic field produced self-consistently by thepoint-particles themselves and described by the Poisson equation ∇ · E(r, t) =

4π∑

α

eα

∫fα

p (r, t) d3p

(2π)3 .

We note that in the equation of motion of each particle the portion of the fieldsproduced by the particle itself should not be taken into account. We also notethat the Poisson equation can be used in order to determine the exact fields interms of the particle orbits, whereas the set of dynamic equations of motion can besolved in order to determine the exact particle phase-space orbits in terms of thefields. Hence, in this exact self-consistent description, if the fields and the particlephase-space coordinates are known at one time, then we can compute them for allfollowing times.

Differentiation with respect to time, together with the dynamic equations ofmotion and the properties ∂

∂a f(a−b) = − ∂∂b f(a−b), ∂

∂t f [g(t)] = ∂f∂g

∂g∂t , aδ(a−b) =

2.4. THE KLIMONTOVICH EQUATION AND ITS DECOMPOSITION 13

bδ(a − b) gives∂fα

p

∂t=

∑1≤i≤Nα

∂

∂t[δ(r − ri(t))δ(p − pi(t))]

=∑

1≤i≤Nα

δ(r − ri(t))∂

∂tδ(p − pi(t)) +

∑1≤i≤Nα

δ(p − pi(t))∂

∂tδ(r − ri(t))

=∑

1≤i≤Nα

δ(r − ri(t))∂pi(t)

∂t·

∂

∂pi(t)δ(p − pi(t))+

∑1≤i≤Nα

δ(p − pi(t))∂ri(t)

∂t·

∂

∂ri(t)δ(r − ri(t))

= −∑

1≤i≤Nα

δ(r − ri(t))∂pi(t)

∂t·

∂

∂pδ(p − pi(t))−

∑1≤i≤Nα

δ(p − pi(t))∂ri(t)

∂t·

∂

∂rδ(r − ri(t))

= −∑

1≤i≤Nα

δ(r − ri(t))∂p

∂t·

∂

∂pδ(p − pi(t)) −

∑1≤i≤Nα

δ(p − pi(t))∂r

∂t·

∂

∂rδ(r − ri(t))

= −∂r

∂t·

∂

∂r

∑1≤i≤Nα

δ(r − ri(t))δ(p − pi(t)) −∂p

∂t·

∂

∂p

∑1≤i≤Nα

δ(p − pi(t))δ(r − ri(t))

= −∂r

∂t·

∂

∂rfα

p (r, t) −∂p

∂t·

∂

∂pfα

p (r, t)

= −v ·∂

∂rfα

p (r, t) − eαE(r, t) ·∂

∂pfα

p (r, t) ,

which yields the Klimontovich equation(∂

∂t+ v · ∂

∂r+ eαE(r, t) · ∂

∂p

)fα

p (r, t) = 0 . (2.6)

In absence of external fields, for fαp (r, t) = Φα

p(r, t) + δfαp (r, t) and E(r, t) =

δE(r, t) the decomposition scheme will yield the plasma kinetic equation(∂

∂t+ v · ∂

∂r

)Φα

p(r, t) = −eα

⟨δE(r, t) ·

∂δfαp (r, t)∂p

⟩, (2.7)

the quasi-neutrality condition∑α

eα

∫Φα

p(r, t) d3p

(2π)3 =∑

α

eαnα(r, t) = 0 , (2.8)

and the equations for the fluctuating quantities(∂

∂t+ v · ∂

∂r

)δfα

p (r, t) = −eαδE(r, t) ·∂Φα

p(r, t)∂p

, (2.9)



∇ · δE(r, t) = 4π∑

α

eα

∫δfα

p (r, t) d3p

(2π)3 . (2.10)

In the following, for simplicity we will omit the (r, t) dependence from the ensembleaverage quantities.

2.5 The fluctuation equation and the permittivity

We start by Fourier transforming the fluctuating part of the Klimontovich equationin space and time,

(ω − k · v) δfαp,k,ω = −ıeαδEk,ω ·

∂Φαp

∂p. (2.11)

The solution of the above algebraic equation is the sum of the homogeneous andthe particular solution. This separation is carried out not only due to mathemat-ical reasoning but also bears physical grounds; the solution of the homogeneousequation is acquired in the limit of uncharged non-interacting particles, eα → 0in Eqs.(2.9,2.10), and represents natural fluctuations due to particle discreetness,whereas the solution of the inhomogeneous equation is directly proportional to thefield fluctuations and represents induced fluctuations. Due to the self-consistentclosure by the Poisson equation, eventually the field fluctuations are also producedby the natural fluctuations. Therefore, the ultimate source of fluctuations is par-ticle discreteness and by combining Eqs.(2.9,2.10) one can express all fluctuatingquantities as linear functions of the natural fluctuations in Fourier space.

To proceed we regard longitudinal fields only, i.e δEk,ω = kk δEk,ω and also

add an infinitesimally small positive imaginary part +ı0 in ω − k · v [Klimon-tovich, 1998]. The addition of the infinitesimally positive imaginary part in thefrequency is done in order to regularize the divergence of the integral at the res-onance ω = k · v; by going into the complex domain Cauchy’s formula can beutilized by an integration contour encircling the pole with the direction of de-touring chosen in accordance with the causality principle. This technique is alsoknown as adiabatic switching of the interaction, since ω = ω + ı0 means thatthe field fluctuations are switched on slowly from t = −∞ instead of switched onabruptly on t = 0, e.g by assuming a monochromatic fluctuation we see that themagnitude lim

(ν,t)→(0+,−∞)|A exp (−ıω t + ν t)| = lim

(ν,t)→(0+,−∞)A exp (ν t) = 0, and

increases slowly until t = 0, where it reaches its nominal value. The reason for thechoice of the sign in +ı0 can also be seen by artificially introducing an infinitesimaldissipation/growth in the system by means of the relaxation time approximation∓ν(fα

p − NαΦα,eqp ), respectively. In that case, the integral responses would be of

2.5. THE FLUCTUATION EQUATION AND THE PERMITTIVITY 15

the form

limν→0+

∫f(x)

x − x0 ± ıνdx = lim

ν→0+

∫f(x)(x − x0 ∓ ıν)

(x − x0)2 + ν2 dx

= limν→0+

∫f(x)(x − x0)

(x − x0)2 + ν2 dx ∓ ı limν→0+

∫f(x)ν

(x − x0)2 + ν2 dx

=∫

limν→0+

f(x)(x − x0)(x − x0)2 + ν2 dx

∓ ıπ

∫f(x) lim

ν→0+

( 1π

ν

(x − x0)2 + ν2

)dx

= P.V∫

f(x)x − x0

dx ∓ ıπ

∫f(x) lim

ν→0+

( 1π

ν

(x − x0)2 + ν2

)dx

= P.V∫

f(x)x − x0

dx ∓ ıπ

∫f(x)δ(x − x0) dx

= P.V∫

f(x)x − x0

dx ∓ ıπ f(x0) ,

with the minus sign in the final formula corresponding to the physically expecteddissipation and +ı0.

δfαp,k,ω = δf

α,(0)p,k,ω + δf

α,(ind)p,k,ω = δf

α,(0)p,k,ω − ıeα

k

(1

ω − k · v + ı0k ·

∂Φαp

∂p

)δEk,ω .

(2.12)We now integrate over the momentum space, define the fluctuating plasma densi-ties δn

α,(0)k,ω =

∫δf

α,(0)p,k,ω

d3p(2π)3 and δn

α,(ind)k,ω =

∫δf

α,(ind)p,k,ω

d3p(2π)3 , and also define the

plasma species susceptibilities χαk,ω = 4πe2

α

k2

∫ 1ω−k·v+ı0

(k · ∂Φα

p

∂p

)d3p

(2π)3 .

δnαk,ω = δn

α,(0)k,ω + δn

α,(ind)k,ω = δn

α,(0)k,ω − ı k

4πeαχα

k,ωδEk,ω . (2.13)

The Fourier transform of the fluctuating part of the Poisson equation yields

ıkδEk,ω = 4π∑

α

eα

∫δfα

pk,ω

d3p

(2π)3

ıkδEk,ω = 4π∑

α

eαδnα,(0)k,ω − ı k

∑α

χαk,ωδEk,ω

ık

(1 +

∑α

χαk,ω

)δEk,ω = 4π

∑α

eαδnα,(0)k,ω

δEk,ω = 4π

ıkϵk,ω

∑α

eα

∫δf

α,(0)p,k,ω

d3p

(2π)3 , (2.14)

with

ϵk,ω = 1 +∑

α

χαk,ω = 1 + 4π

k2

∑α

e2α

∫1

ω − k · v + ı0

(k ·

∂Φαp

∂p

)d3p

(2π)3



denoting the permittivity of the system (the longitudinal part of the dielectrictensor).

Finally, by substituting Eq.(2.14) into Eq.(2.13) we can also express the totalplasma density fluctuations as a linear function of the natural fluctuations,

δnαk,ω =

(1 −

χαk,ω

ϵk,ω

)∫δf

α,(0)p,k,ω

d3p

(2π)3 −χα

k,ω

eαϵk,ω

∑β =α

eβ

∫δf

β,(0)p,k,ω

d3p

(2π)3 . (2.15)

2.6 The plasma kinetic equation and the collision integral

The plasma kinetic equation has the form

(∂

∂t+ v · ∂

∂r

)Φα

p = Jαp , (2.16)

where Jαp = −eα

∂∂p · ⟨δE(r, t)δfα

p (r, t)⟩ is the collision integral. Since for eachspecies we have both induced and natural fluctuating parts, the collision integralwill consist of two terms, Jα,1

p = −eα∂

∂p · ⟨δE(r, t)δfα,(0)p (r, t)⟩ and Jα,2

p = −eα∂

∂p ·⟨δE(r, t)δf

α,(ind)p (r, t)⟩.

For the first term: (i) we inverse Fourier transform in space and time, (ii) wesubstitute for the electric field fluctuations as functions of natural fluctuations only,applying Eq.(2.14), (iii) we use the properties of the natural statistical correlatorto discard the exponentials and the trivial integrations, (iv) we use the realityof the collision integral in real space, (v) we use the Sokhotskyi-Plemelj formula,

1x−x0+ı0 = p.v

(1

x−x0

)− ıπδ(x − x0), to separate the permittivity into real and

imaginary parts, for the imaginary part we get ℑϵk,ω = − 4π2

k2

∑β

e2β

∫δ(ω − k ·

2.6. THE PLASMA KINETIC EQUATION AND THE COLLISIONINTEGRAL 17

v′)

(k ·

∂Φβp′

∂p′

)d3p′

(2π)3 . Hence, overall

Jα,1p = −eα

∂

∂p·∫

k

k⟨δEk,ωδf

α,(0)p,k′,ω′⟩ e−ı(ω+ω′)teı(k+k′)·rdωdω′d3kd3k′

= −∑

β

eαeβ∂

∂p·∫

k

k

4π

ı kϵk,ω⟨δf

β,(0)p′,k,ωδf

α,(0)p,k′,ω′⟩ e−ı(ω+ω′)t×

eı(k+k′)·rdωdω′d3kd3k′ d3p′

(2π)3

= +∑

β

eαeβ∂

∂p·∫

k

k24πı

ϵk,ωδαβΦα

pδ(p − p′)δ(ω + ω′)δ(k + k′)δ(ω − k · v)×

e−ı(ω+ω′)teı(k+k′)·rdωdω′d3kd3k′ d3p′

(2π)3

= +e2α

∂

∂p·∫

ık

2π21

k2ϵk,ωΦα

pδ(ω − k · v) dωd3k

= +e2α

∂

∂p·∫

k

2π2

ıϵ∗k,ω

k2|ϵk,ω|2Φα


= +e2α

∂

∂p·∫

k

2π2ı(ℜϵk,ω − ıℑϵk,ω)

k2|ϵk,ω|2Φα


= −e2α

∂

∂p·∫

k

2π21

k2|ϵk,ω|2Φα

pδ(ω − k · v) 4π2

k2 ×

∑β

e2βδ(ω − k · v′)

(k ·

∂Φβp′

∂p′

)dωd3k

d3p′

(2π)3

= −2∑

β

e2βe2

α

∂

∂p·∫

k1

k4|ϵk,k·v′ |2Φα

pδ(k · v − k · v′)

(k ·

∂Φβp′

∂p′

)d3k

d3p′

(2π)3

= +∑

β

2e2βe2

α

∫ (k · ∂

∂p

)δ(k · v − k · v′)

k4|ϵk,k·v′ |2

(−Φα

pk ·∂Φβ

p′

∂p′

)d3p′ d3k

(2π)3 .

For the second term: (i) we inverse Fourier transform both fluctuating quantitiesin space and time, (ii) we substitute for the induced fluctuations as functions of theelectric field fluctuations, applying Eq.(2.12), (iii) we substitute for the electric fieldfluctuations as functions of natural fluctuations only, applying Eq.(2.14), (iv) we usethe properties of the natural statistical correlator to discard the exponentials andthe trivial integrations, (v) we use the reality of the collision integral in real space,(vi) in the presence of δ(k +k′)δ(ω +ω′) we have 1

ω′−k′·v+ı0 = − 1ω−k·v−ı0 , (vii) the

total permittivity is a real quantity is the (r, t) space, thus for the Fourier transformwe have ϵ−k,−ω = ϵ∗

k,ω and hence ϵ−k,−ωϵk,ω = |ϵk,ω|2, (viii) we decompose ıω−k·v−ı0

into imaginary and real parts with the Sokhotskyi-Plemelj formula, the real part



will be −πδ(ω − k · v).

Jα,2p = −eα

∂

∂p·∫

k

k⟨δEk,ωδf

α,(ind)p,k′,ω′ ⟩ e−ı(ω+ω′)teı(k+k′)·rdωdω′d3kd3k′

= +e2α

∂

∂p·∫

k

k

ı

k′

(1

ω′ − k′ · v + ı0k′ ·

∂Φαp

∂p

)⟨δEk,ωδEk′,ω′ ⟩×

e−ı(ω+ω′)teı(k+k′)·rdωdω′d3kd3k′

= −∑

β

∑γ

eγeβe2α

∂

∂p·∫

k

k

ı

k′16π2

kk′ϵk,ωϵk′,ω′

(1

ω′ − k′ · v + ı0k′ ·

∂Φαp

∂p

)×

⟨δfβ,(0)p′′,k,ω

δfγ,(0)p′,k′,ω′ ⟩e−ı(ω+ω′)teı(k+k′)·rdωdω′d3kd3k′ d3p′′

(2π)3d3p′

(2π)3

= −∑

β

∑γ

eγeβe2α

∂

∂p·∫

k

k

ı

k′16π2


(1

ω′ − k′ · v + ı0k′ ·

∂Φαp

∂p

)δβγΦβ

p′ ×

δ(p′ − p′′)δ(ω + ω′)δ(k + k′)δ(ω − k · v′) e−ı(ω+ω′)teı(k+k′)·rdωdω′d3kd3k′ d3p′′

(2π)3d3p′

(2π)3

= −∑

β

e2βe2

α

∂

∂p·∫

k

k

ı

k

16π2

k2ϵk,ωϵ−k,−ω

(ı

ω − k · v − ı0k ·

∂Φαp

∂p

)×

Φβp′ δ(ω − k · v′)dωd3k

d3p′

(2π)6

= −∑

β

e2βe2

α

∂

∂p·∫

k

k

ı

k

16π2

k2|ϵk,ω |2

(ı

ω − k · v − ı0k ·

∂Φαp

∂p

)Φβ

p′ δ(ω − k · v′)dωd3kd3p′

(2π)6

= +∑

β

e2βe2

α

∂

∂p·∫

k

k

ı

k

16π2

k2|ϵk,ω |2

(πδ(ω − k · v) k ·

∂Φαp

∂p

)Φβ

p′ δ(ω − k · v′)dωd3kd3p′

(2π)6

= +2∑

β

e2βe2

α

∫ (k ·

∂

∂p

) 1k4|ϵk,ω |2

(k ·

∂Φαp

∂p

)Φβ

p′ δ(ω − k · v)δ(ω − k · v′)dωd3kd3p′

(2π)3

= +2∑

β

e2βe2

α

∫ (k ·

∂

∂p

)δ(k · v − k · v′)

k4|ϵk,k·v′ |2

(Φβ

p′ k ·∂Φα

p

∂p

)d3p′ d3k

(2π)3 .

Combining the above we get

Jαp = 2

∑β

e2βe2

α

∫ (k · ∂

∂p

)δ(k · v − k · v′)

k4|ϵk,k·v′ |2

(Φβ

p′k ·∂Φα

p

∂p− Φα

pk ·∂Φβ

p′

∂p′

)d3p′d3k

(2π)3 .

(2.17)Equivalently using Einstein’s convention of summation over repeated indices we


have

Jαp = 1

2∂

∂pi

∑β

∫kikj

(4eαeβ(2π)3

k4|ϵk,k·v′ |2δ(k · v − k · v′)

)

×

[Φβ

p′

∂Φαp

∂pj− Φα

p

∂Φβp′

∂p′j

]d3p′

(2π)3d3k

(2π)3 . (2.18)

Hence, we end up with the Lenard-Balescu form of the plasma kinetic equation[Lenard, 1960; Balescu, 1960](

∂

∂t+ v · ∂

∂r

)Φα

p = 2∑

β

e2βe2

α

∫ (k · ∂

∂p

)δ(k · v − k · v′)

k4|ϵk,k·v′ |2×

[Φβ

p′

(k ·

∂Φαp

∂p

)− Φα

p

(k ·

∂Φβp′

∂p′

)]d3p′ d3k

(2π)3 . (2.19)

Below we discuss the physics lying behind the structure of the collision integraland its main properties;

1. The collision integral can be conveniently rewritten in the Fokker-Planck formand therefore it can be interpreted as the superposition of a drift process inmomentum space (due to a friction force F α

p) and of a diffusive process inmomentum space (with a diffusion coefficient tensor D

α

p). The friction forcestems from the correlator −eα

∂∂p ·⟨δE(r, t)δf

α,(0)p (r, t)⟩ and the diffusion term

from the correlator −eα∂

∂p · ⟨δE(r, t)δfα,(ind)p (r, t)⟩ that involves the spectral

density of the electric field fluctuations ⟨δEk,ωδEk′,ω′⟩. The Fokker-Planckform is [Ichimaru, 1973; Aleksandrov et al., 1983]

Jαp = ∂

∂p· D

α

p ·∂Φα

p

∂p+ ∂

∂p

(F α

pΦαp

), (2.20)

with the components of the diffusion coefficient given by

Dαp,i,j = 1

2∑

β

4e2αe2

β(2π)3∫

kikj

k4|ϵk,k·v′ |2δ(k · v − k · v′)Φβ

p′d3p′

(2π)3d3k

(2π)3 ,

(2.21)and the components of the friction force given by

F αp,i = −1

2∑

β

4e2αe2

β(2π)3∫

ki

k4|ϵk,k·v′ |2δ(k · v − k · v′)

×

(k ·

∂Φβp′

∂p′

)d3p′

(2π)3d3k

(2π)3 . (2.22)



2. The δ−function, δ(k·v−k·v′), in the collision integral is a consequence of theconservation of energy in the collision process. Let us assume two particlescolliding with initial velocities v , v′ and that the electrostatic interactiontakes place through the intermediary of a field, i.e a virtual wave that carriesmomentum hk and energy hω. We also assume small momentum transfer,|hk| ≪ |p|, and non-relativistic velocities. For the first particle conservationof energy/momentum gives [Tsytovich, 1995]

ϵp = ϵp−k + hω

p2

2m= (p − hk)2

2m+ hω

h(k · p)m

= hω ⇒ ω = k · v . (2.23)

Similarly for the second particle we have ϵp′ = ϵp′+k − hω, which resultsin ω = k · v′. Combining we get k · v = k · v′. Hence, the presence ofδ(k · v − k · v′) expresses energy conservation in Coulomb collisions in case ofsmall momentum transfer.

3. The factor 1k4|ϵk,k·v′ |2 in the collision integral expresses the dynamic screening

of the fields of the colliding particles, that is the screening that occurs dueto all the other particles through the dependence of the permittivity on theregular distribution functions of all species. A simple demonstration of thisargument can be done by considering the electrostatic self-field of a unitcharge moving with constant velocity v in a dispersive dielectric medium withcomplex permittivity ϵk,ω. Gauss electric law will read as ∇D = 4πρq, wherethe charge density is ρq(r, t) = δ(r − vt), the displacement field is D(r, t) =−ϵ(r, t)∇ (G(r, t)) and G(r, t) is the Green function of the potential of theself-field. A Fourier transform in space and time will yield the solution Gk,ω =δ(ω−k·v)2π2k2ϵk,ω

and the integral over the frequencies will give Gk =∫

Gk,ωdω =1

2π2k2ϵk,k·v. The presence of |Gk|2 ∝ 1

k4|ϵk,k·v′ |2 , is due to the screening of thefields of both particles. The appearance of the permittivity in the collisionintegral clearly expresses collective effects in collisions.

4. The probability for a collision to occur can be determined by methods of de-tailed balancing in phase-space and comparison with the collision integral.It is of great interest whether the collision integral can be found by a pro-cedure similar to the one adopted by Boltzmann in his derivation of thekinetic equation of dilute gases, especially since the omission of high or-der fluctuation terms is close to Boltzmann’s assumption of molecular chaos[Landau and Lifshitz, 1980]. The answer is no; the collective effects in col-lisions can only be taken into account properly through fluctuation theoryand two particle-scattering processes will inevitably ignore such effects. How-ever, such methods are important for the estimation of the collision prob-ability. Let us denote the probability for a binary collision per unit time


to occur between a particle of a species α with initial momentum p and aparticle of species β with initial momentum p′ with a momentum transfer k

within the range k, k + dk by wαβ

p,p′(k) d3k(2π)3 . Then it is straightforward

that the decrease per unit time of particles of a species α with momen-tum p will be equal to the probability wαβ

p,p′(k) multiplied by the numberof particles α with momentum p and by the number of particles β with mo-

mentum p′,[

δΦαp

δt

]c−

=∑

β

∫wαβ

p,p′(k)ΦαpΦβ

p′d3p′

(2π)3d3k

(2π)3 . While the inverse

process will lead to increase per unit time of particles α with momentum p,[δΦα

p

δt

]c+

=∑

β

∫wαβ

p−k,p′+k(−k)Φα

p−kΦβp′+k

d3p′

(2π)3d3k

(2π)3 . Symmetry consid-

erations in the collision probability and first order expansions in the smallmomentum transfer k both in the probability and the distribution functionswill result in a collision integral of the form

Jαp =

[δΦα

p

δt

]c+

−[

δΦαp

δt

]c−

= 12

∂

∂pi

∑β

∫kikjw

αβ(0)p,p′ (k)×

[Φβ

p′

∂Φαp

∂pj− Φα

p

∂Φβp′

∂p′j

]d3p′

(2π)3d3k

(2π)3 . (2.24)

Finally, by comparing with Eq.(2.18) we get the following expression for thezero order probability for a collision

wαβ(0)p,p′ = 4eαeβ(2π)3

k4|ϵk,k·v′ |2δ(k · v − k · v′) . (2.25)

5. The particle number, the momentum and the energy are conserved locally incollisions, integration of the related moments of the collision integral over themomentum space gives zero. The quantities 1, p, p2

2m are known as colli-sional invariants and can be used also for the construction of solutions ofthe kinetic equation. It is quite straightforward to prove that the number ofparticles of each kind is conserved, while the total (for all species α) localmomentum and kinetic energy are conserved [Liboff, 1990],∫

Jαp

d3p

(2π)3 = 0 , (2.26)

∑α

∫pJα

p

d3p

(2π)3 = 0 , (2.27)

∑α

∫p2

2mαJα

p

d3p

(2π)3 = 0 . (2.28)



6. As far as the solutions of the plasma kinetic equation are concerned, theybear a number of remarkable properties: [Nicholson, 1983; Ichimaru, 1992](I) For Φα

p ≥ 0 at t = 0, then Φp ≥ 0 for all successive times. (II) AnyMaxwellian distribution is a stationary solution, for thermal distributions thedrift and diffusion processes in momentum space balance each other. (III)As the time approaches infinity, any solution approaches a Maxwellian dis-tribution. Hence, these are the only stationary solutions. This property isstrongly connected with the second law of thermodynamics and Boltzmann’sH-theorem, the entropy will increase in time until the closed system reachesthermodynamic equilibrium.

2.7 The spectral densities of fluctuations

Combining the expressions of the fluctuating quantities as functions of the naturalfluctuations with the natural statistical correlator, we can compute the spectraldensities of the fluctuating quantities. These have the general form ⟨δ∆k,ωδ∆k′,ω′⟩and are measurable quantities that can provide vast information on fundamentalplasma parameters [Ichimaru, 1964].

In this section we will compute the spectral density of electrostatic field fluc-tuations SE

k,ω = ⟨δEk,ωδEk′,ω′⟩ (measurable as the voltage power spectrum at theterminals of dipole or monopole antennas), the spectral density of electrostatic po-tential fluctuations Sϕ

k,ω = ⟨δϕk,ωδϕk′,ω′⟩ (measurable as the floating potential ofconventional electrostatic probes) and the spectral densities of plasma density fluc-tuations Sα

k,ω = ⟨δnα,(0)k,ω δn

α,(0)k′,ω′⟩ (measurable as the ion/electron saturation currents

of Langmuir probes).We point out that for the experimental measurement of the spectral densities of

fluctuations (that are related to the discrete nature of the plasma components), qui-escent plasma configurations are needed so that the fluctuation level is not maskedby turbulence, waves or instabilities. This is not typical in laboratory discharges,since plasmas that require an electric field for their generation are usually unstableand exhibit pronounced spatial fine structures. Rare exceptions are the magneticcusp device [Spinicchia et al., 2006], the brush cathode discharge [Persson, 1965;Bingham, 1967] and its variants, such as the reflex brush cathode [Allison andChambers, 1966], the inverse brush cathode [Musal, 1966], the large V-groove cath-ode [Caron, 1971]. On the other hand space plasmas can be quiescent, there thespectral densities of electrostatic field fluctuations have been used for the determina-tion of plasma parameters (electron density, electron temperature, ion bulk speed)for decades through the established diagnostic of Quasi-thermal noise spectroscopy[Meyer-Vernet, 1979; Meyer-Vernet and Perche, 1989].

Before proceeding to the calculations, it is necessary to define the term naturalspectral functions, these are integrals of the natural correlator of any species overthe momentum space i.e

∫µ(v)⟨δf

α,(0)p,k,ωδf

α,(0)p′,k′,ω′⟩ d3p

(2π)3d3p′

(2π)3 with µ(v) a dimension-less quantity that is function of the phase-space speed. For multi-component full-

2.7. THE SPECTRAL DENSITIES OF FLUCTUATIONS 23

ionized plasmas, µ(v) = 1, and the only spectral functions are of the form Sα,(0)k,ω =∫

⟨δfα,(0)p,k,ωδf

α,(0)p′,k′,ω′⟩ d3p

(2π)3d3p′

(2π)3 . In this case they can also be interpreted as the spec-tral density of the natural plasma density fluctuations S

α,(0)k,ω = ⟨δn

α,(0)k,ω δn

α,(0)k′,ω′⟩

since δnα,(0)k,ω =

∫δf

α,(0)p,k,ω

d3p(2π)3 .

We start by calculating the spectral densities of the electrostatic field fluctua-tions. Using Eq.(2.14), the natural correlator δ-function properties and the realitycondition for the permittivity we have

SEk,ω = ⟨δEk,ωδEk′,ω′⟩

= −⟨ 4π

kϵk,ω

4π

k′ϵk′,ω′

∑α

(eα

∫δf

α,(0)p,k,ω

d3p

(2π)3

) ∑β

(eβ

∫δf

β,(0)p′,k′,ω′

d3p′

(2π)3

)⟩

= − 16π2


∑α

∑β

eαeβ

∫⟨δf

α,(0)p,k,ωf

β,(0)p′,k′,ω′⟩

d3p

(2π)3d3p′

(2π)3

= − 16π2


∑α

e2α

∫⟨δf

α,(0)p,k,ωf

α,(0)p′,k′,ω′⟩

d3p

(2π)3d3p′

(2π)3

= 16π2

k2ϵk,ωϵ−k,−ω

∑α

e2α

∫⟨δf

α,(0)p,k,ωf

α,(0)p′,k′,ω′⟩

d3p

(2π)3d3p′

(2π)3

= 16π2

k2ϵk,ωϵ∗k,ω

∑α

e2α

∫⟨δf

α,(0)p,k,ωf

α,(0)p′,k′,ω′⟩

d3p

(2π)3d3p′

(2π)3

= 16π2

k2|ϵk,ω|2∑

α

e2αS

α,(0)k,ω . (2.29)

For the spectral density of the electrostatic potential fluctuations, we simplyuse the definition δE = −∇δϕ and Eq.(2.29) to obtain

Sϕk,ω = ⟨δϕk,ωδϕk′,ω′⟩ = −⟨ 1

kk′ δEk,ωδEk′,ω′⟩

= 1k2 ⟨δEk,ωδEk′,ω′⟩ = 1

k2 SEk,ω

= 16π2

k4|ϵk,ω|2∑

α

e2αS

α,(0)k,ω . (2.30)

The spectral densities of plasma density fluctuations will contain both naturaland induced parts. We use Eq.(2.15), the natural correlator δ-function properties



and the reality condition for the permittivity and the susceptibilities to obtain

Sαk,ω = ⟨δnα

k,ωδnαk′,ω′⟩

= ⟨

(

1 −χα

k,ω

ϵk,ω

)δn

α,(0)k,ω −

χαk,ω

eαϵk,ω

∑β =α

eβδnβ,(0)k,ω

×

(

1 −χα

k′,ω′

ϵk′,ω′

)δn

α,(0)k′,ω′ −

χαk′,ω′

eαϵk′,ω′

∑β =α

eβδnβ,(0)k′,ω′

⟩

=(

1 −χα

k,ω

ϵk,ω

)(1 −

χαk′,ω′

ϵk′,ω′

)⟨δn

α,(0)k,ω δn

α,(0)k′,ω′⟩+

χαk,ω

eαϵk,ω

χαk′,ω′

eαϵk′,ω′

∑β =α

e2β⟨δn

β,(0)k,ω δn

β,(0)k′,ω′⟩

=(

1 −χα

k,ω

ϵk,ω

)(1 −

χα−k,−ω

ϵ−k,−ω

)S

α,(0)k,ω +

χαk,ω

eαϵk,ω

χα−k,−ω

eαϵ−k,−ω

∑β =α

e2β S

β,(0)k,ω

=(

1 −χα

k,ω

ϵk,ω

)(1 −

χαk,ω

ϵk,ω

)∗

Sα,(0)k,ω +

χαk,ω

eαϵk,ω

χα∗k,ω

eαϵk,ω∗∑β =α

e2β S

β,(0)k,ω

=∣∣∣∣1 −

χαk,ω

ϵk,ω

∣∣∣∣2 Sα,(0)k,ω +

∣∣∣∣χαk,ω

ϵk,ω

∣∣∣∣2 ∑β =α

e2β

e2α

Sβ,(0)k,ω

=

(1 − 2ℜ

χαk,ω

ϵk,ω +


ϵk,ω

∣∣∣∣2)

Sα,(0)k,ω +


ϵk,ω

∣∣∣∣2 ∑β =α

e2β

e2α

Sβ,(0)k,ω

=(

1 − 2ℜχα

k,ω

ϵk,ω)

Sα,(0)k,ω +


ϵk,ω

∣∣∣∣2 Sα,(0)k,ω +


ϵk,ω

∣∣∣∣2 ∑β =α

e2β

e2α

Sβ,(0)k,ω

=(

1 − 2ℜχα

k,ω

ϵk,ω)

Sα,(0)k,ω +


ϵk,ω

∣∣∣∣2 ∑β

e2β

e2α

Sβ,(0)k,ω . (2.31)

Finally, we compute the spectral density of the natural plasma density fluctua-tions

Sα,(0)k,ω =

∫ ∫⟨δf

α,(0)p,k,ωδf

α,(0)p′,k′,ω′⟩

d3p

(2π)3d3p′

(2π)3

=∫ ∫

Φαpδ(ω + ω′)δ(k + k′)δ(p − p′)δ(ω − k · v) d3p

(2π)3d3p′

(2π)3

= 1(2π)3

∫Φα

pδ(ω − k · v) d3p

(2π)3

= 1(2π)3

∫Φα(v)δ(ω − k · v)d3v . (2.32)


In case of Maxwellian distributions we have Φα(v) = nα

(mα

2πTα

)3/2exp

(− mαv2

2Tα

).

For the computation we choose spherical coordinates and k//z, the azimuthalintegration is trivial, while for the integration over the elevation angle we use thetransformation x = cos θ. Furthermore, we use the property δ(ax) = 1

|a| δ(x) ofDirac’s function, the property H(|a|x) = H(x) of Heaviside’s step function and∫ +b

−b

δ(x − |x0|)dx =

1, |x0| ≤ b0, |x0| > b

= H(b − |x0|) .

For the speed integral we use the transformation y = mαv2

2Tαand ymin = mα

2Tα

ω2

k2 ,

Sα,(0)k,ω = 1

(2π)3

∫Φα(v)δ(ω − k · v)d3v

= 2πnα

(2π)3

(mα

2πTα

)3/2 ∫ ∞

0

∫ π

0v2 exp

(−mαv2

2Tα

)δ(ω − kv cos θ) sin θdvdθ

= 2πnα

(2π)3

(mα

2πTα

)3/2 ∫ ∞

0

∫ 1

−1v2 exp

(−mαv2

2Tα

)δ(ω − kvx)dvdx

= 2πnα

(2π)3

(mα

2πTα

)3/2 ∫ ∞

0v2 exp

(−mαv2

2Tα

)∫ 1

−1δ[−kv(x − ω

kv)]dxdv

= 1(2π)3

2πnα

k

(mα

2πTα

)3/2 ∫ ∞

0v exp

(−mαv2

2Tα

)∫ 1

−1δ(x − ω

kv)dxdv

= 1(2π)3

2πnα

k

(mα

2πTα

)3/2 ∫ ∞

0v exp

(−mαv2

2Tα

)H(1 − ω

kv)dv

= 1(2π)3

2πnα

k

(mα

2πTα

)3/2 ∫ ∞

0v exp

(−mαv2

2Tα

)H(v − ω

k)dv

= 1(2π)3

2πnα

k

(mα

2πTα

)3/2 ∫ ∞

ωk

v exp(

−mαv2

2Tα

)dv

= 1(2π)3

nα

k

√mα

2πTα

∫ ∞

ymin

e−ydy

= 1(2π)7/2

nα

kvT α

∫ ∞

ymin

e−ydy

= 1(2π)7/2

nα

kvT αexp

(− ω2

2k2v2T α

). (2.33)

Chapter 3

The Klimontovich Description ofComplex Plasmas

Ideal multi-component plasmas consist of species with constant charge over massratio, eα/mα. On the other hand, in complex plasmas the dust species bears thecharacteristic of charge variability. The dust charge depends on the local plasmaparameters and is determined by the balance between plasma fluxes flowing towardsthe grain and charged particles emitted by the grain. The dust charge and itsvariability essentially determine all the novel features of complex plasma systems[Tsytovich, 1999; Tsytovich et al., 2008; Fortov et al., 2009],

• Open systems: Thermodynamically, a system is defined as "open" when itexchanges energy with the ambient in order to maintain its equilibrium. Infact, dust grains constantly collect plasma particles and radiation in order tomaintain their equilibrium charges, these have to be sustained by externalsources. A demonstration of the openness of the complex plasma systems isthe fact that the Maxwellian distribution is not a stationary solution of theplasma kinetic equations. This can be intuitively understood for electrons; intypical configurations with negatively charged dust grains, energetic electronsovercome the repulsive potential barrier and get absorbed by the grains, whichwill lead to the depletion of the high energy tails of thermal distributions.

• Non-hamiltonian systems: Due to charge variability the electrostatic forcesacting on the grains are of non-potential nature, i.e ∇ × (ZdE) = 0. Strictlyspeaking, such a system cannot be described by a Hamiltonian, due to thenon-conservative interaction between dust particles that is ultimately causedby the dynamic dust-plasma interactions.

• Dissipative systems: There is strong and continuous absorption of plasmafluxes on the grains. The dissipative nature of the system is clearly exhibitedby the presence of ω − k · v + ıνd,α(v) instead of ω − k · v + ı0 in the plasma

27

28CHAPTER 3. THE KLIMONTOVICH DESCRIPTION OF COMPLEX

PLASMAS

responses. High dissipation together with the openness of the system lead toan enhanced probability for the formation of self-organized structures.

• Systems with highly charged constituents: In contrast to ordinary plas-mas, where the components are rarely multiply charged, grain charges typi-cally reach hundreds or even thousands of elementary charges, which causesstrong grain-plasma and grain-grain interactions which lead to phase transi-tions and strongly coupled states.

However, the presence of dust does not always imply that it should be treatedas a distinct plasma species [de Angelis, 1992]. This can be determined by thecomparison of two fundamental lengths, the mean inter-grain separation rav =(

34πnd

)1/3and the plasma Debye radius λD. In the case rav ≫ λD, the grain

self-field is completely screened out by the plasma before its closest neighboringgrain can "feel" its presence and then the grains are essentially isolated (e.g dustin fusion, dust in magnetospheres/comets). In the case rav < λD the effects of theneighboring grain interaction as well as other collective effects can be importantand dust should be treated as an additional plasma component. Finally, in casesof strongly coupled dusty systems and rav ≪ λD one can even treat only dust as aspecies while treating the plasma as the ambient medium providing grain chargingand potential screening [Fortov et al., 2005].

In addition when treating the effect of dust particles in plasma kinetics, infrequency regimes typical of ion or electron dynamics, one can also avoid treatingdust dynamically (through kinetic theory or hydrodynamic approximations) due tothe fast temporal scales and short spatial scales involved.

3.1 The effect of charge variability in the structure of theKlimontovich kinetic scheme

The variability of dust charge has deep imprints on the structure of the Klimon-tovich kinetic scheme [Tsytovich and de Angelis, 1999];(i) Foremost, the charge can be regarded as a new phase space variable for dust.The charge extended phase space will be 7-dimensional r , p , q which will leadto a charge derivative term in the Liouville and the Klimontovich equations, yetthey will both still be in a continuity form.(ii) Since any new phase space variable should be accompanied by a dynamic equa-tion for the deterministic description of the system to be viable, the charging equa-tion will now complement the equations of motion.(iii) Moreover, it is necessary to obtain a new relation for the natural statisticalcorrelator for dust, ⟨δf

d,(0)p,k,ω(q)δf

d,(0)p′,k′,ω′(q′)⟩ to account for the charge. This will be

found from first principles and from the homogeneous solution of the fluctuatingpart of the dust Klimontovich equation in absence of fields.(iv) Furthermore, the dust species serves as a sink for plasma particles which also

3.2. BASIC COMPLEX PLASMA PARAMETERS 29

brings out the necessity for a source to sustain the plasma. A rigorous kinetic de-scription should provide results for the absorption cross-sections by itself throughdedicated collision integrals [Schram et al., 2000]. However, one can also adopt asink term description with cross-sections given by pair-particles collision models.(v) The microscopic phase space densities for the dust particles will still be in theform of a sum of products of δ-functions, including a term for the charge. Onthe other hand in the microscopic phase space densities for the plasma particlesthe product of δ-functions should contain extra Heaviside (step) functions that ac-count for the particles vanishing in phase space after absorption or appearing aftergeneration.

It is worth noticing that one can define more microscopic variables for dust:the angular momentum (spinning dust in presence of strong magnetic fields), thedust surface temperature (in case of inhomogeneous plasma fluxes), the mass ofthe grains (in systems with strong neutral absorption), the internal energy of thegrains. However, unlike the dust charge, they can only be important in very specificscenarios.

One question that arises, though, is what is the reason that makes all these ki-netic variables appear? What makes dust so special? The answer is quite intuitive,when approximating a constituent that has a classical inner structure with a pointparticle, then self-consistency demands that all the inner degrees of freedom shouldcomplement r , p as phase-space variables.

An established example can be found in the application of the Klimontovich ap-proach for plasma-molecular systems [Klimontovich et al., 1989]. There moleculesare treated classically as strongly coupled subsystems consisting of pairs of elec-trons/ions bound by a harmonic potential, the point particle description of moleculesthen leads to a 12-dimensional phase space instead of a 6-dimensional, with the mi-croscopic variables corresponding to position/momentum of the motion of the pairas a whole with the total momentum and to position/momentum of a fictitiousparticle with reduced mass in the center of mass frame.

3.2 Basic complex plasma parameters

In this section, we introduce the dusty plasma parameter notations and some con-ventions used in the forthcoming presentation of the kinetic model.

The notations we will follow for the equilibrium dust charge is qeq = −eZd,where Zd is the characteristic charge number (with values near 1000 for sub-microndust grains). The dust radius will be denoted by a. We also introduce averagekinetic particle energies and denote them by Te , Ti , Td for electrons, ions and dustparticles respectively, these quantities are useful in order of magnitude estimationsregarding the range of validity of our main assumptions. Common laboratory valuesare Te ≃ (1 − 3)eV, Ti ≃ 0.03eV, Td ≃ Ti.

The main parameters describing the system are the following dimensionless


PLASMAS

quantities:

z = Zde2

aTe, P = ndZd

ne, τ = Ti

Te, τd = Td(1 + P )

TiZdP(3.1)

The dimensionless charge parameter, z, has values of the order of unity, this impliesthat the equilibrium dust charge is proportional to the dust size. The ion to electrontemperature ratio, τ , has the values τ ≃ 1 in Q-machines and tokamaks and τ ≃0.01 in most discharge plasmas. The dimensionless density parameter, P , not tobe confused with the Havnes parameter defined by a normalization on ion densityPhav = ndZd/ni, gives the quasineutrality condition ni = ne(1 + P ).

We also introduce the thermal velocities for each species vT α =√

Tα

mα, the

Debye lengths λ2Dα = Tα

4πe2αnα

and the plasma frequencies ω2pα = 4πnαe2

α

mα. The total

plasma Debye length will be given by λ−2D =

∑γ

λ−2Dγ . We should also note that

the subscript α implies only the plasma species α = i, e. All fluctuations willbe denoted by δ in front of the physical quantity, the normal components of thedistributions will be denoted by Φ.

3.3 Basic kinetic assumptions and their critical assessment

1. The system is considered infinite and consists of electrons, ions and dustparticles.We refer to systems of large size compared to the length scale of the physicalphenomena of interest. Thus, boundary effects can be ignored and the Fouriertransform can be used in the treatment of the fluctuating quantities.

2. All components are in the gaseous state, where the kinetic energy substantiallyexceeds the interaction energy.This assumption is strongly related with the existence of small parameters andthe omission of high order terms in fluctuations. For the plasma components itis always satisfied, since it coincides with the definition of plasma and namelythe existence of many particles inside the Debye screening sphere. However,for the dust component, due to large value of the dust charge, this is notalways self-evident, especially for micron-dust.We assume that the mean inter-grain distance is rav ≃ n

−1/3d and that the

interaction potential is a Yukawa potential with the effective screening lengthin small distances equal to the ion Debye length (λscr ≃ λDi). Then, the ratioof interaction to kinetic energy for the dust grains, known as the coupling

parameter, can be written as Γ ≃ Z2de2

n−1/3d

Td

e−

n−1/3dλDi and the condition for the

dust component to be in the gaseous state will simply be Γ < 1.

3. The external electric and magnetic fields are considered to be zero.In absence of external electric and magnetic fields and for the mean thermal

3.3. BASIC KINETIC ASSUMPTIONS AND THEIR CRITICALASSESSMENT 31

velocity of the particles much less than the speed of light, a number of sim-plifications is possible. We use the Ampere-Maxwell equation and the Gausselectric law for the fine-grained electromagnetic fields B(r, t) and E(r, t) withthe plasma/dust particles acting a sources, γ = i, e, d ,

∇ × B(r, t) = 1c

∂E(r, t)∂t

+ 4π

c

∑γ

eγ

∫vfγ

p (r, t) d3p

(2π)3 ,

∇ · E(r, t) = 4π∑

γ

eγ

∫fγ

p (r, t) d3p

(2π)3 .

Roughly, we have |B||E| = |δB|

|δE| ∼ |v|c ∼ vT e

c ≪ 1 , which implies that themagnetic field fluctuations can be ignored compared to the electric field fluc-tuations. Moreover, from Faraday’s law

∇ × E(r, t) = −1c

∂B(r, t)∂t

we conclude that the rotational part of the electric field follows the scaling|Erot|

|Elong| ∼(

vT e

c

)2 ≪ 1 and therefore the electric field can be considered longi-tudinal and purely potential, i.e

∇ × E(r, t) = 0 , E(r, t) = −∇ϕ(r, t) .

4. The dust grains are fully characterized by position, momentum and charge.The dust charge is an independent variable and is generally not a function ofthe position of the grain.The dust particles are assumed spherical, mono-disperse, non-spinning withconstant mass, internal energy and surface temperature. The variables thatfully characterize any possible state of the dust grains are therefore position(r), moment (p) and charge (q).

5. The size of the dust grains is small compared to the plasma Debye radius.The dust grains can then be treated as point particles and their exact dis-tribution function will be a sum of δ-functions. This assumption makes itnecessary to use model cross-sections for plasma absorption on dust, that arederived by pair-particle collision approaches. Note that such cross-sectionswill be dependent on the radius and hence some deliverables of the modellike the dust fluctuations, the interaction potential, the forces between dustparticles will be explicitly dependent on the radius too. The finite size of theradius is very important and there exist forces essential for the grain dynamicsthat vanish for zero grain radius. Only in their electromagnetic fluctuationtreatment are the grains behaving as point particles, cross-sections or collisionfrequencies used outside the kinetic model can and should be dependent onthe radius.


PLASMAS

The condition a ≪ λD is nearly always satisfied in engineered dusty plasmaground or micro-gravity experiments and in naturally produced laboratorydust (typically sub-micron). It is also also typically satisfied in astrophysicalenvironments, since there the Debye lengths are large, due to the depletedplasma densities.

6. The dust velocity is much less than the electron/ion velocities.In any exact computation of the charging cross-sections we should have aresult of the form σα(q, vr) with vr = |vα − vd| the magnitude of the relativevelocity. Due to the massive dust grains the characteristic dust velocities aremuch smaller than the plasma velocities, therefore vd ≪ vα and for the cross-sections σα(q, vα), which simplifies the derivations significantly. Moreover,the dynamic screening factor |ϵeff

k,k·v′ |−2 in the dust-dust collision integralscan be evaluated at zero frequencies.

7. The sources of plasma particles and the currents emitted by the grain are con-sidered non-fluctuating.The plasma sources, s(r, t), can be varying in space and time but only slowlywhen compared to the temporal and spatial scales of the fluctuations. Theexistence of clearly distinguishable hierarchical scales is what justifies the de-composition into regular and fluctuating parts. Such sources can be e.g con-stant radiation sources leading to photo-ionization of neutrals. The currentsemitted by the grain (photo-emission, thermionic emission, field emission orsecondary emission) are also considered non-fluctuating and are denoted byIext(r, t).

8. Only the discreteness of the dust component is taken into account while theelectrons and ions are treated as continuous Vlasov fluids in phase space.This means that in the treatment of fluctuations, for the fluctuating partof the dust distribution function we use δfd

p(q, r, t) = δfd,(ind)p (q, r, t) +

δfd,(0)p (q, r, t), while for the electrons and ions we only take into account

the fluctuating parts induced by electric field fluctuations, i.e δfαp (r, t) =

δfα,(ind)p (r, t) [Tsytovich and de Angelis, 1999]. In fact, this procedure has

been described as a two-step averaging [Tsytovich et al., 2004]; a complete selfconsistent averaging procedure would involve simultaneous ensemble averag-ing over the total discreteness of the system in which the collision integralsdescribing the charging process would emerge through the fluctuation the-ory alone, on the contrary here in the first step we average over the plasmadiscreteness and use charging model cross-sections for plasma absorption ondust (like the Orbit Motion Limited approach), while in the second step weaverage over the dust discreteness.Physically, the omission of plasma discreteness is connected to the fact thatthe frequencies related to dust discreteness are much smaller than those re-lated to plasma discreteness due to the much larger grain mass, smaller char-

3.3. BASIC KINETIC ASSUMPTIONS AND THEIR CRITICALASSESSMENT 33

acteristic dust velocities and number densities. The assumption limits thevalidity of the kinetic model in the low frequency regime of dust dynamics,that can roughly be defined by ω ≪ kvT i or more accurately by ω/k < ΛαvT d

with Λα typically of the order of a few and slightly dependent on plasma/dustparameters. The assumption also implies further restrictions in the model; (i)omission of plasma discreteness leads to omission of plasma-plasma collisionintegrals in the kinetic equations which means that plasma binary collisionsshould be neglected compared to Coulomb/inelastic collisions with dust, (ii)omission of plasma discreteness leads to the omission of non-collective dustcharge fluctuations, which should be smaller than collective dust charge fluc-tuations. These consequences will be addressed in the last two assumptions.This assumption is also crucial mathematically, leading to great simplifica-tions, since now it is possible to express all fluctuating quantities through thenatural dust fluctuations and compute collision integrals and spectral densi-ties with the aid of the natural statistical correlator for dust.

9. The dust density parameter must be large enough for the electron/ion binaryCoulomb collisions to be neglected when compared to dust/plasma elastic orinelastic collisions.Coulomb collisions between plasma species can be of three types: ion-ioncollisions, ion-electron collisions and electron-electron collisions, each with itsown mean collision frequency. The largest collision frequency is the ion-ioncollision frequency given by

νii = 43

√πnie

4Λ√

miT3/2i

with Λ =∫ λD

bmin

dbb = ln

(λD

bmin

)the Coulomb logarithm representing the cu-

mulative effect of all Coulomb collisions within a Debye sphere for impact pa-rameters ranging from the distance of closest approach to the Debye length.For Coulomb collisions of ions with dust one can similarly acquire

νid = 43

√πndZ2

de4Λ′

√miT

3/2i

where the Coulomb logarithm will be different in general, due to moderate oreven strong ion-dust coupling. However, the differences will be small and itcan also be demonstrated that neither large-angle scattering nor non-linearscattering can change the above result significantly. Demanding νii < νid

for binary plasma collisions to be negligible we end up with the conditionPZd > 1 [Tsytovich, 1998].Such a condition clearly sets severe limitations; On one hand the dust densitiesmust be high enough so that collisions with dust dominate over plasma binarycollisions, whereas on the other hand the dust densities must be low enough


PLASMAS

so that the dust component is in the gaseous state. This restricts the validityof the kinetic model in engineered weakly coupled dusty plasma experiments.

10. The charge on the grain is sufficiently large (Zd ≫ 1) and the dust chargefluctuations are small.Dust grains embedded in a plasma are generally multiply charged, Zd ≫ 1.However, for grain sizes below tens of nanometers Zd could be relatively low,such cases cannot be treated classically, since then quantum mechanical ef-fects for electrons in charging usually become important: thermionic emission,field-assisted tunneling of grain surface electrons to the surrounding plasma,quantum tunneling onto the grain of plasma electrons overcoming the repul-sive potential barrier.Due to fluctuations associated with plasma or dust discreteness the surfacepotential and the charge of the dust grain will also fluctuate. Dust chargefluctuations can be either non-collective (referring to individual dust grains)or collective (referring to ensembles of dust grains) [Tsytovich et al., 2002].The source of non-collective dust charge fluctuations is the discreteness ofthe charging process itself associated with discrete impacts of individual elec-trons and ions on the grain. It can be theoretically treated as a one-stepMarkov process with the accompanying master equation describing the gen-eration and depletion of dust particles with characteristic charge number q/ethat can only vary in a ±1 stepwise fashion [Matsoukas and Russell, 1995].Alternatively, it can be treated by considering the Klimontovich decomposi-tion of the charging equation with the natural fluctuating parts of the plasmadistribution functions taken into account only, together with the natural sta-tistical correlator and the use of form factors in the capture cross-sections[Tsytovich and de Angelis, 2002]. Both approaches are equivalent and in caseof O.M.L cross-sections the result is ⟨(δZd)2⟩

Zd= τ+z

z(1+τ+z) which for τ ≪ 1

and z ≃ 1 results in ⟨(δZd)2⟩Z2

d

≃ 1Zd

1(1+z) ≃ 1

Zd. We therefore conclude that

provided that Zd ≫ 1 the dust charge fluctuations are small compared to thequasi-equilibrium dust charge, i.e ⟨(δq)2⟩ ≪ q2

eq.The source of collective dust charge fluctuations is the discreteness of thedust component. Results from the Klimontovich approach have revealed thatinduced plasma fluctuations have a negligible effect, hence one can take intoaccount only induced and natural dust fluctuations [Tsytovich and de An-gelis, 2002]. An approximate result in terms of dusty plasma parameters is⟨(δZd)2⟩

Z2d

≃ nda2λDiz2

τ2(1+z)2 , which for typical values is always much less thanunity (but simultaneously larger than the non-collective part).We have so far demonstrated that for grains with sizes above tens of nanome-ters, the kinetic assumption ⟨(δq)2⟩ ≪ q2

eq strictly holds. We know investigatethe necessity for such an assumption: Owing to the charge expansion of thephase-space, the Liouville and consequently the Klimontovich equation fordust contains an extra charge derivative term. Thus, the Fourier transformed

3.4. THE KLIMONTOVICH EQUATIONS FOR THE DUST/PLASMACOMPONENTS 35

fluctuation equation for dust will be a first order inhomogeneous differentialequation with respect to the charge instead of an algebraic equation. This willalter both the natural statistical correlator and the treatment of the induceddust fluctuations. The approximation of small deviations from the equilib-rium charge leads to physical results and less cumbersome Green’s functions.

3.4 The Klimontovich equations for the dust/plasmacomponents

The Liouville phase-space for a system consisting of Nd dust particles will be 7Nd-dimensional and the exact dust particle density in such a phase-space will have the

form N(r1, p1, q1, ..., rNd, pNd

, qNd, t) =

Nd∏i=1

δ(ri − Xi(t))δ(pi − pi(t))δ(qi − Qi(t)).

Liouville’s equation, the expression of conservation of probability in the phase space,will now obtain the form

∂N

∂t+

Nd∑i=1

∇ri(riN) +Nd∑i=1

∇pi(piN) +

Nd∑i=1

∂

∂qi(qiN) = 0 (3.2)

The density N represents the joint probability that the dust particle 1 has coor-dinates between (r1, p1, q1) and (r1 + dr1, p1 + dp1, q1 + dq1), the dust particle 2has coordinates between (r2, p2, q2) and (r2 + dr2, p2 + dp2, q2 + dq2)... and thedust particle Nd has coordinates between (rNd

, pNd, qNd

) and (rNd+ drNd

, pNd+

dpNd, qNd

+ dqNd). Hence, if we integrate over the phase-space coordinates of all

particles except one, we will end up with the reduced probability distribution ofone particle fd(r, p, q, t) that is the same with the Klimontovich microscopic phase-space density and will satisfy the equation [Tsytovich and de Angelis, 1999]

∂fdp(r, q, t)

∂t+ ∇r(rfd

p(r, q, t)) + ∇p(pfdp(r, q, t)) + ∂

∂q(qfd

p(r, q, t)) = 0 . (3.3)

So far, we only did an abstract extension to the phase-space. The physical behaviorof the dust particles is governed by a set of three dynamic equations. These are thedefinition of momentum , Newton’s second law of motion for the Lorentz force andthe charging equation.

r = p/md ; p = qE + (q/md)p × B ; I = q = Iext +∑

α

Iα . (3.4)

If we use the fact that momentum/space are independent variables, neglect themagnetic field and substitute for all terms, we end up with the generalized form ofthe Klimontovich equation for dust particles,(

∂

∂t+ v ·

∂

∂r+ qE ·

∂

∂p

)fd

p (r, q, t) +∂

∂q

[(Iext(r, t) +

∑α

Iα(q, r, t)

)fd

p (r, q, t)

]= 0 . (3.5)


PLASMAS

The current Iα(q, r, t) is the current of plasma particles collected by the grain, itwill be the sum of current fluxes on the grain, thus we just integrate the currentflux eασα(q, v)v all over the distribution function of the plasma species,

Iα(q, r, t) =∫

eασα(q, v)vfαp (r, t) d3p

(2π)3 . (3.6)

In case of the plasma species, the phase space will remain 6Nα-dimensional,the difference with the multi-component kinetic theory is that probability is notconserved in the usual way, ions/electrons are lost due to the charging process withthe dust particles acting as sinks for the plasma species distribution and hence, anexternal source of plasma particles will be added, which we assume that only has aregular component. The generalized Klimontovich equation for the plasma specieswill be given by [Tsytovich and de Angelis, 1999](

∂

∂t+ v ·

∂

∂r+ eαE ·

∂

∂p

)fα

p (r, t) = sα(r, t) −(∫

σα(q, v)vfdp′ (r, q, t)dq

d3p

(2π)3

)fα

p (r, t) .

(3.7)Finally, the system of equations is self-consistently closed by Maxwell’s equa-

tions with the particles acting as the sources of the fine-grained electromagneticfields. In case of un-magnetized plasmas and in absence of external´electric fieldsthese degenerate in the Poisson equation, that will have the form

∇ · E(r, t) = 4π

(∑α

eα

∫fα

p (r, t) d3p

(2π)3 +∫

qδfdp′(r, q, t)d3p′dq

(2π)3

). (3.8)

While in multi-component models the Klimontovich equations are coupled onlythrough the microscopic electrostatic fields, here there is additional coupling dueto the charging equation and the sink term. This introduces the expressions for thefluctuating quantities and the permittivity, introduces new collision integrals andin general results in cumbersome algebra.

In the latter we will omit the r, t dependencies for simplicity. Moreover, thedust momentum will be denoted by p′ when in the same expression with the plasmamomentum p.

3.5 The decomposition of the Klimontovich equations

The quantities are decomposed into regular (ensemble averaged) and fluctuating(with zero average) parts. Initially, the Klimontovich equations are ensemble aver-aged, which results in the kinetic equation for the regular part. The regular part issubtracted from the Klimontovich equation and second order fluctuation terms areneglected as well as their averages, which results in the equation for the fluctuatingpart. This equation is then Fourier transformed in space and time, where due tothe different space-time scales the regular components can be considered constantwhen treating the fluctuating terms.

3.5. THE DECOMPOSITION OF THE KLIMONTOVICH EQUATIONS 37

We apply the methodology for the dust particles, with E = δE, fdp(q) = Φd

p(q)+δfd

p(q), Iα(q) = ⟨Iα(q)⟩ + δIα(q), where the fluctuations in the particle current arerelated with the fluctuations in the particle density distribution via

δIα(q) = eα

∫vσα(q, v)δfα

p

d3p

(2π)3 . (3.9)

Moreover,the external currents are considered to have a regular part only. For thedust regular component we obtain

∂

∂t+ v · ∂

∂r+ ∂

∂q

[(Iext +

∑α

⟨Iα(q)⟩

)]Φd

p(q) = −q∂

∂p· ⟨δEδfd

p(q)⟩

− ∂

∂q

∑α

⟨δIα(q)δfdp(q)⟩ .

(3.10)

The terms on the left side of the equation: vary smoothly in the r, p, q phase-space and represent collective effects, the third term describes collective effectsstemming from the charging process and is absent in the multi-component kineticmodel. The terms on the right side of the equation: are spiky quantities andrepresent collisional effects, the first term describes dust-dust collisions and dust-plasma particle collisions in the presence of dust charge fluctuations, the secondterm describes the collisional effects of the charging process and has no analogy inprevious kinetic models.

For the random dust component, we Fourier transform in space and time (∂/∂t →−ıω, ∇ → ık) and we use the separate time/space-scale assumption to avoid con-volutions in the Fourier transforms of products of functions. The result will be:

ı(ω − k · v)δfdp,k,ω(q) − ∂

∂q

[(Iext +

∑α

⟨Iα(q)⟩)

δfdp,k,ω(q)

]= qδEk,ω · ∂

∂pΦd

p(q)

+ ∂

∂q

(∑α

δIαk,ω(q)Φd

p(q)

). (3.11)

Homogeneous equation: when the right hand side is zero, we have a differentialequation that depends on the spiky quantity δfd

p,k,ω(q) only, the solution will givethe free dust particle fluctuations due to discreetness. Inhomogeneous equation:when the right hand side is non-zero, we have to find a particular solution, this willbe dependent on the spiky quantities δIα

k,ω(q), δEk,ω and will describe fluctuationsinduced by the fluctuating electric field and by fluctuations of the currents in thecharging of dust, the term δIα

k,ω(q) will couple this equation to the random plasmacomponent equation. The total solution will be the sum of the homogeneous andthe particular solution, δfd

p,k,ω(q) = δfd,(0)p,k,ω(q) + δfd,ind

p,k,ω(q), in analogy with themulti-component kinetic models.


PLASMAS

We apply the methodology for the plasma species, with E = δE, fdp(q) =

Φdp(q) + δfd

p(q), fαp = Φα

p + δfαp , we assume that the external source does not

have a fluctuated part. We also define the capture collision frequency throughνd,α(v) =

∫vσα(q′, v)Φd

p′(q′) dq′d3p′

(2π)3 and the fluctuating part of the capture fre-quency in Fourier space through δνd,α(ω, k, v) =

∫vσα(q′, v)δfd

p′,k,ω(q′) dq′d3p′

(2π)3 . Forthe regular component we have the equation:[

∂

∂t+ v · ∂

∂r

]Φα

p = sα − νd,α(v)Φαp − eα⟨δEk,ω

∂

∂pδfα

p,k,ω⟩

−∫

vσα(q′, v)⟨δfαp,k,ωδfd

p′,k,ω(q′)⟩dq′d3p′

(2π)3 . (3.12)

The first four terms vary smoothly in the phase space and describe collective effects,with the last two terms describing the effect of sinks/sources. The last two termsare the modified collision integral, they contain collisions of plasma particles andcollisions of plasma particles with dust particles and also inelastic collisions withdust grains.

For the random fluctuating component we obtain:

δfαp,k,ω = 1

ı(ω − k · v + ıνd,α(v))

(eαδEk,ω ·

∂Φαp

∂p+ δνd,α(ω, k, v)Φα

p

). (3.13)

Thus, in this model, the presence of sinks due to the inelastic collisions with dustleads to an effective damping of the fluctuations. We also remind that there are nonatural fluctuations but only fluctuations induced by the electric field or by dustdiscreteness.

Finally, the regular part of the Poisson equation together with the regular part ofthe charging equation will just reproduce the quasi-neutrality condition,

∑α

eαnα+

qeqnd = 0, while the fluctuating part of the Poisson equation will simply be

ıkδEk,ω = 4π

∑α

eα

∫δfα

p,k,ω

d3p

(2π)3 +∫

qδfdp′,k,ω(q) d3p′dq

(2π)3

. (3.14)

3.6 The assumption of small deviations from the dustequilibrium charge

In this part, we will use the approximation that the deviations from the equilibriumcharge in both the regular and the fluctuating parts of the dust particle distributionfunction are small [Tsytovich and de Angelis, 1999]. The quasi-equilibrium chargeis given by the condition that the average net particle flux on the dust particlesvanishes. Hence, we regard the charging equation for the ensemble averaged parts

3.6. THE ASSUMPTION OF SMALL DEVIATIONS FROM THE DUSTEQUILIBRIUM CHARGE 39

and set the total current dq/dt equal to zero, this yields

Iext +∑

α

⟨Iα(qeq)⟩ = 0 ⇒ Iext +∑

α

∫eασα(qeq, v)vΦα

p

d3p

(2π)3 = 0 (3.15)

We assume small deviations of the dust charges around the equilibrium value,q =qeq + ∆q. We use the Taylor expansion for the particle current on the dust particleand we keep the first order terms only.

∑α

⟨Iα(q)⟩ =∑

α

⟨Iα(qeq)⟩ + ∆q∂

∂q

(∑α

⟨ Iα(qeq)⟩)

+ 12!

(∆q)2 ∂2

∂q2

(∑α

⟨Iα(qeq)⟩)

≃∑

α

⟨Iα(qeq)⟩ + ∆q∂

∂q

(∑α

⟨ Iα(qeq)⟩

)≃∑

α

⟨Iα(qeq)⟩ − νch∆q , (3.16)

where we defined the charging frequency νch = − ∂∂q

(∑α

⟨ Iα(q)⟩

)|q=qeq . We

substitute q = qeq + ∆q in the charging equation and use the Taylor expansion andthe quasi-equilibrium condition, together with Eq.(3.15),

∂

∂t(qeq + ∆q) = Iext +

∑α

⟨Iα(qeq)⟩ − νch∆q ⇒

∂

∂t(∆q) = −νch∆q ⇒ ∂(∆q)

∆q= −νch∂t ⇒∫ t

0

∂(∆q)∆q

= −νcht ⇒ ln ∆q(t)∆q0

= −νcht ⇒ ∆q(t) = ∆q0 exp (−νcht) , (3.17)

where ∆q0 is the initial value of the deviation at t = 0. It is obvious that thecharges are relaxing to their equilibrium values with a time constant τrel = 1/νch.Furthermore, the initial sign of the deviations does not change until equilibrium isreached. Therefore, by normalizing the deviations with their initial value we willalways obtain a positive quantity, q = ∆q/∆q0 > 0 .

We can now return to the equation of the fluctuating part of the dust distributionfunction and apply the above approximation. Initially, we regard the homogeneousdifferential equation in order to acquire a relation for the dust natural statisticalcorrelator,

ı(ω − k · v)δfd,(0)p,k,ω(q) − ∂

∂q

[(Iext +

∑α

⟨Iα(q)⟩

)δf

d,(0)p,k,ω(q)

]= 0

ı(ω − k · v)δfd,(0)p,k,ω(∆q) + ∂

∂∆q

(νch∆qδf

d,(0)p,k,ω(∆q)

)= 0. (3.18)


PLASMAS

The above equation is a first order homogeneous differential equation of the form,

A(x)f(x) + d

dx(B(x)f(x)) = 0

A(x)f(x) + B(x)df(x)dx

+ f(x)dB(x)dx

= 0

B(x)df(x)dx

= −[A(x) + dB(x)

dx

]f(x)

df(x)f(x)

= −[

A(x)B(x)

+ B′(x)B(x)

]dx

ln f(x) = − ln B(x) −∫

A(x)B(x)

dx

f(x) = C ′

B(x)exp

(−∫

A(x)B(x)

dx

),

where C ′ is the integration constant. For x = ∆q , A(x) = ı(ω − k · v) , B(x) =νch∆q , C = C′

νchthis gives

δfd,(0)p,k,ω(∆q) = C

∆qexp

[ln (∆q)−ı(ω−k·v)/νch

]δf

d,(0)p,k,ω(∆q) = C

∆qexp

[− ı(ω − k · v)

νchln (∆q)

](3.19)

In the case of ∆q < 0 we should find an alternative expression for the solution due tothe undefinable negative logarithm. Since sgn(∆q) = sgn(∆q0), we can normalizewith respect to the initial deviation, q = ∆q/∆q0 > 0 and resolve the ill-definedlogarithm issue,

ı(ω − k · v)δfd,(0)p,k,ω(∆q) + ∂

∂∆q

(νch∆q0

∆q

∆q0δf

d,(0)p,k,ω(∆q)

)= 0

ı(ω − k · v)δfd,(0)p,k,ω(∆q) + ∂

∂(

∆q∆q0

) [νch

(∆q

∆q0

)δf

d,(0)p,k,ω(∆q)

]= 0

ı(ω − k · v)δfd,(0)p,k,ω(q) + ∂

∂q

[νchqδf

d,(0)p,k,ω(q)

]= 0

δfd,(0)p,k,ω(q) = C

qexp

[− ı(ω − k · v)

νchln (q)

](3.20)

Moreover, we can choose the constant C to correspond to the position of the chargeat t = 0 being r = r0, this delta function condition is equivalent to e−ık·r0 in theFourier space. Overall, we have

δfd,(0)p,k,ω(q) = e−ık·r0

qexp

[− ı(ω − k · v)

νchln q

]. (3.21)


This expression for the natural dust fluctuations can be expressed in an approximatebut more convenient form if we switch back to the time-space domain. We use theinverse Fourier transform, split the resulting quadruple integral into a product ofintegrals and use the properties of the delta function δ(t) = 1

2π

∫ +∞−∞ e−ıωtdω ,

δ(r) = 1(2π)3

∫ +∞−∞

∫ +∞−∞

∫ +∞−∞ e+ık·rd3k and δ(at) = 1/|a|δ(t) ,

δfd,(0)p (r, q, t) ∝ 1

q

1(2π)4

∫ +∞

−∞e−ık·r0 exp

[− ı(ω − k · v)

νchln q

]e−ıωte+ık·rdωd3k

∝ 1q

12π

∫ +∞

−∞exp

[−ıω(t + ln q

νch)]dω×

1(2π)3

∫ +∞

−∞exp

[+ık · (r − r0 + v ln q

νch)]d3k

∝ 1q

δ

(t + ln q

νch

)δ

(r − r0 + v ln q

νch

)∝ 1

qδ

(t + ln q

νch

)δ (r − r0 − vt)

∝ νch

qδ (ln q + νcht) δ (r − r0 − vt)

∝ νch

qδ [ln (q exp (+νcht))] δ (r − r0 − vt) (3.22)

One of the properties of the delta function states that δ(f(x)) =∑

i

δ(x − xi)f ′(xi)

where the function f(x) is continuously differentiable, the summation is over allthe roots of f(x) and xi is a simple root of the function f(x). We use this propertyfor our case with f(q) = ln [qe+νcht]. It has a single root at q = e−νcht and thederivative is simply 1

q . We now have

δfd,(0)p (r, q, t) ∝ δ(∆q − ∆q0 exp (−νcht))δ (r − r0 − vt) (3.23)

The conclusions from this relation are: any charge moves in space with constantvelocity v and reaches its equilibrium value ∆q = 0 rapidly with the time deter-mined by the ratio 1/νch, which is small. If we have low-frequency fluctuations infrequencies less than the charging frequency, we can neglect the exponential factorin Eq.(3.23), hence for these fluctuations the equilibrium charge is reached almostinstantaneously,

δfd,(0)p (r, q, t) ∝ δ(∆q)δ (r − r0 − vt) (3.24)


PLASMAS

We return to Fourier space,

δfd,(0)p,k,ω(q) ∝ δ(∆q)

∫ +∞

−∞

∫ +∞

−∞

∫ +∞

−∞

∫ +∞

−∞δ (r − r0 − vt) e+ıωte−ık·rdtd3r

∝ δ(∆q)∫ +∞

−∞

∫ +∞

−∞

∫ +∞

−∞δ (r − r0 − vt) e−ık·rd3r

∫ +∞

−∞e+ıωtdt

∝ δ(∆q)∫ +∞

−∞e+ıωte−ık·(r0+vt)dt

∝ δ(∆q)e−ık·r0

∫ +∞

−∞e−ı(k·v−ω)tdt

∝ e−ık·r0δ(∆q)δ(ω − k · v) . (3.25)

It is obvious that this solution still satisfies the homogeneous equation of fluctua-tions, the first term will vanish due to δ(ω − k · v) and the second due to δ(∆q).One could also reach this solution by using a limiting procedure for lim (νch) → 0in Eq.(3.21) and some properties of generalized functions.We can now average the product of two free particle fluctuations with respectto the charge position r0, we use that δ(∆q)δ(∆q′) = δ(q − qeq)δ(q′ − qeq) =δ(q − q′)δ(q − qeq) and ⟨eık·r0eık′·r0⟩ =

∫ ∫ ∫exp (ı(k + k′) · r0)d3r0 = δ(k + k′),

⟨δfd,(0)p′,k′,ω′ (q′)δf

d,(0)p,k,ω

(q)⟩ ∝ δ(∆q)δ(∆q′)δ(ω − k · v)δ(ω′ − k′ · v′)δ(p − p′)

∝ δ(q − q′)δ(q − qeq)δ(ω − k · v)δ(ω′ − k′ · v′)δ(p − p′)δ(k + k′)∝ δ(q − q′)δ(q − qeq)δ(ω − k · v)δ(ω′ − k′ · v)δ(p − p′)δ(k + k′)∝ δ(q − q′)δ(q − qeq)δ(ω − k · v)δ(ω′ + k · v)δ(p − p′)δ(k + k′)∝ δ(q − q′)δ(q − qeq)δ(ω − k · v)δ(ω′ + ω)δ(p − p′)δ(k + k′) . (3.26)

Finally, the proper normalization of the average should be the average part of thedust distribution function, which yields

⟨δfd,(0)p′,k′,ω′ (q′)δf

d,(0)p,k,ω(q)⟩ = Φd

p(q) δ(q − q′)δ(q − qeq)δ(ω − k · v)δ(ω′ + ω)δ(p − p′)δ(k + k′) .(3.27)

This is a generalization of the natural statistical correlator of the multi-componentkinetic theory,

⟨δfα,(0)p′,k′,ω′δf

β,(0)p,k,ω⟩ = Φα

pδαβδ(p − p′)δ(k + k′)δ(ω + ω′)δ(ω − k · v) , (3.28)

where the differences are: the omitted Kronecker delta due to the fact that there isonly one kind of dust particles and the addition of charge related delta functions.

We now find Green’s function of the inhomogeneous equation for the fluctuatingpart of the dust distribution function. It can be conveniently rewritten as[

ıω − k · v

νch+ ∂

∂qq

]δfd

p,k,ω(q) = 1νch

Rp,k,ω(q), (3.29)

where Rp,k,ω(q) refers to the inhomogeneous part, that is dependent on the electricfield and particle current fluctuations. For a general solution, we should first find


the Green’s function that satisfies the differential equation[ıω − k · v

νch+ ∂

∂qq

]G(q, q′, ω − k · v) = 1

νchδ(q − q′). (3.30)

The solution will be the same as in the homogeneous equation, but the constantwill be dependent on q′, thus

G(q, q′) = C(q′)q

exp(

−ıω − k · v

νchln q

)(3.31)

In the theory of Green’s functions in the second order differential equations, theGreen’s function is continuous but not differentiable. In our case, we have a firstorder differential equation, this leads to the conclusion that the Green’s functionwill not be continuous due to the necessary delta jump on q′.

G(q, q′) =

C1(q′)

q exp(

−ı ω−k·vνch

ln q)

, q′ > q

C2(q′)q exp

(−ı ω−k·v

νchln q)

, q′ < q(3.32)

Now let ϵ > 0, with ϵ being arbitrarily small,then q′+ = q′ + ϵ,q′

− = q′ − ϵ and weintegrate the differential equation in the interval (q′

−, q′+) using the discontinuity of

the Green’s function,

ıω − k · v

νch

∫ q′+

q′−

G(q, q′)dq +∫ q′

+

q′−

∂

∂qqG(q, q′)dq = 1

νch

∫ q′+

q′−

δ(q − q′)dq ⇒

ıω − k · v

νch

∫ q′+

q′−

G(q, q′)dq +∫ q′

+

q′−

∂

∂qqG(q, q′)dq = 1

νch⇒[qG(q, q′)

]q′+

q′−

= 1νch

⇒

q′ [G+(q′, q′) − G−(q′, q′)] = 1νch

⇒ C1(q′) − C2(q′) = − 1νch

exp(

+ıω − k · v

νchln q′

)(3.33)

Causality arguments lead to the conclusion that for q′ < q the response must bezero. Hence, G−(q, q′) = 0 ⇒ C2(q′) = 0, and with the use of the above relationC1(q′) = − 1

νchexp

(+ı ω−k·v

νchln q′

). Overall, we have

G(q, q′) =

− 1

νchq exp(

−ı ω−k·vνch

[ln (q) − ln (q′)])

, q′ > q

0, q′ < q(3.34)

Knowledge of the Green’s function of the problem means that the solution of theinhomogeneous equation with an inhomogeneous term Rp,k,ω(q) will be

δfd,(ind)p,k,ω (q′) =

∫G(q′, q′′)Rp,k,ω(q′′)dq′′ . (3.35)


PLASMAS

We note that the inhomogeneous term consists of a "non-charge derivative" sourceterm (RI

p,k,ω(q) = qδEk,ω · ∂∂p Φd

p(q)) and a "full charge derivative" source term

(RIIp,k,ω(q) = ∂

∂q

(∑α

δIαk,ω(q) Φd

p(q)

)).

We can now investigate the basic properties of the Green’s function. In thetreatment of fluctuations and the collision integrals various integrals of the inducedfluctuating part of the dust distribution function will appear. In general they canbe categorized into two groups. In the first case, the integral will have the formIcol1 =

∫A(q, q′)δf

d,(ind)p,k,ω (q′)dq′, with A(q, q′) some weighting function and the

source term not a full derivative with respect to the charge, then we can use theapproximation q′ ≃ qeq in the weighting function, A(q, q′) ≃ A(q, qeq). In thesecond case, the integral will have the form Icol2 =

∫D(q, q′)δf

d,(ind)p,k,ω (q′)dq′, with

D(q, q′) some weighting function and the source term a full derivative with respectto the charge, RII

p,k,ω(q) = ∂B(q)∂q . Use of the same approximation, would lead to a

zero result, to avoid the trivial result, we use the Taylor expansion of the weightingfunction around the equilibrium charge and keep the two first linear terms.For the first case, we substitute Eq.(3.35) in the integral, separate the multipleintegrals, use the equilibrium approximation and substitute for the Green’s function

Icol1 =∫

A(q, q′)δfd,(ind)p,k,ω (q′)dq′

=∫

RIp,k,ω(q′′)dq′′

∫A(q, q′)G(q′, q′′)dq′

=∫

RIp,k,ω(q′′)A(q, qeq)dq′′

∫G(q′, q′′)dq′

= −∫


∫1

νchq′ exp(

−ıω − k · v

νch[ln (q′) − ln (q′′)]

)dq′

= −∫


∫1

νchq′ exp(

−ıΩ

νch[ln (q′) − ln (q′′)]

)dq′,

(3.36)

where we have set Ω = ω − k · v for convenience. The Green’s function is zerofor q′′ < q′ and non-zero for q′′ > q′, hence the upper limit in the dq′ integrationmust be q′′. Since the normalized charge is always positive the lower boundary ofintegration will be zero. Moreover, use of the transformation ln q′ = u in the same


integral will result in du = dq′

q′ and the integration limits will become (−∞, ln q′).

Icol1 = −∫


∫ q′′

0

1νchq′ exp

(−ı

Ωνch

[ln (q′) − ln (q′′)])

dq′

= − 1νch

∫RI

p,k,ω(q′′)A(q, qeq) exp (ı Ωνch

ln q′′)dq′′∫ ln q′′

−∞exp (−ı

Ωνch

u)du

= 1ıΩ

∫RI


ln q′′)dq′′[exp (−ı

Ωνch

u)]ln q′′

−∞

= 1ıΩ

∫RI


ln q′′) exp (−ıΩ

νchln q′′)dq′′

= 1ıΩ

∫RI

p,k,ω(q′′)A(q, qeq)dq′′

= A(q, qeq)ı(ω − k · v + ı0)

∫RI

p,k,ω(q′′)dq′′ , (3.37)

where +ı0 was added in the denominator due to causality.For the second case, let us use the same approximation for the weighting functionD(q, q′) ≃ D(q, qeq) with RII

p,k,ω(q) = ∂B(q)∂q . We use the property of Eq.(3.37) and

obtain

Icol2 = A(q, qeq) 1ı(ω − k · v)

∫∂B(q′′)

∂q′′ dq′′

Icol2 = A(q, qeq) 1ı(ω − k · v)

[B(q′′)]+∞−∞ = 0 ,

(3.38)

since the physical quantity B(q) must vanish at infinity. This implies that in thezeroth order approximation the integral is zero, thus, we must use the first orderapproximation for the weighting function D(q, q′) to avoid the trivial result. Weexpand in Taylor series and set ∂D(q,q′)

∂q′ |q′=qeq = C(q),

D(q, q′) = D(q, qeq + ∆q0q′) ⇒ D(q, q′) ≃ D(q, qeq) + ∆q0q′ ∂D(q, q′)∂q′ |q′=qeq ⇒

D(q, q′) ≃ D(q, qeq) + ∆q0q′C(q).(3.39)

When we substitute in the integral, following the above discussion the zeroth orderterm will give a trivial result, following the same procedure as with the first case,


PLASMAS

and using integration by parts at the end we have:

Icol2 =∫

RIIp,k,ω(q′′)dq′′

∫∆q0C(q)q′G(q′, q′′)dq′

= −∆q0C(q)∫


∫ q′′

0

q′

νchq′ exp[−ı

Ωνch

(ln q′ − ln q′′)]dq′

= −∆q0C(q)∫


∫ q′′

0

1νchq′ exp

[−ı

Ωνch

(ln q′ − ln q′′) + ln q′]dq′

= −∆q0C(q)νch

∫RII

p,k,ω(q′′) exp(

ıΩ

νchln q′′

)dq′′∫ ln q′′

−∞exp[(

−ıΩ

νch+ 1)

u]du

= −∆q0C(q)νch

νch

−ıΩ + νch

∫RII


ıΩ

νchln q′′

)dq′′

exp[(

−ıΩ

νch+ 1)

u]ln q′′

−∞

= −∆q0C(q)νch

νch

−ıΩ + νch

∫RII


ıΩ

νchln q′′

)exp(

−ıΩ

νchln q′′

)exp (ln q′′)dq′′

= − C(q)−ıΩ + νch

∫RII

p,k,ω(q′′)q′′dq′′


∫∂B(q′′)

∂q′′ q′′dq′′


[B(q′′)q′′]+∞

−∞+ C(q)

−ıΩ + νch

∫B(q′′)dq′′

= C(q)−ıΩ + νch

∫B(q′′)dq′′

= − C(q)ı(ω − k · v + ıνch)

∫B(q′′)dq′′

= − 1ı(ω − k · v + ıνch)

∂D(q, q′)∂q′ |q′=qeq

∫B(q′′)dq′′ . (3.40)

The only responses that are a full derivative of the charge, are the ones relatedto the charging process. From Eq.(3.40) we see that in all expressions related tothe charging process the Green’s function reduces to expressions containing thecharging frequency. This conclusion is important when we evaluate the effects ofeach process.

Finally, by combining Eqs.(3.37,3.40), for F (q, q′) an arbitrary differentiablefunction of the charge we end up with the property∫

F (q, q′)δfd,(ind)p,k,ω

(q′)dq′ =F (q, qeq)

ı(ω − k · v + ı0)

∫q′′δEk,ω ·

∂Φdp(q′′)∂p

dq′′

−1

ı(ω − k · v + ıνch)

(∂F (q, q′)

∂q′

)q′=qeq

∫ ∑α

δIαk,ω(q′′)Φd

p(q′′)dq′′.

(3.41)

An additional property can be derived by exploiting the narrowness of the regularpart of the dust distribution function around the equilibrium dust charge, using the

3.7. THE PERMITTIVITY 47

reduced dust distribution function Φdp defined through Φd

p(q) = δ(q − qeq)Φdp and

integrating Eq.(3.41) over the dust momentum space∫F (q, q

′)δfd,(ind)p,k,ω

(q′)dq

′ =F (q, qeq)

ı(ω − k · v + ı0)

∫q

′′δEk,ω ·

∂Φdpδ(q′′ − qeq)

∂pdq

′′−

1ı(ω − k · v + ıνch)

(∂F (q, q′)

∂q′

)q′=qeq

∫ ∑α

δIαk,ω(q

′′)Φdpδ(q

′′ − qeq)dq′′

∫F (q, q


(q′)dq

′ =F (q, qeq)

ı(ω − k · v + ı0)qeq

k

k·

∂Φdp

∂pδEk,ω

−1

ı(ω − k · v + ıνch)

(∂F (q, q′)

∂q′

)q′=qeq

∑α

δIαk,ω(qeq)Φd

p∫F (q, q


(q′)dq

′ = −ı k

4π qeq

F (q, qeq)

(4π q2

eq

k21

ω − k · v + ı0k ·

∂Φdp

∂p

)δEk,ω

+ ı

(∂F (q, q′)

∂q′

)q′=qeq

∑α

δIαk,ω(qeq)

(1

ω − k · v + ıνch

Φdp

)

∫F (q, q


(q′)

d3pdq′

(2π)3 = −ı kF (q, qeq)

4π qeq

[4π q2

eq

k2

∫1

ω − k · v + ı0

(k ·

∂Φdp

∂p

)d3p

(2π)3

]δEk,ω

+∂F (q, qeq)

∂q′

∑α

δIαk,ω(qeq)

(∫ı

ω − k · v + ıνch

Φdp

d3p

(2π)3

)∫

F (q, q′)δf

d,(ind)p,k,ω

(q′)

d3pdq′

(2π)3 = −ı kχd,eq

k,ω

4π qeq

F (q, qeq)δEk,ω +(

∂F (q, q′)∂q′

)q′=qeq

χd,chk,ω

∑α

δIαk,ω(qeq) ,

(3.42)

where we also used the definition of the dust susceptibility (χd,eqk,ω ) and the dust

charging process response (χd,chk,ω )

χd,eqk,ω =

4π q2eq

k2

∫1

ω − k · v + ı0

(k ·

∂Φdp

∂p

)d3p

(2π)3 , (3.43)

χd,chk,ω =

∫ı

ω − k · v + ıνchΦd

p

d3p

(2π)3 . (3.44)

3.7 The permittivity

In order to acquire the permittivity we must express all fluctuating quantities asfunctions of the natural dust fluctuations. We start from the expression for thefluctuating part of the particle currents to the grains δIk,ω(q) and substitute for theinduced plasma fluctuations δf

α,(ind)p,k,ω , the fluctuating part of the capture frequency


PLASMAS

δνd,α(k, ω, v) and the longitudinal field δEk,ω = kk δEk,ω.

δIk,ω(q) =∑

α

∫eαvσα(q, v)δf

α,(ind)p,k,ω

d3p

(2π)3

=∑

α

∫eαvσα(q, v)

ı(ω − k · v + ıνd,α(v))

(eαδEk,ω ·

∂Φαp

∂p+ δνd,α(k, ω, v)Φα

p

)d3p

(2π)3

=

(∑α

∫e2

αvσα(q, v)ı(ω − k · v + ıνd,α(v))

k

k·

∂Φαp

∂p

d3p

(2π)3

)δEk,ω

+∑

α

∫eαvσα(q, v)

ı(ω − k · v + ıνd,α(v))δνd,α(k, ω, v)Φαp

d3p

(2π)3

= Sk,ω(q)δEk,ω

+∫ (∑

α

∫eαv2σα(q, v)σα(q′, v)ı(ω − k · v + ıνd,α(v))Φα

pd3p

(2π)3

)δfd

p′,k,ω(q′)d3p′dq′

(2π)3

= Sk,ω(q)δEk,ω +∫

Sk,ω(q, q′)δfdp′,k,ω(q′)d3p′dq′

(2π)3

= Sk,ω(q)δEk,ω +∫

Sk,ω(q, q′)δfd,(ind)p′,k,ω

(q′)d3p′dq′

(2π)3

+∫

Sk,ω(q, q′)δfd,(0)p′,k,ω

(q′)d3p′dq′

(2π)3 , (3.45)

where we used the definition of the mixed responses

Sk,ω(q) =∑

α

∫e2

αvσα(q, v)ı(ω − k · v + ıνd,α(v))

k

k·

∂Φαp

∂p

d3p

(2π)3 , (3.46)

Sk,ω(q, q′) =∑

α

∫eαv2σα(q, v)σα(q′, v)ı(ω − k · v + ıνd,α(v))

Φαp

d3p

(2π)3 . (3.47)

For the evaluation of the second adder of the above expression we will apply theproperty of Eq.(3.42) for F (q, q′) = Sk,ω(q, q′),∫

Sk,ω(q, q′)δfd,(ind)p′,k,ω (q′)d3p′dq′


k,ω

4π qeqSk,ω(q, qeq)δEk,ω

+ ∂Sk,ω(q, q′)∂q′ χd,ch

k,ω

∑α

δIαk,ω(qeq)

= −ı kχd,eq

k,ω

4π qeqSk,ω(q, qeq)δEk,ω

+ S′k,ω(q, qeq)χd,ch

k,ω

∑α

δIαk,ω(qeq) (3.48)


with the mixed response S′k,ω(q, q′) given by

S′k,ω(q, q′) =

∑α

∫eαv2σα(q, v)σ′

α(q′, v)ı(ω − k · v + ıνd,α(v))

Φαp

d3p

(2π)3 . (3.49)

Overall, we get an algebraic equation for δIk,ω(qeq) by setting q = qeq

δIk,ω(q) = Sk,ω(q)δEk,ω −ı kχd,eq

k,ω

4π qeqSk,ω(q, qeq)δEk,ω + S′

k,ω(q, qeq)χd,chk,ω

∑α

δIαk,ω(qeq)

+∫

Sk,ω(q, q′)δfd,(0)p′,k,ω

(q′)d3p′dq′

(2π)3

δIk,ω(qeq) = Sk,ω(qeq)δEk,ω −ı kχd,eq

k,ω

4π qeqSk,ω(qeq , qeq)δEk,ω + S′

k,ω(qeq , qeq)χd,chk,ω

δIk,ω(qeq)

+∫

Sk,ω(qeq , q′)δfd,(0)p′,k,ω

(q′)d3p′dq′

(2π)3

δIk,ω(qeq) =

(Sk,ω(qeq) −

ı kχd,eqk,ω

4π qeqSk,ω(qeq , qeq)

)×(

1 − S′k,ω(qeq , qeq)χd,ch

k,ω

)−1δEk,ω

+∫ (

1 − S′k,ω(qeq , qeq)χd,ch

k,ω

)−1Sk,ω(qeq , q′)δf

d,(0)p′,k,ω

(q′)d3p′dq′

(2π)3

δIk,ω(qeq) = βk,ωδEk,ω +∫

γk,ω(q′)δfd,(0)p′,k,ω

(q′)d3p′dq′

(2π)3 (3.50)

with the auxiliary responses βk,ω and γk,ω defined by

βk,ω =Sk,ω(qeq) − ı kχd,eq

k,ω

4π qeqSk,ω(qeq, qeq)

1 − S′k,ω(qeq, qeq)χd,ch

k,ω

, (3.51)

γk,ω(q) = Sk,ω(qeq, q)1 − S′

k,ω(qeq, qeq)χd,chk,ω

. (3.52)

The latter result simplifies the property of Eq.(3.42) expressing the integral as afunction of the electric field fluctuations and the natural dust fluctuations only,∫

F (q, q′)δfd,(ind)p,k,ω

(q′)d3pdq′

(2π)3 =

[(∂F (q, q′)

∂q′

)q′=qeq

χd,chk,ω

βk,ω −ı kχd,eq

k,ω

4π qeqF (q, qeq)

]δEk,ω+

(∂F (q, q′)

∂q′

)q′=qeq

χd,chk,ω

∫γk,ω(q′)δf

d,(0)p′,k,ω

(q′)d3p′dq′

(2π)3 .

(3.53)

We now return to the fluctuating parts of the plasma distribution functions, weintegrate them over the plasma momentum space, δni

k,ω =∫

δfαp,k,ω

d3p(2π)3 , use the


PLASMAS

property of Eq.(3.53) and acquire

δnαk,ω = −

ı k

4π eα

[4πe2

α

k2

∫1

ω − k · v + ıνd,α

(k ·

∂Φαp

∂p

)d3p

(2π)3

]δEk,ω

+∫

v

ı(ω − k · v + ıνd,α(v))Φα

p

d3p

(2π)3

∫σα(q′, v)δfd


(2π)3

= −ı kχα

k,ω

4π eαδEk,ω +

∫v


p

d3p

(2π)3

∫σα(q′, v)δfd


(2π)3

= −ı kχα

k,ω

4π eαδEk,ω +

∫v


p

d3p

(2π)3

∫σα(q′, v)δf

d,(0)p′,k,ω

(q′)d3p′dq′

(2π)3 +∫v


p

d3p

(2π)3

∫σα(q′, v)δf

d,(ind)p′,k,ω

(q′)d3p′dq′

(2π)3

= −ı kχα

k,ω

4π eαδEk,ω +

∫v


p

d3p

(2π)3

∫σα(q′, v)δf

d,(0)p′,k,ω

(q′)d3p′dq′

(2π)3

−ı kχd,eq

k,ω

4π qeq

∫vσα(qeq , v)


p

d3p

(2π)3 δEk,ω

+ χd,chk,ω

βk,ω

∫vσ′

α(qeq , v)ı(ω − k · v + ıνd,α(v))

Φαp

d3p

(2π)3 δEk,ω

+ χd,chk,ω

∫vσ′

α(qeq , v)ı(ω − k · v + ıνd,α(v))

Φαp

d3p

(2π)3

∫γk,ω(q′)δf

d,(0)p′,k,ω

(q′)d3p′dq′

(2π)3

= −ı kχα

k,ω

4π eαδEk,ω +

∫v


p

d3p

(2π)3

∫σα(q′, v)δf

d,(0)p′,k,ω

(q′)d3p′dq′

(2π)3

−ı kχd,eq

k,ω

4π qeq

qαk,ω(qeq)

eαδEk,ω + χd,ch

k,ωβk,ω

βαk,ω(qeq)

eαδEk,ω

+ χd,chk,ω

βαk,ω(qeq)

eα

∫γk,ω(q′)δf

d,(0)p′,k,ω

(q′)d3p′dq′

(2π)3

= −ı kχα

k,ω

4π eαδEk,ω −

ı kχd,eqk,ω

4π qeq

qαk,ω(qeq)

eαδEk,ω + χd,ch

k,ωβk,ω

βαk,ω(qeq)

eαδEk,ω+∫

qαk,ω(q′)

eαδf

d,(0)p′,k,ω

(q′)d3p′dq′

(2π)3 + χd,chk,ω

βαk,ω(qeq)

eα

∫γk,ω(q′)δf

d,(0)p′,k,ω

(q′)d3p′dq′

(2π)3 ,

(3.54)

where we used the definitions of the responses χαk,ω denoting the plasma suscep-

tibilities altered from the multi-component expressions due to the presence of thedissipative frequency νd,α(v) in the denominator, and of the charging responsesqα

k,ω(q) and βαk,ω(q)

χαk,ω = 4πe2

α

k2

∫1

ω − k · v + ıνd,α

(k ·

∂Φαp

∂p

)d3p

(2π)3 , (3.55)

qαk,ω =

∫eαvσα(q, v)


p

d3p

(2π)3 , (3.56)


βαk,ω =

∫eαvσ′

α(q, v)ı(ω − k · v + ıνd,α(v))

Φαp

d3p

(2π)3 , (3.57)

Another important quantity to be evaluated is∫

q′δfd,(ind)p′,k,ω

d3p′dq′

(2π)3 . With the useof Eq.(3.53) we obtain

∫q′δf

d,(ind)p′,k,ω

d3p′dq′

(2π)3 =

[χd,ch

k,ω βk,ω −ı kχd,eq

k,ω

4π

]δEk,ω

+ χd,chk,ω

∫γk,ω(q′)δf

d,(0)p′,k,ω(q′)d3p′dq′

(2π)3 . (3.58)

The fluctuating part of the Poisson equation will now lead to an expression ofthe electrostatic field fluctuations as a function of the natural dust fluctuations only.Consequently all other fluctuating quantities can now be expressed as functions ofthe dust discreteness. Using Eqs.(3.54,3.59) and βk,ω =

∑α

βαk,ω , qk,ω =

∑α

qαk,ω

we get

ıkδEk,ω = 4π

(∑α

∫eαδf

αp,k,ω

d3p

(2π)3 +

∫q

′δf

d,(0)p′,k,ω

(q′)

d3p′dq′

(2π)3 +

∫q

′δf

d,(ind)p′,k,ω

(q′)

d3p′dq′

(2π)3

)

= 4π

(∑α

eαδnαk,ω +

∫q

′δf

d,(0)p′,k,ω

(q′)

d3p′dq′

(2π)3 +

∫q

′δf

d,(ind)p′,k,ω

(q′)

d3p′dq′

(2π)3

)

= −ı k

∑α

χαk,ωδEk,ω −

ı kχd,eqk,ω

qeq

qk,ω(qeq)δEk,ω

+ 4πχd,chk,ω

βk,ωβk,ω(qeq)δEk,ω +[

4πχd,chk,ω

βk,ω − ı kχd,eqk,ω

]δEk,ω

+ 4πχd,chk,ω

βk,ω(qeq)

∫γk,ω(q

′)δfd,(0)p′,k,ω

(q′)

d3p′dq′

(2π)3 + 4π

∫q

′δf

d,(0)p′,k,ω

(q′)

d3p′dq′

(2π)3

+ 4π

∫qk,ω(q


(q′)

d3p′dq′

(2π)3 + 4πχd,chk,ω

∫γk,ω(q


(q′)

d3p′dq′

(2π)3

= −ık

∑α

χαk,ω + χ

d,eqk,ω

+qk,ω(qeq)

qeq

χd,eqk,ω

−4π

ı kχ

d,chk,ω

βk,ωβk,ω −4π

ı kχ

d,chk,ω

βk,ω

δEk,ω

+ 4π

∫ q

′ + qk,ω(q′) + χ

d,chk,ω

γk,ω(q′) + χ

d,chk,ω

βk,ω(qeq)γk,ω(q′)

δfd,(0)p′,k,ω

(q′)

d3p′dq′

(2π)3

ık

1 +∑

α

χαk,ω + χ

d,eqk,ω

+qk,ω(qeq)

qeq

χd,eqk,ω

−4π

ı kχ

d,chk,ω

βk,ωβk,ω(qeq) −4π

ı kχ

d,chk,ω

βk,ω

δEk,ω =

+4π

∫ q

′ + qk,ω(q′) + χ

d,chk,ω

γk,ω(q′) + χ

d,chk,ω

βk,ω(qeq)γk,ω(q′)

δfd,(0)p′,k,ω

(q′)

d3p′dq′

(2π)3 .


PLASMAS

For the second term in brackets we have

qeffk,ω (q) = q + qk,ω(q) + χd,ch

k,ω γk,ω(q) + χd,chk,ω βk,ω(qeq)γk,ω(q)

= q + qk,ω(q) + χd,chk,ω γk,ω(q)

(1 + βk,ω(qeq)

)= q + qk,ω(q) +

Sk,ω(qeq, q)χd,chk,ω


k,ω

.(

1 + βk,ω(qeq))

,

while for the first term in brackets

ϵk,ω = 1 +∑

α

χαk,ω + χ

d,eqk,ω

+qk,ω(qeq)

qeq

χd,eqk,ω

−4π

ı kχ

d,chk,ω

βk,ωβk,ω(qeq) −4π

ı kχ

d,chk,ω

βk,ω

= 1 +∑

α

χαk,ω + χ

d,eqk,ω

(1 +

qk,ω(qeq)qeq

)−

4π

ı kχ

d,chk,ω

βk,ω

(1 + βk,ω(qeq)

)= 1 +

∑α

χαk,ω + χ

d,eqk,ω

(1 +

qk,ω(qeq)qeq

)

−4π

ı kχ

d,chk,ω

Sk,ω(qeq) −ı kχ

d,eqk,ω

4π qeqSk,ω(qeq, qeq)

1 − S′k,ω

(qeq, qeq)χd,chk,ω

(1 + βk,ω(qeq)

)= 1 +

∑α

χαk,ω + χ

d,eqk,ω

(1 +

qk,ω(qeq)qeq

)+

χd,chk,ω

qeq

χd,eqk,ω

Sk,ω(qeq, qeq)

1 − S′k,ω


(1 + βk,ω(qeq)

)+

4πı

kχ

d,chk,ω

Sk,ω(qeq)

1 − S′k,ω


(1 + βk,ω(qeq)

)= 1 +

∑α

χαk,ω + χ

d,eqk,ω

[1 +

qk,ω(qeq)qeq

+χd,ch

k,ω

qeq

Sk,ω(qeq, qeq)

1 − S′k,ω


(1 + βk,ω(qeq)

)]+

4πı

kχ

d,chk,ω

Sk,ω(qeq)

1 − S′k,ω


(1 + βk,ω(qeq)

)= 1 +

∑α

χαk,ω +

χd,eqk,ω

qeq

[qeq + qk,ω(qeq) +

Sk,ω(qeq, qeq)χd,chk,ω

1 − S′k,ω


(1 + βk,ω(qeq)

)]+

4πı

kχ

d,chk,ω

Sk,ω(qeq)

1 − S′k,ω


(1 + βk,ω(qeq)

)= 1 +

∑α

χαk,ω +

qeffk,ω

(qeq)

qeq

χd,eqk,ω

+4πı

k

Sk,ω(qeq)χd,chk,ω

1 − S′k,ω


(1 + βk,ω(qeq)

).

Therefore, overall we have

δEk,ω = 4π

ıkϵk,ω

∫qeff

k,ω (q)δfd,(0)p′,k,ω(q) d3p′dq

(2π)3 , (3.59)

where the effective charge is defined by

qeffk,ω (q) = q + qk,ω(q) +

Sk,ω(qeq, q)χd,chk,ω


k,ω

.(

1 + βk,ω(qeq))

(3.60)

3.8. THE COLLISION INTEGRALS 53

and the permittivity is defined by

ϵk,ω = 1 +∑

α

χαk,ω +

qeffk,ω (qeq)

qeqχd,eq

k,ω + 4πı

k



k,ω

(1 + βk,ω(qeq)

).

(3.61)The latter quantities determine the interactions in dusty plasmas.

In order to simplify the cumbersome expressions for the collision integrals, wedefine a number of new responses. The effective permittivity, that determines thedynamic screening of the fields of colliding particles, is defined by

ϵeffk,ω (q) = qeqϵk,ω

qeffk,ω (q)

, (3.62)

The effective particle permittivity is defined through ϵeffk,ω (qeq) = ϵ

eff,(P )k,ω + χd,eq

k,ω .In terms of integral responses it will be given by

ϵeff,(P )k,ω = qeq

qeffk,ω (qeq)

1 +

∑α

χαk,ω + 4πı

k



k,ω

(1 + βk,ω(qeq)

).

(3.63)The charging response Λk,ω, that is related to the charging process only, is definedby

Λk,ω = ıγk,ωϵeffk,ω (qeq) + 4πqeq

kβk,ω . (3.64)

3.8 The collision integrals

We start with the presentation of the dust collision integral [Tsytovich and deAngelis, 1999]

Jdp(q) = −q

∂

∂p· ⟨δEδfd

p(q)⟩ − ∂

∂q

(∑α

⟨δIα(q)δfdp(q)⟩

), (3.65)

The first term describes the interaction of the electrostatic micro-field fluctuationswith the fluctuating part of the dust distribution function and hence refers to thedust-dust and dust-plasma collisions in presence of dust-charge fluctuations, sucha term is also present in multicomponent plasmas. The second term describesthe effect of the charging process in dust kinetics and has no analogy in multi-component kinetic models.Expressions of all fluctuating quantities as functions of the natural correlator, useof the natural statistical correlator and Green’s function (but without substitutingfor it) will yield the explicit form

Jdp(q) =

∂

∂p·

(∫D

d

p(q, q′) ·

∂Φdp(q′)∂p

dq′

)+

∂

∂p·(

FdpΦd

p(q))

+∂

∂p·

(q

∫F

d,qp (q, q

′)∂Φd

p(q′)∂q′ dq

′

)+

∂

∂q

(∫q

′F

q,dp (q, q

′) ·∂Φd

p(q′)∂p

dq′

)−

∂

∂q

(δ⟨I⟩Φd

p(q))

+∂

∂q

(∫Ich(q, q

′)∂Φd

p(q′)∂q′ dq

′

)(3.66)


PLASMAS

The first adder describes diffusion in momentum space with the diffusion tensorbeing an integral over the charge and the integrand given by the tensor

Ddp,l,m(q, q′) = − 2

πℜ

∫klkmqq′q2

eq

k4|ϵeffk,k·v′(qeq)|2

G(q, q′, k · v′ − k · v)Φdp′

d3p′d3k

(2π)3

.

(3.67)The structure of the diffusion tensor resembles strongly the diffusion tensor formulti-component plasmas with some major differences: (i) the diffusion tensor isintegrated over the charge space in order to account for dust charge variability dur-ing the interaction, (ii) dust variability also leads to q2

eqqq′ for the charge insteadof q4

eq that would be present in absence of dust charge fluctuations, (iii) the usualδ(k ·v −k ·v′) function is substituted by G(q, q′, k ·v −k ·v′), this is to be expectedsince δ(k · v − k · v′) is Green’s function for free streaming particles in a r, pphase-space and G(q, q′, k ·v −k ·v′) is Green’s function for free streaming particlesin an extended r, p, q phase-space, (iv) the dynamic screening of the fields ofcolliding particles is now represented by 1

k4|ϵeff

k,k·v′ (qeq)|2 instead of 1k4|ϵk,k·v′ |2 .

The second adder stems from the ensemble average −q ∂∂p · ⟨δEδf

d,(0)p (q)⟩ and rep-

resents a drift process in momentum space. The relevant friction force is givenby

F dp,l(q) = qeqq

2π2 ℜ

∫kl

ı k2ϵeff∗k,k·v′(qeq)

d3k

. (3.68)

It is the same friction force that is present in multi-component kinetic models.There, however the imaginary (dissipative) part of the permittivity enables the useof the Plemelj-Sokhotskyi formula for further simplifications and ultimately themore compact form of the Lennard-Balescu collision integral.The third and fourth adders involve mixed second order partial derivatives overthe momentum and the charge and they stem from the −q ∂

∂p · ⟨δEδfdp(q)⟩ and

− ∂∂q

(∑α


)ensemble averages respectively. They appear due to the

fact that in dust dust collisions both the momenta and the charges of the dustparticles change. Notice that in a stochastic description this implies a correlationbetween the stochastic momentum term of the Langevin equation and the noiseterm of the charging equation. The friction forces are given by

F d,qp,l (q, q′) = 1

(2π)2 ℜ∫

klqeq

k2ϵeffk,k·v′(qeq)

(4π qeq

kϵeff∗k,k·v′(qeq)

βk,k·v′(qeq) + ıγk,k·v′(qeq)

)

× G(q, q′, k · v′ − k · v) Φdp′

d3p′d3k

(2π)3 , (3.69)

3.8. THE COLLISION INTEGRALS 55

F q,dp,l (q, q′) = 1

(2π)2 ℜ∫

klqeq

k2ϵeff∗k,k·v′(qeq)

(4π qeq

kϵeffk,k·v′(qeq)

βk,k·v′(qeq) + ıγk,k·v′(qeq)

)

× G(q, q′, k · v′ − k · v) Φdp′

d3p′d3k

(2π)3 , (3.70)

The fifth adder involves a first order derivative with respect to the charge and stems

from the ensemble average − ∂∂q

(∑α


). The physics behind this

term will become more transparent if we regard the charge derivative term in the lefthand side of the kinetic equation and apply the approximation of small deviations

from the equilibrium charge. We will then obtain ∂∂q

[(∑α

Iα(q) + Iext

)Φd

p(q)

]=

− ∂∂q

[νch(q − qeq)Φd

p(q)], which when combined with the present adder results in

+ ∂∂q

[(νch(q − qeq) − δ⟨I⟩) Φd

p(q)]. It is now obvious that it describes collective

corrections to the charging currents due to shadowing of plasma fluxes due to thepresence of neighboring grains. It is given by

δ⟨I⟩ = ℜ∫ (

γk,k·v′ + 4π qeq

ıkϵeffk,k·v′(qeq)

βk,k·v′

)d3k

(2π)3

. (3.71)

The sixth adder involves a second order derivative in the charge and essentiallydescribes a diffusion process in the charge space. The diffusion coefficient will be ascalar given by

Ich(q, q′) =1

(2π)3 ℜ

∫ ∣∣∣∣∣ 4πqeq

ı kϵeffk,k·v′ (qeq)

βk,k·v′ + γk,k·v′

∣∣∣∣∣2

G(q, q′, k · v′ − k · v)Φdp′

d3kd3p′

(2π)3

.

(3.72)Combining all the above, the kinetic equation of the regular part of the dust dis-tribution function will be

∂

∂t+ v ·

∂

∂r

Φd

p(q) =∂

∂p·

(∫D

d

p(q, q′) ·

∂Φdp(q′)∂p

dq′

)+

∂

∂p·(

FdpΦd

p(q))

+

∂

∂p·

(q

∫F

d,qp (q, q

′)∂Φd

p(q′)∂q′ dq

′

)+

∂

∂q

(∫q

′F

q,dp (q, q

′) ·∂Φd

p(q′)∂p

dq′

)+

∂

∂q

[(νch(q − qeq) − δ⟨I⟩) Φd

p(q)]

+∂

∂q

(∫Ich(q, q

′)∂Φd

p(q′)∂q′ dq

′

).

(3.73)

We now proceed to the presentation of the plasma collision integral [Tsytovichand de Angelis, 1999]

Jαp = −eα

∂

∂p· ⟨δEδfα

p ⟩ − ⟨δνd,α(v)δfαp ⟩

= −eα∂

∂p· ⟨δEδfα

p ⟩ −∫

vσα(q, v)⟨δfαp δfd

p′(q)⟩d3p′dq

(2π)3 . (3.74)


PLASMAS

The second ensemble average is new compared to multi-component theories and isobviously a consequence of dust acting as a sink of plasma particles. The explicitform of the collision integral is

Jαp = ∂

∂p· D

α,d

p ·∂Φα

p

∂p+ ∂

∂p·(

F α,dp Φα

p

)+ νfl

d,α(v)Φαp . (3.75)

The first adder describes a diffusion process in momentum space with the diffusiontensor given by

Dα,dp,l,m = 2e2

αq2eq(2π)3

∫klkm

k4|ϵeffk,k·v′ (qeq)|2

1π

νd,α(v)(k · v − k · v′)2 + ν2

d,α(v)

Φd

p′d3p′d3k

(2π)6 .

(3.76)

Compared to the diffusion tensor of multi-component plasmas we notice: (i) due tothe omission of plasma discreteness the tensor consists of one instead of three adders,since only collisions of plasma particles with dust are taken into account, (ii) the δ-function δ(k·v−k·v′) which represents energy/momentum conservation in Coulombcollisions is substituted by the function 1

πνd,α(v)

(k·v−k·v′)2+ν2d,α

(v) which expresses theinelasticity of collisions due to plasma absorption. It is a Lorentz line nascent δ-

function with the property limνd,α(v)→0

(1π

νd,α(v)(k · v − k · v′)2 + ν2

d,α(v)

)= δ(k · v − k ·

v′). The width of the Lorentzian broadening is, as expected, determined by thedissipation, i.e the plasma capture frequency νd,α(v), (iii) the dynamic screening ofthe fields of the colliding particles is now represented by the effective permittivityin the expression 1

k4|ϵeff

k,k·v′ (qeq)|2 , (iv) due to the characteristic dust velocities being

much smaller than the plasma phase space velocities one can neglect the dustvelocity v′ in the expression k · v − k · v′ and also evaluate the responses in theirstatic limit.The second adder describes a drift in momentum space with the friction force givenby

F d,αp,l = −eαqeqvσα(qeq, v)

2π2 ℜ∫

kl

k2|ϵeffk,k·v′(qeq)|2

(k · v − k · v′ + ıνd,α(v))(k · v − k · v′)2 + ν2

d,α(v)

×(

ϵeff,(P )k,k·v′ +

4πq2eqσ′

α(qeq, v)kσα(qeq, v)

χd,chk,k·v′βk,k·v′

)Φd

p′d3p′d3k

(2π)3 . (3.77)

In contrast to multi-component plasmas the friction force is determined completelyby the charging process. This is partly due to the omission of the plasma dis-creteness, since in its presence there would at least be one additional non-vanishingcontribution from the ensemble average −eα

∂∂p ⟨δEδf

α,(0)p,k,ω⟩ which would be of the

form F αp,l = e2

α

2π2 ℜ∫

kl

ı k2ϵeff,i∗k,k·v′

d3k

.


The third adder is the collective absorption frequency νfld,α(v). This term competes

with the −νd,α(v) Φαp term of the kinetic equation, which describes absorption pro-

cesses where individual particles participate and are significantly altered in thepresence of many particles, e.g shadowing of fluxes to the dust grains due to ab-sorption / scattering from other neighboring grains. Notice that now the totalcapture frequency of the plasma kinetic equation will be

(νd,α(v) − νfl

d,α(v))

, withrelevant implications for hydrodynamic descriptions.Combining all the above, the kinetic equation of the regular part of the plasmadistribution function will be[

∂

∂t+ v · ∂

∂r

]Φα

p = sα −(

νd,α(v) − νfld,α(v)

)Φα

p

+ ∂

∂p· D

α,d

p ·∂Φα

p

∂p+ ∂

∂p·(

F α,dp Φα

p

). (3.78)

3.9 The spectral densities of fluctuations

For the spectral density of the electrostatic field fluctuations we use Eq.(3.59) andthe natural statistical dust correlator to acquire

SEk,ω =

16π2q2eq

k2|ϵeffk,ω (qeq)|2

Sd,(0)k,ω . (3.79)

Similarly for the spectral density of the electrostatic potential fluctuations we useδϕk,ω = ıδEk,ω

k and get

Sϕk,ω =

16π2q2eq

k4|ϵeffk,ω (qeq)|2

Sd,(0)k,ω , (3.80)

where we notice the spectral enhancement in low frequencies due to the presenceof Z2

d in the nominator of both expressions.For the spectral density of the plasma density fluctuations we use Eqs.(3.54,3.59)

to express the fluctuating part of the plasma densities as a function of dust naturalfluctuations only,

δnαk,ω =

∫Nα

k,ω(q)δfd,(0)p′,k,ω(q) d3p′dq

(2π)3 , (3.81)

with Nαk,ω(q) = − qeq

eαϵeffk,ω

(q)Mα

k,ω(q) + Λαk,ω(q) where

Mαk,ω(q) = χα

k,ω +qα

k,ω(qeq)qeq

χd,eqk,ω + 4πı

kχd,ch

k,ω βk,ωβαk,ω(qeq) ,

Λαk,ω(q) =

qαk,ω(q)eα

+βα

k,ω(qeq)eα

γk,ω(q)χd,chk,ω .


PLASMAS

Hence, with the use of the natural statistical dust correlator we get

Sαk,ω = |Nα

k,ω(qeq)|2Sd,(0)k,ω .

We note the dependence of Sαk,ω on Z2

d , since Nαk,ω(q) ∝ Zd and Λα

k,ω(q) ∝ Zd

the latter through qαk,ω(qeq)

eα∝ Zd and γk,ω(qeq) ∝ Zd. This also demonstrates

the significant enhancement of the plasma fluctuation spectra for low frequenciestypical of dust dynamics.

Finally, for the spectral densities of dust density fluctuations we can integratethe equation for δfp′,k,ω(q)d,(ind) over the dust momentum / charge space or alter-natively we can set F (q, q′) = 1 in the general property of Eq.(3.42),∫

δfd,(ind)p′,k,ω (q) d3p′dq


k,ω

4πqeqδEk,ω

δnd,(ind)k,ω = −

ı kχd,eqk,ω

4πqeq

4πqeq

ık

∫1

ϵeffk,ω (q)

δfd,(0)p′,k,ω(q) d3p′dq

(2π)3

δnd,(ind)k,ω = −

∫χd,eq

k,ω

ϵeffk,ω (q)

δfd,(0)p′,k,ω(q) d3p′dq

(2π)3

δnd,(ind)k,ω + δn

d,(0)k,ω = −

∫ (χd,eq

k,ω

ϵeffk,ω (q)

− 1

)δf

d,(0)p′,k,ω(q) d3p′dq

(2π)3

δndk,ω = −

∫ (χd,eq

k,ω

ϵeffk,ω (q)

− 1

)δf


(2π)3

δndk,ω =

∫Nd

k,ω(q) δfd,(0)p′,k,ω(q) d3p′dq

(2π)3 (3.82)

with Ndk,ω(q) = −

[χd,eq

k,ω

ϵeffk,ω

(q)− 1]. The natural statistical correlator will now yield

Sdk,ω = |Nd

k,ω(qeq)|2 Sd,(0)k,ω . (3.83)

3.10 Kinetic phenomena unique in complex plasmas

Below we will focus on a number of novel results that stem from the kinetic modelof complex plasmas and manifest the effects of dust charge and dust charge fluctu-ations.

Stochastic heating as a consequence of dust charge variability: One of the basic con-sequences of dust charge fluctuations is that the energy is not conserved in dust-dustinteractions. This can lead to a growth of the mean local energy ("temperature")

3.10. KINETIC PHENOMENA UNIQUE IN COMPLEX PLASMAS 59

of the dust particles in time, coined as stochastic heating of dust particles. The en-ergy source for this instability is ultimately the external source of plasma particles,necessary for dust grains to maintain their equilibrium charges [de Angelis et al.,2005].The generic nature of Eq.(3.73) can be demonstrated by integrating over the dustcharge space and defining the reduced distribution function Φd

p =∫

Φdp(q) [Tsy-

tovich and de Angelis, 2001]. This will yield a kinetic equation for Φdp that is

relatively simplified, since the last terms of Eq.(3.73) vanish being full derivativeswith respect to the charge. In a zero order approximation with respect to ω

kvT α≪ 1

and a first order approximation with respect to ωνch

≪ 1, together with the appli-cation of the Plemelj-Sokhotskyi formula and the properties of Green’s functionG(q, q′, ω − k · v) it can be demonstrated that the first two terms will express en-ergy conservation and provide relaxation to equilibrium containing δ(k · v′ − k · v).On the other hand the last remaining term will be non-conservative and will expressthe effect of dust charge fluctuations.In that case the equation for the mean energy ϵ =

∫p2

2m Φdp

d3p(2π)3 will be

dϵ

dt=∫

p2

2mINC

dd (p) d3p

(2π)3

with

INCdd (p) = ∂

∂p·∫

qF d,qp (q, q′)

∂Φdp(q′)

∂q′ dqdq′ ,

that can ultimately be approximated by dϵdt = νϵϵ with νϵ > 0.

It is worth reporting that similar results have been reported from a Fokker-Planckapproach to systems with variable charges [Ivlev et al., 2004] and from the stochastictheory of a harmonic oscillator with random frequency [Marmolino, 2011]. Stochas-tic heating has been recently implemented to address the acceleration of small grainsin the interstellar medium, where it is particularly important for the processes ofshattering and coagulation of dust [Ivlev et al., 2010]. In a totally different pa-rameter regime, the presence of hyper-velocity particles in tokamak edge plasmas[Ratynskaia et al., 2008] can possibly be explained by the same mechanism [Mar-molino et al., 2008].

Dust charge distributions: Integration of Eq.(3.73) over the dust momentum spacetogether with the introduction of the reduced dust distribution function (normal-ized to unity) Φd(r, q, t) =

∫Φd

p(r, q, t) d3p(2π)3 = ndfd(q) will yield an equation for

the evolution of dust charge distribution.. The first three terms of Eq.(3.73) vanishbeing full derivatives with respect to the momentum and in the stationary and


PLASMAS

homogeneous case we have [Tsytovich and de Angelis, 2002]

∂

∂q((νch(q − qeq) − δ⟨I⟩)fd(q)) + ∂

∂q

∫q′F d

p(q, q′) ·∂Φd

p(q′)∂p

d3p

(2π)3 dq′

+ ∂

∂q

∫Ich(q, q′)∂Φp(q′)

∂q′ dq′ d3p

(2π)3 = 0.

Approximate solutions, in the parameter regime of laboratory applications showthat fd(q) has a Lorentz shape, with the width coinciding with the width of dustcharge fluctuations, i.e

fd(q) = 1π

√⟨(∆q)2⟩

q2eq

(q − qeq)2 + ⟨(∆q)2⟩q2

eq

.

Therefore, within the approximation of small deviations from the equilibrium dustcharge, the above function will reduce to a δ-function δ(q − qeq), which brings outthe approximation Φd

p(r, q, t) = Φdp(r, t)δ(q − qeq).

Collective dust charge fluctuations are related with the dust discreteness and canbe found by the present theory by the ensemble average of the square of the first∆q moment of the fluctuating dust charge distribution function (normalized by thedust density) [Tsytovich and de Angelis, 2002],

⟨(∆q)2⟩ = 1n2

d

⟨∫∆qδfd(r, q, t)sq

∫∆q′δfd(r, q′, t)dq′

⟩.

With use of δfd(r, q′, t) =∫

δfdp(r, q′, t) d3p

(2π)3 , Fourier transforms in space / timeand decomposition in natural and induced parts one can find approximate relationsfor the collective dust charge fluctuations. It also be shown that they usually exceedthe non-collective dust charge fluctuations related to plasma discreteness, a resultthat might be important for the system’s condensation to a strongly coupled state.

Spectral densities as a diagnostic tool for sub-micron dust: Due to the large numberof elementary charges residing on dust (Zd ≫ 1), there is an important enhance-ment of the spectral densities of plasma and electrostatic field fluctuations in thelow frequency regime (Z2

dSd,(0)k,ω ). The spectral enhancement is roughly proportional

to nd

vT dZ2

d which implies a proportionality nda7/2 and therefore strong dependenceon the dust density and radius. Similarly, the region of spectral enhancement isstrongly depending on the radius of the dust grains. Hence, the measurementof the spectral densities and comparison with the theoretical results can providerich information about the dust density and composition [Ratynskaia et al., 2007;Ratynskaia et al., 2010]. Such a diagnostic is particularly important for in situdetection of sub-micron dust particles that cannot be monitored by fast cameras.

Chapter 4

The Klimontovich Description ofPartially Ionized Complex Plasmas

Complex plasmas are a thermodynamically open system: When embedded intoplasma, the dust grains are fast charged to an equilibrium value with a time constantequal to the inverse of the charging frequency. In order to maintain that charge, theycontinuously absorb plasma particles/radiation, depleting the system. Therefore,plasma or radiation sources are needed to replenish the plasma density and thesystem exchanges particles/energy with the background medium [Tsytovich et al.,2008]. These sources can be either constant in the temporal/ spatial scales of thefluctuations (e.g radiation sources) or fluctuating (e.g electron impact ionization ofneutrals).

Laboratory complex plasmas are engineered in low temperature discharges,where in most cases the background gas in not fully ionized. In a full kineticmodel of partially ionized complex plasmas, the system consists of four distinctspecies interacting with each other: ions, electrons, dust particles and neutral gas.For a self-consistent description, the Klimontovich equations of all species shouldbe analyzed and closure should be provided by the Maxwell equations [Tsytovichet al., 2005].

4.1 The Bhatnagar-Gross-Krook collision integral

Before proceeding to the Klimontovich description of the system we present theBhatnagar-Gross-Krook (BGK) collision integral [Bhatnagar et al., 1954]. This isa model collision integral that cannot be derived from first principles. However,it can be viewed as an approximation of the Boltzmann collision term for systemsclose to thermodynamic equilibrium [Liboff, 1990].

The Boltzmann equation describes the evolution of the smooth average part ofthe single particle distribution function of a rarefied gas within the assumptions,(i) three body and higher correlations are neglected, (ii) the collision time is much

61

62CHAPTER 4. THE KLIMONTOVICH DESCRIPTION OF PARTIALLY

IONIZED COMPLEX PLASMAS

larger than the duration of collisions, (iii) collisions are elastic and their dynamicsare well-described by classical mechanics,

∂

∂t+ v · ∂

∂r+ F · ∂

∂p

f1 =

∫dΩ∫

d3p2σ(Ω, |v1 − v2|)|v1 − v2|(f ′1f ′

2 − f1f2) ,

(4.1)with F denoting the external force acting on the particles, where f1 = f(r, p1, t)refers to the first colliding particle before the collision event, f2 = f(r, p2, t) refersto the second colliding particle before the collision event and the primes refer tothe particles after the collision event with p1 + p2 = p′

1 + p′2 and |p1|2 + |p2|2 =

|p′1|2 + |p′

2|2, whereas σ(Ω, |v1 −v2|)dΩ is the differential cross-section for scatteringin a solid angle Ω in the center of mass frame. In a compact form the Boltzmanncollision term can be rewritten as J(f1) =

∫f ′f ′

1dµ1 −∫

ff1dµ1 with dµ1 =σ(Ω, |v−v1|)|v − v1|dΩ d3p1.

Near equilibrium the system is close to a local Maxwellian state. As inferredby the H-theorem, the main effect of collisions is to cause a quick relaxation toequilibrium and therefore the primed components (referring to the after-collisioninterval) can be assumed as local Maxwellians,

∫f ′f ′

1dµ1 ≃∫

f0′f0′

1 dµ1. Since lo-cal Maxwellians nullify the collision integral we also get J(f0) = 0 ⇒

∫f0′

f0′

1 dµ1 =∫f0f0

1 dµ1. Moreover, due to the conservation of particles, momentum and kineticenergy in the collision f0

1 and f1 will have the first three moments equal, whichleads us to the reasonable assumption

∫f1dµ1 =

∫f0

1 dµ1. Overall, we have

J(f1) =∫

f ′f ′1dµ1 −

∫ff1dµ1

≃∫

f0′f0′

1 dµ1 −∫

ff1dµ1

≃∫

f0f01 dµ1 −

∫ff1dµ1

≃ f0∫

f01 dµ1 − f

∫f1dµ1

≃ f0∫

f01 dµ1 − f

∫f0

1 dµ1

≃∫

f01 dµ1

(f0 − f

)≃ ν(v)

(f0 − f

),

where the collision frequency is defined by ν(v) =∫

f01 dµ1 =

∫f0

1 σ(Ω, |v−v1|)|v −v1|dΩ d3p1. This is the form of the BGK collision integral, which we rewrite as

δf

δt= −ν(v)

(f − f0(v)

). (4.2)

Its physical meaning becomes transparent by assuming a spatially homogeneousgas in absence of external forces and with an initial velocity distribution f(0). In

4.2. THE EFFECT OF NEUTRALS IN THE STRUCTURE OF THEKLIMONTOVICH EQUATIONS 63

that case the Boltzmann equation within the BGK approximation will yield

∂f

∂t= −ν(f − f0)

∂f

∂t+ νf = νf0

f(t) = f(0)e−ν t + f0(1 − e−ν t) ,

which in the limit to infinity will lead to f(∞) = f0. Therefore the BGK collisionintegral expresses; (i) the destruction of phase of ordered motion due to collisions,(ii) transition to thermodynamic equilibrium with a time scale depending on thefrequency of collisions only, (iii) transition to Maxwellian distributions independentof the initial distribution of the gas.

The defect of such a model is that the particles are not conserved instantaneouslydue to the velocity dependence of ν(v). This can easily be remedied by using anaverage collision frequency defined via ν = 1

n

∫ν(v)f0(v)d3v. Thus,∫

δf

δtd3v = −ν

∫f(v)d3v + ν

∫f0(v)d3v = −ν n + ν n = 0 .

Up to this point, we have discussed the form of the BGK collision integral forthe smooth averaged part of the distribution function. In order to apply it to theKlimontovich approach, we need to consider it for the exact microscopic phase-spacedensities. Its form, in this case, will be given by

δfp(r, t)δt

= −ν(fp(r, t) − N(r, t)Φeq

p

), (4.3)

where N(r, t) =∫

fp(r, t) d3p(2π)3 = n(r, t)+δn(ind)(r, t)+δn(0)(r, t) is the total par-

ticle density average and fluctuating, where Φeqp is the average distribution function,

normalized to unity, making the collision integral zero. In that sense, collisions aredescribed as simultaneous birth / death processes in the phase space, where a par-ticle of fp(r, t) is substituted by a particle with Φeq

p . The latter makes the BGKcollision integral particularly attractive for the description of charge exchange col-lisions, where a fast ion "receives" a valence electron from an atom, essentiallycreating a slow ion.

4.2 The effect of neutrals in the structure of theKlimontovich equations

In a classical statistical description, neutral atoms can be treated as strongly cou-pled subsystems consisting of electrons and a multiply charged ion bound by somemodel potential. In that sense, collisions with ions / electrons such as short rangepolarization scattering can be treated collectively, but not collisions of quantum-mechanical nature like ion-atom resonant charge transfer collisions and electron



impact ionization, which also happen to be the dominant collisional processes intypical discharges [Liebermann and Lichtenberg, 1994].

However, neutrals can also be treated as point particles without internal struc-ture provided that the collisional frequencies are given by external models (e.g pairparticle approaches). Moreover, for relatively large pressures, one can safely assumethat neutral-neutral, ion-neutral and dust-neutral collisions are frequent enough tokeep the neutrals in thermodynamic equilibrium. To simplify things more and en-able focusing on plasma / dust kinetics, neutrals can be treated as continuous fluids(omission of their discrete nature) following a Maxwellian distribution ΦM

p (normal-ized to unity) with an average kinetic energy Tn. They are essentially the ambientmedium not only providing dissipation and relaxation to equilibrium through col-lisions but also providing the source of plasma particles through electron impactionization [Tsytovich et al., 2005].

Collisions of neutrals with electrons can be neglected, with their frequenciesbeing typically two orders of magnitude less than those of ion-neutral collisions.Collisions with ions and dust grains, will be treated by implementing the BGKrelaxation time approximation in the Klimontovich equations. In case of ions, theaddition will be −νn,i(f i

p − NiΦMp ), where Ni = ni + δn

(ind)i and it is assumed that

the thermalization is towards the neutral distribution function due to Ti ≃ Tn andmi ≃ mn, while νn,i is velocity independent and usually νn,i = nnvT iσn,i with σn,i

the velocity independent charge exchange collision cross-sections. In case of dust,the addition will be −νn,d(fd

p(q) − NdΦd,eqp ), where Nd = nd + δn

(0)d + δn

(ind)d and

νn,d a velocity independent collision frequency usually described by the Epsteinkinetic model due to the large radius of the grains compared to the size of theneutrals, with Φd,eq

p the normalized to unity distribution function making the dustcollision integral zero. Notice that there is a degree of arbitrariness in the choice ofΦd,eq

p ; there is a variety of collisional processes each leading the dust distributionto its equilibrium form with a different rate, for example, in case dust-neutralcollisions are dominating, Φd,eq

p can be reasonably assumed to be the Maxwellian

Φd,eqp =

(md

2π Td

)3/2exp

(− mdv2

2Tdn

)with Tdn = mnTn+mdTd

mn+md.

Electron impact ionization of neutrals will be treated as an instantaneous pro-cess generating new ions assumed to have the same velocity distribution with their"parent" neutrals ΦM

p (normalized to unity). The related source term in the ionswill have the form siz = νIni(r, t)ΦM

p , where νI = 1ne(r,t)

∫νI(v)fe

pd3p

(2π)3 . Sinceequal numbers of electrons and ions are generated by ionization we end up withsiz = νIΦM

p , where νI =∫

νI(v)fep

d3p(2π)3 . It is obvious that the source of plasma

particles is fluctuating following the fluctuations of the electron distribution func-tion fe

p.Finally, neutrals alter the plasma capture cross-sections on dust σα(q, v). This

implies that the equilibrium dust charge, the charging frequency and the responsesrelated to charging and absorption will be altered. This is not apparent from thestructure of the Klimontovich equations and their decomposition though, since they

4.3. THE KLIMONTOVICH EQUATIONS FOR THE DUST/PLASMACOMPONENTS 65

can be formulated with arbitrary cross-sections. Notice that the above assumptionsregarding the description of neutrals complement the basic kinetic assumptionsanalyzed in the previous chapter.

4.3 The Klimontovich equations for the dust/plasmacomponents

Due to (i) continuous absorption of ions on dust grains, (ii) generation of ions inelectron impact ionization of neutrals and (iii) inelastic collisions with neutrals, thedistribution function of the ion species is no longer constant in time as measuredalong the orbit of a hypothetical particle in phase space, D

Dt fi(r, p, t) = 0. TheKlimontovich equation should also contain sink and source terms due to theseprocesses D

Dt fi(r, p, t) = siz+sin+sid. In the un-magnetized case the Klimontovichequation for the ion-species will have the form, omitting the (r, t) dependence,

∂

∂t+ v · ∂

∂r+ eiE · ∂

∂p

f i

p =(∫

νI(v)fep

d3p

(2π)3

)ΦM

p − νn,i(f ip − NiΦM

p )

−(∫

σi(q, v)vfdp′(q)dqd3p′

(2π)3

)f i

p . (4.4)

Since there are no sinks/sources for the dust particles, the Klimontovich equa-tion for dust should have a continuity form in the phase-space. There are two maindifferences from the standard form, (i) The charge variability leading to a p, r, qaugmentation of the Hamiltonian phase space, which adds a charge derivative termto the Liouville and subsequently the Klimontovich equation (ii) The presence ofdust-neutral collisions: treated approximately via the BGK collision integral. Inthe un-magnetized case the Klimontovich equation for the dust species has the form

∂

∂t+ v · ∂

∂r+ qE · ∂

∂p

fd

p(q) + ∂

∂q

[(Iext +

∑α

Iα(q)

)fd

p(q)

]=

−νn,d

(fd

p(q) − NdΦd,eqp (q)

). (4.5)

The Klimontovich equation for the electron species, will essentially have thesame form with the Klimontovich equation for the ion species, with the exceptionthat electron-neutral collisions will be due to elastic polarization scattering. Due tothe very different temporal and spatial scales of the electron dynamics compared tothe dynamics of the massive ions/dust particles, in low frequencies, typical of dustdynamics, a full kinetic description can be avoided. Instead of the Klimontovichequation, it can be assumed that the electron distribution function has a dependenceon the local potential of the form

fep(r, t) = Φe

p(r, t)[1 + eϕ(r, t)

Te(r, t)

], (4.6)



where Φep(r, t) is the equilibrium distribution function at the local mean energy

Te(r, t) (adiabatic electrons). Since there is no equation for Φep(r, t), its form has

to be assumed a priori. In case it is assumed Maxwellian, it is obvious that fep(r, t)

is just the linearized Maxwell-Boltzmann distribution.Since we are describing the motion of charged particles in their own fields,

self-consistent closure of the system will stem from the Poisson equation withplasma/dust charges treated as sources,

∇ · E(r, t) = 4π

(∑α

eα

∫fα

p

d3p

(2π)3 +∫

q′fdp′(q′)dq′d3p′

(2π)3

). (4.7)

4.4 Decomposition in regular and fluctuating parts

The strategy is to decompose the Klimontovich equations for all species togetherwith the Poisson equation, then express all fluctuating quantities as a function of thenatural dust fluctuations (which can be considered as the only discreet component ofthe system in sufficiently low frequencies). This way one can derive expressions forthe total permittivity and the effective charge, quantities that define interactions incomplex plasmas. Afterwards, using the natural statistical correlator, the collisionintegrals can be computed in the Boltzmann kinetic equation of each componentand also the spectral densities of fluctuations.

The equation for the fluctuating part of the dust distribution function will be(∂

∂t+ v′ · ∂

∂r

)δfd

p′(q) + ∂

∂q

[(Iext +

∑α

⟨Iα(q)⟩)

δfdp′(q)

]+

qδE · ∂

∂p′ Φdp(q) + ∂

∂q

[∑α

δIαΦdp′(q)

]= −νn,d

[δfd

p′(q) − δndΦd,eqp′ (q)

].

Since dust discreteness is taken into account, the dust particle density fluctuationswill have both natural and induced parts, i.e δnd = δn

(0)d + δn

(ind)d . We Fourier

transform in space and time, with −ıω + ık · v′ + νn,d = −ı (ω − k · v′ + ıνn,d) wehave

−ı (ω − k · v′ + ıνn,d) δfdp′,k,ω(q) + ∂

∂q

(Iext +

∑α

⟨Iα⟩

)δfd

p′,k,ω(q)

=

−qδEk,ω · ∂

∂p′ Φdp′(q) − ∂

∂q

[∑α

δIαk,ωΦd

p′(q)

]+ νn,d

(δn

d,(0)k,ω + δn

d,(ind)k,ω

)Φd,eq

p′ (q) .

We know that δndk,ω =

∫ ∫δfd

p′,k,ω(q) dqd3p′

(2π)3 , thus the BGK description of colli-sions clearly introduces a feedback in the equation. In order to express δn

d,(ind)k,ω , as

a function of the processes that induce it (electric field, charging plasma current,

4.4. DECOMPOSITION IN REGULAR AND FLUCTUATING PARTS 67

natural fluctuations) we need to integrate the equation over the charge and mo-mentum space. Furthermore, since we take into account the inhomogeneous partof the equation, we have

∫ ∫δfd


(2π)3 = δnd,(ind)k,ω only. When integrating

over the charge variable, terms that are full derivatives of the charge will vanish,because the distribution function of dust in any physical complex plasma systemmust tend to zero at q → ±∞ faster than any powers of the charge,∫

δfdp′,k,ω(q)dq =

∫q

ı (ω − k · v′ + ıνn,d)δEk,ω · ∂

∂p′ Φdp′(q)dq

− νn,d

(δn

d,(0)k,ω + δn

d,(ind)k,ω

) ∫ Φd,eqp′ (q)

ı (ω − k · v′ + ıνn,d)dq .

Since we are examining longitudinal fields δEk,ω = kδEk,ω = kk δEk,ω . We also

use the narrowness of the dust distribution function around the equilibrium chargeΦd

p′(q) = Φdp′δ(q − qeq) in the evaluation of the charge integrals,

∫ ∫δfd

p′,k,ω(q)d3p′dq

(2π)3 = − ıqeqδEk,ω

k

∫ k · ∂∂p′ Φd

p′(qeq)ω − k · v′ + ıνn,d

d3p′

(2π)3

+ ıνn,d

(δn

d,(0)k,ω + δn

d,(ind)k,ω

) ∫ Φd,eqp′ (qeq)

ω − k · v′ + ıνn,d

d3p′

(2π)3 .

We define the responses χd,eqk,ω , deq

k,ω and andk,ω

χd,eqk,ω =

4π q2eq

k2

∫1

ω − k · v′ + ıνn,d

(k ·

∂Φdp′(q)

∂p′

)d3p′

(2π)3 , (4.8)

deqk,ω =

∫ Φd,(eq)p′ (qeq)

ω − k · v′ + ıνn,d

d3p′

(2π)3 , (4.9)

an,dk,ω = 1

1 − ıνn,ddeqk,ω

, (4.10)

and end up with

δnd,(ind)k,ω = − ık

4πqeqχd,eq

k,ω δEk,ω + ıνn,ddeqk,ω

(δn

d,(0)k,ω + δn

d,(ind)k,ω

)(

1 − ıνn,ddeqk,ω

)δn

d,(ind)k,ω = − ık

4πqeqχd,eq

k,ω δEk,ω + ıνn,ddeqk,ωδn

d,(0)k,ω

δnd,(ind)k,ω = − ık

4πqeqand

k,ωχd,eqk,ω δEk,ω + ıνn,ddeq

k,ωandk,ωδn

d,(0)k,ω . (4.11)



Returning to the equation for the dust component and substituting for the induceddust density fluctuations, we now have

−ı(ω − k · v′ + ıνn,d

)δfd

p′,k,ω(q) + ∂

∂q

(Iext +

∑α

Iα(q)

)δfd

p′,k,ω(q)

= −Rp′,k,ω(q) ,

(4.12)where the inhomogeneous term is given by

Rp′,k,ω(q) = qδEk,ω · ∂

∂p′ Φdp′ (q) + ∂

∂q

[∑α

δIαk,ωΦd

p′ (q)

]− νn,d

(δn

d,(0)k,ω + δn

d,(ind)k,ω

)Φd,eq

p′ (q)

= qδEk,ω

kk · ∂

∂p′ Φdp′ (q) + ∂

∂q

[∑α

δIαk,ωΦd

p′ (q)

]

+ ıνn,d

(k

4πqeqand

k,ωχd,eqk,ω δEk,ω − νn,ddeq

k,ωandk,ωδn

d,(0)k,ω + ıδn

d,(0)k,ω

)Φd,eq

p′ (q)

= qδEk,ω

kk · ∂

∂p′ Φdp′ (q) + ∂

∂q

[∑α

δIαk,ωΦd

p′ (q)

]

+ ıνn,dandk,ω

(k

4πqeqχd,eq

k,ω δEk,ω + ıδnd,(0)k,ω

)Φd,eq

p′ (q) , (4.13)

due to −νn,dandk,ωdeq

k,ω + ı = −νn,ddeqk,ω

1−ıνn,ddeqk,ω

+ ı = −νn,ddeqk,ω

+ı+νn,ddeqk,ω

1−ıνn,ddeqk,ω

= ı andk,ω . The

homogeneous equation has the form

−ı (ω − k · v′ + ıνn,d) δfdp′,k,ω(q) + ∂

∂q

(Iext +

∑α

Iα(q)

)δfd

p′,k,ω(q)

= 0 .

It is a first order differential equation with respect to the charge. It can be solvedanalytically, even for the external and plasma currents being arbitrary functionsof the charge. Yet, a more intuitive approach can be found using the approxima-tion of small deviations of dust charges from their equilibrium values: We use thecharging equation dq

dt = Iext +∑

α

Iα(q), we assume q = qeq + ∆q with ∆q small,

we Taylor expand the plasma current keeping the first order term only and we usethe equilibrium condition Iext +

∑α

⟨Iα(qeq)⟩ = 0, the result will be

∆q(t) = ∆q0 exp (−νcht) ,

with the charging frequency given by νch = − ∂∂q

(∑α

⟨Iα⟩

)|q=qeq , expressing the

time constant at which the charges relax to their equilibrium value. Following the


same Taylor expansion in the homogeneous equation, will result to

ı (ω − k · v′ + ıνn,d) δfdp′,k,ω(∆q) + ∂

∂∆q

(νch∆qδfd

p′,k,ω(∆q))

= 0 .

The only difference with the case of fully ionized complex plasmas is the presenceof neutral-dust collision frequency, since it is part of a constant factor in the chargedifferential equation, all previous results can be extended to our case with thesubstitution ω − k · v′ → ω − k · v′ + ıνn,d. The solution of the homogeneousequation will be given by Eq.(3.21), the Green’s function of the problem will begiven by Eq.(3.34), the correlator of the free particle fluctuations will be givenby Eq.(3.27) and the properties of integrals of the induced dust distribution overthe charge described by Eqs.(3.37,3.40). Knowledge of the Green’s function (deltaresponse) of the problem, implies that the solution of the inhomogeneous equationwith a source term Rp′,k,ω(q) will be δf

d,(ind)p′,k,ω (q) =

∫G(q, q′)Rp′,k,ω(q′)dq′. Let

F (q) be an arbitrary differentiable weighting function of the charge, let us wishto compute the integral of the form

∫F (q)δf

d,(ind)p′,k,ω (q)dq . It is obvious that it

should be evaluated for charges close to the equilibrium charge. In case the sourceterm is not a full derivative of the charge, evaluation at q = qeq would suffice,the Green’s function would then reduce to 1

ı(ω−k·v′+ıνn,d) (property of Eq.(3.37)).In case the source term is a full derivative of the charge, evaluation at q = qeq

gives zero contribution and first order deviations from equilibrium should also betaken into account, the Green’s function would then reduce to − 1

ı(ω−k·v′+ıνn,d+ıνch)(property of Eq.(3.40)). From Eq.(4.13) we notice that the actual source term canbe decomposed in both terms,∫

F (q)δfd,(ind)p′,k,ω

(q)dq =∫

F (q)G(q, q′)Rp′,k,ω(q′)dqdq′

=∫

F (q)G(q, q′)R1,p′,k,ω(q′)dqdq′

+∫

F (q)G(q, q′)R2,p′,k,ω(q′)dqdq′

= F (qeq)ı (ω − k · v′ + ıνn,d)

∫R1,p′,k,ω(q′)dq′

−∂F (q)

∂q|q=qeq

ı (ω − k · v′ + ıνn,d + ıνch)

∫R2,p′,k,ω(q′)dq′

= F (qeq)ı (ω − k · v′ + ıνn,d)δEk,ω

∫q′

kk · ∂

∂p′ Φdp′ (q′)dq′

+ ıνn,dandk,ω

(k

4πqeqχd,eq

k,ω δEk,ω + ıδnd,(0)k,ω

)∫Φd,eq

p′ (q′)dq′

−∂F (q)

∂q|q=qeq

ı (ω − k · v′ + ıνn,d + ıνch)

∫ ∑α

δIαk,ω(q′)Φd

p′ (q′)dq′ .

(4.14)



An additional property can be derived by using the narrowness of the average partof the dust distribution function and by integrating the above relation over themomentum space, together with the definitions of the response χd,ch

k,ω ,

χd,chk,ω =

∫ı

ω − k · v′ + ıνch + ıνn,dΦd

p′d3p′

(2π)3 , (4.15)

we have∫F (q)δfd,ind

p′,k,ω(q)

d3p′dq

(2π)3 =F (qeq)

ıkqeqδEk,ω

∫k · ∂

∂p′ Φdp′(

ω − k · v′ + ıνn,d

) d3p′

(2π)3 + F (qeq)νn,dandk,ω

×∫ Φd,eq

p′(ω − k · v′ + ıνn,d

) d3p′

(2π)3 (k

4πqeqχd,eq

k,ωδEk,ω + ıδn

d,(0)k,ω

)

+∂F (q)

∂q|q=qeq

∑α

δIαk,ω(qeq)

∫ıΦd

p′(ω − k · v′ + ıνn,d + ıνch

) d3p′

(2π)3

= −ıF (qeq)kχd,eq

k,ω

4πqeqδEk,ω + F (qeq)

kχd,eqk,ω

4πqeqδEk,ωνn,dand

k,ωdeqk,ω

+ ıF (qeq)νn,dan,dk,ω

deqk,ω

δnd,(0)k,ω

+∂F (q)

∂q|q=qeq

∑α

δIαk,ω(qeq)χd,ch

k,ω

= −ıF (qeq)kχd,eq

k,ω

4πqeqδEk,ωand

k,ω + ıF (qeq)νn,dan,dk,ω

deqk,ω

δnd,(0)k,ω

+∂F (q)

∂q|q=qeq

∑α


k,ω. (4.16)

Finally, the kinetic equation for the average part of the dust distribution functionwill be (

∂

∂t+ v′ · ∂

∂r

)Φd

p′(q) = − ∂

∂q

(Iext + ⟨

∑α

Iα⟩)

Φdp′(q)

−

νn,d

(Φd

p′(q) − ndΦd,eqp′ (q)

)− q

⟨δE ·

∂δfdp′(q)

∂p′

⟩− ∂

∂q

⟨∑α

δIαδfdp′(q)

⟩. (4.17)

We notice that the ionization frequency and the absorption frequency dependon the distribution functions of the electrons and the dust particles respectively.Therefore, they have both fluctuating and average parts. We start from νI(v):From the adiabatic assumption for the electron species, after decomposition weacquire,

Φep + δfe

p = Φep + eδϕ

TeΦe

p ⇒ δfep = eδϕ

TeΦe

p ⇒ δfep,k,ω = ıe

TekΦe

p δEk,ω . (4.18)


This relation after integration over the momentum space will result in a relationfor the electron density fluctuations,∫

δfep,k,ω

d3p

(2π)3 = ıe

Tek

∫Φe

p

d3p

(2π)3 δEk,ω ⇒ δnek,ω = ıene

TekδEk,ω .

We set neνe =∫

νI(v)Φep

d3p(2π)3 for the average part, while we acquire

δνI =∫

νI(v)δfep = ıe

Tek

(∫νI(v)Φe

p

d3p

(2π)3

)δEk,ω ⇒ δνI

k,ω = ıeneνe

TekδEk,ω .

For the absorption frequency we directly set

νd,i(v) =∫

σi(q, v)vΦdp′(q) dqd3p′

(2π)3 ,

δνd,ik,ω =

∫σi(q, v)vδfd


(2π)3 .

The equation for the fluctuating part of the ion distribution function will be(∂

∂t+ v · ∂

∂r

)δf i

p+eδE · ∂

∂pΦi

p = δνIΦMp −νn,i

(δf i

p − δniΦMp

)−δνd,iΦi

p−νd,iδf ip .

We Fourier transform in time and space, we use longitudinal fields and gather allδf i

p terms together,

−ı (ω − k · v + ı(νn,i + νd,i)) δf ip,k,ω + eδEk,ω

kk · ∂

∂pΦi

p =

δνIk,ωΦM

p + νn,iδnik,ωΦM

p − δνd,ik,ωΦi

p .

We rearrange the terms, substitute for δνIk,ω , δνd,i

k,ω and solve for the induced fluc-tuations

δf ip,k,ω = − 1

ı (ω − k · v + ı(νn,i + νd,i))−eδEk,ω

kk · ∂

∂pΦi

p + ıeneνe

TekδEk,ωΦM

p

+δnik,ωνn,iΦM

p −∫

σi(q, v)vΦipδfd


(2π)3 .

Since ions are treated as continuous Vlasov fluids in the phase space, there willbe no natural ion density fluctuations, δni

k,ω = δni,(ind)k,ω . We notice again that

treatment of collisions via the BGK collision integral creates a feedback in theequation. We integrate all over the momentum space, to find a solution for theion density fluctuations δni

k,ω =∫

δf ip,k,ω

d3p(2π)3 . We also use the definitions of the

responses χik,ω, Gk,ω and qi

k,ω(q),

χik,ω = 4π e2

k2

∫1

ω − k · v + ıνd,i(v) + ıνn,i

(k ·

∂Φip

∂p

)d3p

(2π)3 , (4.19)



Gk,ω =∫ ΦM

p

ı (ω − k · v + ıνd,i(v) + ıνn,i)d3p

(2π)3 , (4.20)

qik,ω(q) =

∫evσi(q, v)

ı (ω − k · v + ıνd,i(v) + ıνn,i)Φi

p

d3p

(2π)3 , (4.21)

and acquire

δnik,ω = eδEk,ω

ık

∫1

ω − k · v + ı(νn,i + νd,i)k · ∂

∂pΦi

pd3p

(2π)3

− ıeneνe

TekδEk,ω

∫ΦM

p

ı (ω − k · v + ı(νn,i + νd,i))d3p

(2π)3

− νn,iδnik,ω

∫ΦM

p

ı (ω − k · v + ı(νn,i + νd,i))d3p

(2π)3

+∫ (∫

σi(q, v)vΦip

ı (ω − k · v + ı(νn,i + νd,i))d3p

(2π)3

)δfd


(2π)3

= −ıkχi

k,ω

4πeδEk,ω − ıeneνe

TekGk,ωδEk,ω − νn,iGk,ωδni

k,ω +∫

qik,ω(q)

eδfd


(2π)3

= − 11 + νn,iGk,ω

ıkχik,ω

4πeδEk,ω − 1

1 + νn,iGk,ω

ıeneνe

TekGk,ωδEk,ω

+ 11 + νn,iGk,ω

∫qi

k,ω(q)e

δfdp′,k,ω(q) dqd3p′

(2π)3 . (4.22)

Overall, we substitute back in δf ip,k,ω, we also set for our convenience Dk,ω(v) =

ω − k · v + ıνn,i + ıνd,i(v),

δf ip,k,ω = 1

ıDk,ω(v)eδEk,ω

kk ·

∂Φip

∂p− ıeneνeδEk,ω

kTe

ΦMp

ıDk,ω(v)

+ νn,i

1 + νn,iGk,ω

ıkχik,ωδEk,ω

4πe

ΦMp

ıDk,ω(v)

+ νn,i

1 + νn,iGk,ω

ıeneνeGk,ωδEk,ω

Tek

ΦMp

ıDk,ω(v)

− νn,i

1 + νn,iGk,ω

∫qi

k,ω(q)e

δfdp′,k,ω(q) dqd3p′

(2π)3ΦM

p

ıDk,ω(v)

+∫

vσi(q, v)Φip

ıDk,ω(v)δfd


(2π)3 . (4.23)

Finally, the kinetic equation for the average part of the ion distribution function


will be(∂

∂t+ v · ∂

∂r

)Φi

p = neνeΦMp − νd,i(v)Φi

p − νn,i

(Φi

p − niΦMp

)− e

⟨δE ·

∂δf ip

∂p

⟩−∫

vσi(q, v)⟨δf i

pδfdp′(q)

⟩ d3p′dq

(2π)3 . (4.24)

The normal component of the Poisson equation, due to the average electric fieldconsidered zero, will give the modified quasi-neutrality condition (after use of thenarrowness of the dust distribution function around the equilibrium charge),

∇ · ⟨E⟩ = 4π∑

α

eα

∫Φα

p

d3p

(2π)3 + 4π

∫qΦd

p′(q) d3p′dq

(2π)3

4π∑

α

eαnα + 4πqeq

∫Φd

p′d3p′

(2π)3 = 0

4π∑

α

eαnα + 4πqeqnd = 0 .

On the other hand the random component will give an equation for the fluctuatingfield, after Fourier transforming we obtain

ıkδEk,ω = 4π∑

α

eα

∫δfα

p,k,ωd3p

(2π)3 + 4π

∫qδfd

p′,k,ω(q) d3p′dq

(2π)3

= 4πeδnik,ω − 4πeδne

k,ω + 4π

∫qδf

d,(0)p′,k,ω

(q) d3p′dq

(2π)3 + 4π

∫qδf

d,(ind)p′,k,ω

(q) d3p′dq

(2π)3 .

(4.25)

4.5 The permittivity

The relation for the plasma current flowing to the dust particles is∑

α

Iα(q) =

∑α

∫eαvσα(q, v)fα

p

d3p

(2π)3 . Therefore, the fluctuating plasma current will be given

by∑

α

δIα(q) =∑

α

∫eαvσα(q, v)δfα

p

d3p

(2π)3 . We will express∑

α

δIα(q) as a func-

tion of the electric field fluctuations and the natural fluctuations using Eq.(4.18,4.23)and the property described by Eq.(4.16),

∑α

δIα(q) = −∫

evσe(q, v)δfep

d3p

(2π)3 +∫

evσi(q, v)δf ip

d3p

(2π)3 . (4.26)



It is obvious that the final expression consists of seven terms, we shall evaluate eachterm separately,

A1 = −∫

evσe(q, v)δfep

d3p

(2π)3 = −∫

evσe(q, v)ıeΦe

pδEk,ω

kTe

d3p

(2π)3

= − ıe2

kTe

[∫vσe(q, v)Φe

p

d3p

(2π)3

]δEk,ω = SII

k,ω(q)δEk,ω ,

where we used the definition of the response SIIk,ω(q),

A2 =∫

evσi(q, v)eδEk,ω

k

k · ∂Φip

∂p

ıDk,ω(v)d3p

(2π)3

=

[e2∫

vσi(q, v)ıDk,ω(v)

k

k·

∂Φip

∂p

d3p

(2π)3

]δEk,ω = SI

k,ω(q)δEk,ω ,

where we used the definition of the response SIk,ω(q),

A3 = −∫

evσi(q, v) ıeneνe

kTeδEk,ω

ΦMp

ıDk,ω(v)d3p

(2π)3

= − ıeneνe

kTe

[∫evσi(q, v)

ΦMp

ıDk,ω(v)d3p

(2π)3

]δEk,ω

=[− ıeneνe

kTeλi

k,ω(q)]

δEk,ω = SIIIk,ω(q) δEk,ω ,

where we used

λik,ω(q) =

∫evσi(q, v)

ı (ω − k · v + ıνd,i(v) + ıνn,i)ΦM

p

d3p

(2π)3 (4.27)

and the definition of the response SIIIk,ω(q),

A4 =∫

evσi(q, v) νn,i

1 + νn,iGk,ω

ıkχik,ωδEk,ω

4πe

ΦMp

ıDk,ω(v)d3p

(2π)3

=ıkχi

k,ω

4πe

νn,i

1 + νn,iGk,ω

[∫evσi(q, v)ΦM

p

ıDk,ω(v)d3p

(2π)3

]δEk,ω

=

[νn,i

1 + νn,iGk,ω

ıkχik,ω

4πe

]λi

k,ω(q)δEk,ω = ∆SIk,ωλi

k,ω(q)δEk,ω ,


where we used the definition of the response ∆SIk,ω,

A5 =∫

evσi(q, v) νn,i

1 + νn,iGk,ω

ıeneνeGk,ωδEk,ω

Tek

ΦMp

ıDk,ω(v)d3p

(2π)3

= ıeneνeGk,ω

Tek

νn,i

1 + νn,iGk,ω

[∫evσi(q, v)ΦM

p

ıDk,ω(v)d3p

(2π)3

]δEk,ω

=[

νn,i

1 + νn,iGk,ω

ıeneνe

TekGk,ω

]λi

k,ω(q)δEk, ω = ∆SIIk,ωλi

k,ω(q)δEk,ω ,

where we used the definition of the response ∆SIIk,ω,

A6 = −∫

evσi(q, v) νn,i

1 + νn,iGk,ω

∫qi

k,ω(q′)e

δfdp′,k,ω(q′) dq′d3p′

(2π)3ΦM

p

ıDk,ω(v)d3p

(2π)3

= − νn,i

1 + νn,iGk,ω

∫ [∫evσi(q, v)ΦM

p

ıDk,ω(v)d3p

(2π)3

]qi

k,ω(q′)e


(2π)3

= − νn,i

1 + νn,iGk,ω

∫λi

k,ω(q)qi

k,ω(q′)e


(2π)3

=∫ [

− νn,i

1 + νn,iGk,ω

λik,ω(q)

eqi

k,ω(q′)

]δfd


(2π)3

=∫

SIIk,ω(q, q′)δfd


(2π)3 ,

where we used the definition of the response SIIk,ω(q, q′) ,

A7 =∫

evσi(q, v)∫

vσi(q′, v)Φip

ıDk,ω(v)δfd


(2π)3d3p

(2π)3

= e

∫ [∫v2σi(q, v)σi(q′, v)Φi

p

ıDk,ω(v)d3p

(2π)3

]δfd


(2π)3

=∫

SIk,ω(q, q′) δfd


(2π)3 ,

where we used the definition of the response SIk,ω(q, q′).

Overall, we have∑

α

δIα(q) =7∑

i=1Ai, using the definition of the responses Sk,ω(q),

∆Sk,ω and Sk,ω(q, q′) (note that q refers to the charge dependence of the plasma



current, q’ refers to already integrated charge)

Sk,ω(q) = e2∫

vσi(q, v)ı (ω − k · v + ıνd,i(v) + ıνn,i)

(k ·

∂Φip

∂p

)d3p

(2π)3

− ı e2

kTe

∫vσe(q, v)Φe

p

d3p

(2π)3 − ı eneνe

kTeλi

k,ω(q), (4.28)

∆Sk,ω = ıνn,i

1 + νn,iGk,ω

(kχi

k,ω

4π e+ eneνe

kTeGk,ω

),

(4.29)

Sk,ω(q, q′) = e

∫v2σi(q, v)σi(q′, v)

ı (ω − k · v + ıνd,i(v) + ıνn,i)Φi

p

d3p

(2π)3

− νn,i

1 + νn,iGk,ω

λik,ω(q)

eqi

k,ω(q′) , (4.30)

we obtain∑α

δIα(q) =(SI

k,ω(q) + SIIk,ω(q) + SIII

k,ω(q))

δEk,ω +(∆SI

k,ω + ∆SIIk,ω

)λi

k,ω(q)δEk,ω

+∫ (

SIk,ω(q, q′) + SII

k,ω(q, q′))


(2π)3

= Sk,ω(q)δEk,ω + ∆Sk,ωλik,ω(q)δEk,ω +

∫Sk,ω(q, q′)δfd

p′,k,ω(q′)dq′d3p′

(2π)3

=(Sk,ω(q) + ∆Sk,ωλi

k,ω(q))

δEk,ω +∫

Sk,ω(q, q′) δfdp′,k,ω(q′) dq′d3p′

(2π)3 .

But since the dust distribution function has both induced and natural fluctuations,we have∑

α

δIα(q) =(Sk,ω(q) + ∆Sk,ωλi

k,ω(q))

δEk,ω +∫

Sk,ω(q, q′) δfd,(0)p′,k,ω(q′) dq′d3p′

(2π)3

+∫

Sk,ω(q, q′) δfd,(ind)p′,k,ω (q′) dq′d3p′

(2π)3 . (4.31)

In order to evaluate the last adder of Eq.(4.31), we apply the property of Eq.(4.16)by setting q → q′ and F (q′) → Sk,ω(q, q′). We also define the responses βi

k,ω(q) =∂qi

k,ω(q)∂ q and S′

k,ω(q, q′) = ∂Sk,ω(q,q′)∂q′ ,

βik,ω(q) =

∫evσ′

i(q, v)ı (ω − k · v + ıνd,i(v) + ıνn,i)

Φip

d3p

(2π)3 ,

(4.32)


S′k,ω(q, q′) = e

∫v2σi(q, v)σ′

i(q′, v)ı (ω − k · v + ıνd,i(v) + ıνn,i)

Φip

d3p

(2π)3

− νn,i

1 + νn,iGk,ω

λik,ω(q)

eβi

k,ω(q′) , (4.33)

and obtain∫Sk,ω(q, q′) δf

d,(ind)p′,k,ω (q′) dq′d3p′

(2π)3 = −ıSk,ω(q, qeq)kχd,eq

k,ω

4πqeqδEk,ωand

k,ω+

ıSk,ω(q, qeq)νn,dan,dk,ωdeq

k,ωδnd,(0)k,ω + S′

k,ω(q, qeq)∑

α


k,ω .

We substitute in Eq.(4.31), set q = qeq and we solve for the fluctuating plasmacurrent,

∑α

δIαk,ω(qeq) =

[Sk,ω(qeq) + ∆Sk,ωλi

k,ω(qeq)1 − S′


−ıkand

k,ωχd,eqk,ω

4πqeq

Sk,ω(qeq, qeq)1 − S′


]

×δEk,ω +∫

andk,ωSk,ω(qeq, qeq)


k,ω

δf

d,(0)p′,k,ω

d3p′

(2π)3 .

Finally, we use the definitions of γk,ω(q, q′), β0k,ω(q) and βk,ω(q)

γk,ω(q, q′) = Sk,ω(q, q′)1 − S′


andk,ω , (4.34)

β0k,ω(q) =

Sk,ω(q) + λik,ω(q)∆Sk,ω


k,ω

, (4.35)

βk,ω(q) = β0k,ω(q) − ı k

4π qeqχd,eq

k,ω γk,ω(q, qeq) , (4.36)

and acquire∑α

δIα(qeq) = βk,ω(qeq)δEk,ω +∫

γk,ω(qeq, qeq)δfd,(0)p′,k,ω

d3p′

(2π)3 . (4.37)

We shall use the decomposed Poisson equation, Eq.(4.25), together with theexpressions for the induced plasma density fluctuations, the property of Eq.(4.16)and the relation for the plasma current, Eq.(4.37), to express the electric field viathe natural dust fluctuations only.

ıkδEk,ω = 4πeδnik,ω − 4πeδne

k,ω + 4π

∫qδf


(2π)3 + 4π

∫qδf

d,(ind)p′,k,ω (q) d3p′dq

(2π)3 .



Before proceeding in a term by term evaluation, we apply Eq.(4.16) for F (q) = q

and F (q) = qik,ω(q), using the definition of the response βi

k,ω(q) = ∂qik,ω(q)∂q .

∫qδf

d,(ind)p′,k,ω (q)d3p′dq

(2π)3 = −ıkχd,eq

k,ω

4πδEk,ωand

k,ω + ıqeqνn,dan,dk,ωdeq

k,ωδnd,(0)k,ω

+∑

α


k,ω

= −ıkχd,eq

k,ω

4πδEk,ωand

k,ω + ıqeqνn,dan,dk,ωdeq

k,ωδnd,(0)k,ω

+ βk,ω(qeq)χd,chk,ω δEk,ω + γk,ω(qeq, qeq)δn

d,(0)k,ω χd,ch

k,ω

=

βk,ω(qeq)χd,ch

k,ω − ıkχd,eq

k,ω

4πand

k,ω

δEk,ω

+

ıqeqνn,dan,dk,ωdeq

k,ω + γk,ω(qeq, qeq)χd,chk,ω

δn

d,(0)k,ω .

(4.38)

∫qi

k,ω(q)δfd,(ind)p′,k,ω (q)d3p′dq

(2π)3 = −ıqik,ω(qeq)

kχd,eqk,ω

4πqeqδEk,ωand

k,ω

+ ıqik,ω(qeq)νn,dan,d

k,ωdeqk,ωδn

d,(0)k,ω

+ βik,ω(qeq)

∑α


k,ω

= −ıqik,ω(qeq)

kχd,eqk,ω

4πqeqδEk,ωand

k,ω


k,ωdeqk,ωδn

d,(0)k,ω

+ βik,ω(qeq)βk,ω(qeq)δEk,ωχd,ch

k,ω

+ βik,ω(qeq)γk,ω(qeq, qeq)δn

d,(0)k,ω χd,ch

k,ω

=

βi

k,ω(qeq)βk,ω(qeq)χd,chk,ω − ıqi

k,ω(qeq)kχd,eq

k,ω

4πqeqand

k,ω

× δEk,ω + βi

k,ω(qeq)γk,ω(qeq, qeq)χd,chk,ω


k,ωdeqk,ωδn

d,(0)k,ω

(4.39)


We start by evaluating the first adder,

4πeδnik,ω = − 1

1 + νn,iGk,ωıkχi

k,ωδEk,ω − 4πe

1 + νn,iGk,ω

ıeneνe

TekGk,ωδEk,ω

+ 4π

1 + νn,iGk,ω

∫qi

k,ω(q)δfdp′,k,ω(q) dqd3p′

(2π)3

= − 11 + νn,iGk,ω

ıkχik,ωδEk,ω − 4πe

1 + νn,iGk,ω

ıeneνe

TekGk,ωδEk,ω

+ 4π

1 + νn,iGk,ω

∫qi

k,ω(q)δfd,(0)p′,k,ω

(q) dqd3p′

(2π)3

+ 4π

1 + νn,iGk,ω

∫qi

k,ω(q)δfd,(ind)p′,k,ω

(q) dqd3p′

(2π)3

= − 11 + νn,iGk,ω

ıkχik,ωδEk,ω − 4πe

1 + νn,iGk,ω

ıeneνe

TekGk,ωδEk,ω

+ 4π

1 + νn,iGk,ω(∫

qik,ω(q)δf

d,(0)p′,k,ω

(q) dqd3p′

(2π)3 +βi

k,ω(qeq)βk,ω(qeq)χd,chk,ω − ıqi

k,ω(qeq)kχd,eq

k,ω

4πqeqand

k,ω

δEk,ω

+

βik,ω(qeq)γk,ω(qeq, qeq)χd,ch

k,ω + ıqik,ω(qeq)νn,dan,d

k,ωdeqk,ω

δn

d,(0)k,ω )

= −ıkχi

k,ω

1 + νn,iGk,ω+ 4πeGk,ω

1 + νn,iGk,ω

eneνe

Tek2 +and

k,ω

1 + νn,iGk,ω

qik,ω(qeq)χd,eq

k,ω

qeq+

4πı

k

βik,ω(qeq)βk,ω(qeq)χd,ch

k,ω

1 + νn,iGk,ωδEk,ω + 4π

∫

qik,ω(q)

1 + νn,iGk,ω

+βi


1 + νn,iGk,ω+

ıqik,ω(qeq)νn,dan,d

k,ωdeqk,ω

1 + νn,iGk,ω δf

d,(0)p′,k,ω(q) dqd3p′

(2π)3

= −ıkχi

k,ω

1 + νn,iGk,ω+ Gk,ω

1 + νn,iGk,ω

νe

k2λ2De

+and

k,ω

1 + νn,iGk,ω

qik,ω(qeq)χd,eq

k,ω

qeq+

4πı

k


k,ω

1 + νn,iGk,ωδEk,ω + 4π

∫

qik,ω(q)

1 + νn,iGk,ω

+βi


1 + νn,iGk,ω+


k,ωdeqk,ω

1 + νn,iGk,ω δf

d,(0)p′,k,ω

(q) dqd3p′

(2π)3 .

(4.40)

For the second adder we have

−4πeδnek,ω = −4πe

ıene

TekδEk,ω = −ık

4πe2ne

Tek2 δEk,ω = −ık

[1

k2λ2De

]δEk,ω , (4.41)



while the third adder is already in the desired form. The last adder becomes

4π

∫qδf

d,(ind)p′,k,ω (q)d3p′dq

(2π)3 = −ık

χd,eq

k,ω andk,ω + 4πı

kβk,ω(qeq)χd,ch

k,ω

δEk,ω+

4π

∫ ıqeqνn,dan,d

k,ωdeqk,ω + γk,ω(qeq, qeq)χd,ch

k,ω

δf

d,(0)p′,k,ω(q)d3p′dq

(2π)3 . (4.42)

Finally, combining we end up with

ık1 + 1k2λ2

De

+ 1k2λ2

De

νeGk,ω

1 + νn,iGk,ω+

χik,ω

1 + νn,iGk,ω+ 4πı

kβk,ω(qeq)χd,ch

k,ω +

4πı

k


k,ω

1 + νn,iGk,ω+ χd,eq

k,ω andk,ω +

andk,ω

1 + νn,iGk,ω

qik,ω(qeq)χd,eq

k,ω

qeqδEk,ω =

4π

∫q +

qik,ω(q)

1 + νn,iGk,ω+

βik,ω(qeq)γk,ω(qeq, qeq)χd,ch

k,ω

1 + νn,iGk,ω+


k,ωdeqk,ω

1 + νn,iGk,ω

+ıqeqνn,dan,dk,ωdeq

k,ω + γk,ω(qeq, qeq)χd,chk,ω δf

d,(0)p′,k,ω(q)d3p′dq

(2π)3 .

Since, the total permittivity and the effective charge are defined through the relation

ıkϵk,ωδEk,ω = 4π

∫qeff

k,ω (q)δfd,(0)p′,k,ω(q)d3p′dq

(2π)3 , (4.43)

a direct comparison will give us their final expressions,

qeffk,ω (q) = q − qeqχnd

k,ω + χd,chk,ω γk,ω(qeq, qeq) + 1

1 + νn,iGk,ω[qi

k,ω(q)

+ χd,chk,ω γk,ω(qeq, qeq)βi

k,ω(qeq)] , (4.44)

ϵk,ω = ϵpk,ω + and

k,ωχd,eqk,ω

[1 + 1

1 + νn,iGk,ω

qik,ω(qeq)

qeq

], (4.45)

where

χndk,ω = −ıνn,dand

k,ωdeqk,ω

[1 +

qik,ω(qeq)/qeq

1 + νn,iGk,ω

], (4.46)

ϵpk,ω = 1 + 1

k2λ2De

(1 + νe

1 + νn,iGk,ωGk,ω

)+ 1

1 + νn,iGk,ωχi

k,ω

+ 4πı

kχd,ch

k,ω βk,ω(qeq)

[1 +

βik,ω(qeq)

1 + νn,iGk,ω

]. (4.47)

Chapter 5

Charging of Non-emitting Grainsin Presence of Neutrals

The problem of dust charging is central to the field of complex plasmas, sincethe high dust charge values and their fluctuations are responsible for the mostfundamental processes. Here, we are interested in the charging of non-emittingdust grains embedded in isotropic partially ionized plasmas.

The dust charge will be studied in its normalized form z = Zde2

aTewhich is also

the normalized dust surface potential ϕs = − Zdea . For the comprehensive study

of the effect of pressure it is important to define the plasma collisionality αc =λD

ln,i, that is the ratio of the dust screening length approximated by the ion Debye

length to the mean free path in ion collisions with neutrals [Khrapak and Morfill,2009]. It determines whether the ions are collisionless (pair-particle approach) orcollisional (hydrodynamic approach) in the perturbed sphere around the grain. It isimportant to point out, that the collision cross-sections of electrons with neutrals areat least two orders of magnitude smaller than the collision cross-sections of ions withneutrals (typical values for Neon are σn,e = 10−16 cm2 and σn,i = 10−14 cm2), thismeans that there exists a wide range of pressures where ions are hydrodynamicallytreated while electrons are collisionless.

Before proceeding to quantitative results we briefly present a qualitative view ofthe problem; A grain immersed in a plasma will initially become negatively chargeddue to the higher mobility of the light electrons. The quasi-stationary charge willthen be set up by the balance of electron and ion currents flowing to the grain andit will remain negative. For very low pressures, the Orbit Motion Limited (O.M.L.)approach can be implemented for both the electron and the ion species [Allen, 1992;Allen et al., 2000]. As the pressure gradually increases, collisions start to affect theion flux. Effective charge exchange collisions lead to the trapping of low energy ionsin the strong field region around the grain and to eventual absorption on the grainsurface [Lampe et al., 2003]. This leads to an increase in the ion flux and hence adecrease in dust charge. For larger pressures, ion transport will become collision

81

82CHAPTER 5. CHARGING OF NON-EMITTING GRAINS IN PRESENCE OF

NEUTRALS

dominated, ion mobility will become suppressed by collisions with neutrals and theion flux will reduce, leading to an increase in the dust charge, which also implies theexistence of a minimum for ln,i ∼ λD. For even larger pressures, electron transportwill also become collisional, both electron and ion fluxes will simultaneously reduceand the dust charge saturates with respect to the gas pressure.

For a better understanding of the problem we can identify the following regimesaccording to the value of the plasma collisionality,

1. Collisionless regime, ln,e, ln,i ≫ λD.

2. Weakly collisional regime, ln,i > λD.

3. Intermediate collisional regime, ln,i ∼ λD.

4. Strongly collisional regime, ln,i < λD.

5. Fully collisional regime, ln,i, ln,e ≪ λD.

We point out that gas discharges typically operate in the weakly and intermediatecollisional regimes.

5.1 Charging in the collisionless regime

The collisionless regime is defined by ln,e, ln,i ≫ λD, the Orbit Motion Limited(O.M.L) cross-sections can then be used for the computation of the electron andion currents. The basic assumptions of the O.M.L. approach are: (i) the dust grainis isolated, i.e other dust grains do not affect the motion of electrons and ions in itsvicinity, (ii) electrons and ions do not experience collisions during the approach tothe grain, (iii) there are no barriers in the effective potential [Kennedy and Allen,2003].

The heavy grain is considered to be an infinitely massive scattering center for theelectrons and ions colliding with it. Due to the conservation of angular momentumof the plasma particle we have mαvb = mαr2ϕ, where u is the velocity of theparticle when it enters the mean free path sphere and b is the impact parameter.Conservation of the total energy of the particle also imposes the requirement that

12

mαv2 + eαϕ∞ = 12

mαr2 + 12

mαr2ϕ2 + eαϕs, (5.1)

where ϕ∞ ≃ 0 is the potential on the mean free path surface, ϕs is the potentialat the surface of the grain. At the closest approach r = rmin = a and r = 0,eliminating the angular velocity and solving for the impact parameter will result inb2 = b2

max = (1 − 2eαϕs

mαv2 )a2. The absorption cross-sections will be πb2max.

In the case that the currents emitted from the grain are negligible, the grain willbe negatively charged and we will have ϕs < 0. Hence, there will be an attractivepotential for the ions, which implies that an ion with any velocity can be absorbed

5.2. CHARGING IN THE WEAKLY COLLISIONAL REGIME 83

by the grain. On the other hand, there will be a repulsive potential for the electrons,which implies that electrons should have velocity larger than a minimum value inorder to reach the grain. This velocity can be found by equating the initial kineticenergy of the electron with the potential energy of the electron on the grain, inthat case the electron barely reaches the grain surface with zero kinetic energy, theresult will be vmin =

√2e|ϕs|

me. Combining our results we end up with,

σe(v) =

πa2(

1 − 2e|ϕs|mev2

), v >

√2e|ϕs|

me

0, v <√

2e|ϕs|me

(5.2)

σi(v) = πa2(

1 + 2e|ϕs|mev2

). (5.3)

The charge of the particle is related to the surface potential through the relationq = Cϕs, where C is the capacitance of the dust particle in the plasma. For sphericaldust grains with a ≪ λD we have C = 4πϵ0a, and we actually have the potentialof a point particle ϕs = q/a.

Finally, use of Iα(qeq) =∫

evσα(qeq, v)Φαp

d3p(2π)3 and Maxwellian distributions

will yield the O.M.L charging currents

Ie = −√

8πa2enevT e exp (−z) ,

Ii =√

8πa2enivT i(1 + z

τ) . (5.4)

Current balance will finally yield the transcendental equation

nevT e exp (−z) = nivT i(1 + z

τ) . (5.5)

5.2 Charging in the weakly collisional regime

In the weakly collisional regime, ln,i > λD, electrons are still collisionless, whereascollisions of ions with neutrals have an effect on charging. In order to demonstratethis; Let us assume an ion in the perturbed plasma region in the vicinity of thegrain and that it undergoes a resonant charge exchange collision with a neutral. Thehigh-energy orbiting ion will be substituted by a low-energy ion that will be -withhigh efficiency- trapped in the region of high attractive potential and eventuallyabsorbed by the grain (unless it undergoes subsequent collisions). This will leadto an increase in the ion flux to the grain and hence a decrease in the dust charge[Zakrzewski and Kopiczynski, 1974].

We consider the sphere of strong ion-dust interaction around the grain, with itsradius defined by equating the ion kinetic and interaction energies, i.e |U(R0)| = Ti.The expected number of collisions with neutrals can be approximated by R0

ln,i< 1

and the probability of n occurrences will be given by the Poisson distribution


NEUTRALS

P (n) = 1n!

(R0ln,i

)n

exp(

− R0ln,i

), The probability of no collisions will be P (0) =

exp(

− R0ln,i

)≃ 1− R0

ln,i, the probability of a single collision P (1) = R0

ln,iexp

(− R0

ln,i

)≃

R0ln,i

(1 − R0

ln,i

)≃ R0

ln,i, while the probability of multiple collisions P (n > 1) ≃(

R0ln,i

)2≃ 0. We essentially have two independent ion populations

• Collisionless ions appearing with P0 = 1 − R0ln,i

and having an O.M.L chargingcurrent I0 =

√8πea2nivT i(1 + z

τ ).

• Ions colliding once with P1 = R0ln,i

and then being absorbed with 100% effi-ciency, which means that the charging current will be the random current inthe R0 sphere I1 =

√8πeR2

0nivT i.

The total ion current will then be

Ii =∑

i=0,1PiIi =

√8πea2nivT i(1 + z

τ)(

1 − R0

ln,i

)+

√8πeR2

0nivT iR0

ln,i

≃√

8πea2nivT i(1 + z

τ) +

√8πe

R30

ln,inivT i

≃√

8πea2nivT i

(1 + z

τ+ R2

0a2

R0

ln,i

).

In the above equation R0 is still undetermined, it is depending on the potentialaround the grain. The simplest solution can be found by assuming a Yukawapotential with the screening length given by the ion Debye length. In that case,with βT = z

τa

λDithe thermal scattering parameter,

|U(R0)| = Ti ⇒ Zde2

R0exp (− R0

λDi) = Ti

z

τ

a

λDiexp (− R0

λDi) = R0

λDi

βT exp (− R0

λDi) = R0

λDi,

and R0 = λDix0 with x0 the root of the transcendental equation βT e−x = x.A more self-consistent calculation involves the solution of the Poisson equationwith electrons following the Boltzmann distribution and ions following the Gurevichdistribution [Alpert et al., 1965], with the latter taking into account absorption onthe grain (but with the effects of barriers in the effective potential and trapped ionsstill neglected). Numerical solution [Ratynskaia et al., 2006] yielded a fit with goodaccuracy in the Yukawa form with an effective screening length λ = (1+0.2

√βT )λDi

and R0(βT ) = (−0.1 + 0.8√

βT )λDi, for typical βT = 1 − 13.

5.3. CHARGING IN THE INTERMEDIATE COLLISIONAL REGIME 85

Finally, since the electrons are still collisionless, Ie = −√

8πea2nevT ee−z andthe dust charge will be given by the solution of

nivT i

(1 + z

τ+ R2

0a2

R0

ln,i

)= nevT ee−z . (5.6)

5.3 Charging in the intermediate collisional regime

In the intermediate collisional regime, ln,i ∼ λD while the electrons are still col-lisionless. In such a regime neither a pair-particle nor a hydrodynamic approachcan be implemented and therefore the ion current can only be approximated viainterpolation formulas [Khrapak and Morfill, 2008].

The requirements an interpolation formula are (i) in the limit ln,i > λD it shouldcoincide with the ion current for the weakly collisional regime IW C

i , (ii) in thelimit ln,i ≪ λD it should asymptotically reach the hydrodynamic limit ISC

i , (iii) itshould reproduce the theoretically predicted and experimentally verified minimumin the dust charge, therefore Ii should have a pronounced maximum. Such aninterpolation formula is

IICi =

[(1

IW Ci

)γ

+(

1ISC

i

)γ]−1/γ

, (5.7)

where γ > 0 is the charging index. It can either be a function of the plasmaparameters, chosen to best fit numerical results or molecular dynamics data as inHutchinson’s [Hutchinson and Patacchini, 2007] and Zobnin’s fits [Zobnin et al.,2000], or a constant.

5.4 Charging in the strongly collisional regime

In the limit of strong collisionality, ln,i < λD ≪ ln,e, electron transport to the grainis still collisionless, while ion transport to the grain is collision dominated [Su andLam, 1963; Khrapak et al., 2006]. In this case a fluid description can be adopted forthe ion component. In a steady state the momentum equation and the continuityequation will read us

Γi = niµiE − Di∇ ni ,

∇ · Γi = Qsi − QLi ,

where Qsi , Qse represent the source and loss rates of the ions (ionization, volumerecombination, absorption on dust etc), µi and Di are the ion mobility and dif-fusivity respectively (with µi considered independent of the electric field and theEistein relation assumed to hold) and Γi = nivi the ion flux.

By assuming that the characteristic ionization and recombination lengths aremuch larger than the plasma Debye length, the continuity equation will yield ∇ ·


NEUTRALS

Γi = 0 and the ion flux will be conserved. In spherical coordinates it will only havea constant radial component satisfying

Γi = 4πr2(

ni(r)µidϕ(r)

dr+ Di

dni(r)dr

).

On the other hand, for the electron density distribution around the negativelycharged grain, one can assume that it follows a Boltzmann relation

ne(r) = n0 exp (eϕ(r)Te

) ,

where ne0 = ni0 = n0 are the electron/ion densities in the quasi-neutral unper-turbed plasma. The system of equations is closed by the Poisson equation,

d2ϕ

dr2 + 2r

dϕ

dr= −4πe(ni(r) − ne(r)) ,

with the boundary conditions ϕ(∞) = 0, ϕ(a) = − Zdea = ϕs, ni(a) = ne(a) = 0 and

ni(∞) = ne(∞) = n0. We should notice that electron Boltzmann relation producesa slight inconsistency not satisfying the boundary conditions for ϕ(a) and ne(a)simultaneously, it yields ne(a) = n0e−z.

The above system can only be solved numerically. However, in the limit a ≪ λD

and close to the surface of the grain, a major simplification can be made; Since thedensities of the electrons and ions asymptotically tend to zero at the grain surface,it can be assumed that the geometrical terms of the Poisson equation dominate inthe vicinity of the grain. In that case the Poisson equation will take its sphericalvacuum form, which has the bare Coulomb potential as a solution ϕ(r) = − Zde

r .In that case the equations are decoupled and the ion flux conservation relation canbe solved to yield Ii = eΓi = 4πeaniDi

(z/τ)1−exp (−z/τ) . For the collisionless electrons

the O.M.L current will be Ie = −√

8πea2nevT ee−z. The dimensionless charge z asusual will be found from the current balance (we assume ne ≃ ni ≃ n0)

4πean0Di(z/τ)

1 − exp (−z/τ)=

√8πea2n0vT ee−z

√2πDi

(z/τ)1 − exp (−z/τ)

= avT ee−z

zez

1 − exp (−z/τ)= aτvT emiνn,i√

2πTi

zez

1 − exp (−z/τ)= τ√

2π

a

ln,i

vT e

vT i.

In gas discharges typically τ ≃ 0.01 and therefore exp (−z/τ) → 0, which yields

zez = τ√2π

a

ln,i

vT e

vT i. (5.8)

5.5. CHARGING IN THE FULLY COLLISIONAL REGIME 87

The above equation is transcendental, since zez is a monotonically increasing func-tion, for increasing pressure; the ln,i mean free path increases, the surface potentialincreases and hence the dust charge will increase.

5.5 Charging in the fully collisional regime

In the fully collisional regime, ln,i, ln,e ≪ λD, thus, both electron and ion transportto the grain will be collision dominated [Chang and Laframboise, 1976; Khrapaket al., 2006]. In this regime the electron fluxes will also be determined by thehydrodynamic equations. Flux continuity will result in

Γe = 4πr2(

−ne(r)µedϕ(r)

dr+ De

dne(r)dr

),

while the ion flux relation and the Poisson equation will remain the same. Em-ploying the same approximations in the limit of infinitesimally small dust grains,a ≪ λD, we end up with the currents

Ii = eΓi = 4πeaniDi(z/τ)

1 − exp (−z/τ),

Ie = −eΓe = −4πeaneDeze−z

1 − e−z.

Current balance, with the use of exp (−z/τ) → 0, results in (we assume ne ≃ni ≃ n0)

4πean0Di(z/τ)

1 − exp (−z/τ)= 4πean0De

ze−z

1 − e−z

z

1 − exp (−z/τ)= τ

De

Di

ze−z

1 − e−z

1 = τDe

Di

e−z

1 − e−z

z ≃ ln(

1 + τDe

Di

)z ≃ ln

(1 + τ

vT eσn,i

vT iσn,e

).

We conclude that in the fully collisional regime the dust charge is independent ofthe pressure.

5.6 Maximum of the charge as a function of pressure

From the behavior of the charge as a function of the plasma collisionality andits saturation for large pressures, it is obvious that it will be maximum either in


NEUTRALS

Figure 5.1: The dimensionless dust charge as a function of the plasma collisionality.Adopted from Khrapak and Morfill (2009).

the collisionless or the fully collisional regime. The determination of the chargemaximum is not a trivial issue, since the flux balance equation for O.M.L. currentsis transcendental.

The approximate solution for the charge in the fully collisional regime, in thecase ne = ni, can be further simplified by using τ ≃ 0.01, σn,i

σn,e≃ 100 and nevT e ≫

nivT i, with the latter valid for moderate dust densities, so that the electron densityis not severely depleted.

zSC ≃ ln(

1 + τnevT eσn,i

nivT iσn,e

)≃ ln

(1 + nevT e

nivT i

)≃ ln

(nevT e

nivT i

)⇒ e−zSC ≃ nivT i

nevT e.

Let us assume a function of the normalized charge f(z) = nevT ee−z −nivT i

(1 + z

τ

),

Table 5.1: The plasma collisionality and the expressions for the ion and electronfluxes.

regime λD

ln,iIi Ie z(p)

CL ≪ 1√

8πa2enivT i(1 + zτ ) −

√8πea2nevT ee−z const.

WC < 1√

8πea2nivT i

(1 + z

τ + R20

a2R0ln,i

)−

√8πea2nevT ee−z

IC ∼ 1 ´[(

1IW C

i

)γ

+(

1ISC

i

)γ]−1/γ

−√

8πea2nevT ee−z min.SC > 1 4πeaniDi

(z/τ)1−exp (−z/τ) −

√8πea2nevT ee−z

FC ≫ 1 4πeaniDi(z/τ)

1−exp (−z/τ) −4πeaneDeze−z

1−e−z const.

5.6. MAXIMUM OF THE CHARGE AS A FUNCTION OF PRESSURE 89

the first derivative with respect to z is dfdz = −nevT ee−z − nivT i

τ < 0, the functionis monotonically decreasing and hence it can only have one zero. That zero will bydefinition be the O.M.L. charge, zOML. We evaluate the function for z = 0 andz = zSC ,

f(zSC) = nevT ee−zSC − nivT i

(1 + zSC

τ

)= nevT e

nivT i

nevT e− nivT i − nivT i

zSC

τ

= nivT i − nivT i − nivT izSC

τ= −nivT i

zSC

τ= −nivT i

τln(

nevT e

nivT i

)< 0 ,

f(0) = nevT ee0 − nivT i = nevT e − nivT i > 0

Due to the continuity of f(z), by Bolzano’s theorem, for f(0)f(zSC) < 0, the onlyroot of f(z) must lie between (0, zSC), and hence zOML < zSC , and zSC is themaximum of the charge.

Chapter 6

Dust Charging Experiments inPK-4

6.1 The discharge tube and the microgravity experiments

The project Plasma-Kristall 4 (PK-4) is a continuation of the PK series of complexplasma experiments performed onboard the Russian Mir Space Station and theInternational Space Station (ISS). Unlike its predecessors PKE-Nefedov and PK-3plus [Nefedov et al., 2003] that were using a planar rf capacitive discharge, the PK-4experiment is run in a long cylindrical chamber with a combined dc/rf discharge[Fortov et al., 2005].

PK-4 is expected to fly on the ISS after 2013. Earth’s gravity exerts an externalforce on the massive dust component in ground based laboratory experiments. Thisforce prohibits the levitation of the dust grains in the bulk quasi-neutral plasmaand can only be avoided under microgravity conditions. Such conditions can beachieved by parabolic flights, sounding rockets or orbital flights.

The plasma is produced by applying a voltage of about 1000 V to the dc elec-trodes and the operating gas is either neon or argon at pressures between 10 and500 Pa, while the DC current can vary until 5 mA. The dc discharge is produced ina glass tube with a length of 35 cm and a diameter of 3 cm, while an rf dischargecan also be applied by external rf coils. Basic discharge modes are: (i) pure dcdischarge, (ii) two rf inductive discharges, (iii) combinations of the dc dischargewith one or two rf inductive discharges, (iv) combination of the dc discharge withone rf capacitive discharge. Inlet of dust grains in the discharge chamber is possiblethrough a set of four dust dispensers. The dust grains are melamine formaldehydemonodisperse spherical particles of diameters between 0.5 and 11 µm, while nano-particles can also be synthesized directly in the discharge. For their observationthe dust grains are illuminated by a laser sheet and observed by two fast chargecoupled device (CCD) video cameras.

A number of experiments are scheduled to be performed, they include: charging

91

92 CHAPTER 6. DUST CHARGING EXPERIMENTS IN PK-4

of dust particles and dust-acoustic waves, determination of the ion drag force,observation of the critical point in liquid-gas phase transitions, cloud collisionsand lane formation, observations of the transition from laminar flow to turbulenceat a microscopic level, investigation of Laval nozzle flows, experiments on solitons,agglomeration and particle growth [Thoma et al., 2007].

6.2 Dust charging experiments

When inserted into the discharge plasma, the dust particles get negatively chargedand start drifting against the dc electric field. As long as the flow is stable andinter-grain forces can be neglected (which is valid for sufficiently low numbers ofdust particles), the drift velocity of the dust grains is simply set up by the balanceof the forces acting on them: (i) the electrostatic force, (ii) the neutral drag force,(iii) the ion drag force, (iv) the electron drag force. The electron drag force isalways negligible due to the small ion to electron temperature ratio (τ ≤ 0.01),while the ion drag force is usually much smaller than the other two forces. It isevident that measurement of the dust drift velocity by analysis of recorded cameraframes, together with measurement of the plasma parameters by Langmuir probesand the force balance equation can yield experimental values of the charge. Thissimple approach is known as the force balance method [Ratynskaia et al., 2004;Khrapak et al., 2005].

Alternatively, the charge can be experimentally measured by the onset of thedust acoustic wave instability [Ratynskaia et al., 2004]. For large numbers of dustparticles there is an easily identifiable transition from stable flow to unstable flow(with a clear wave behavior). Responsible for this sharp transition (with an ex-perimental error of 1 Pa only) is the relative drift between the grains and the ions,

Figure 6.1: Sketch of the PK-4 experimental facility.

6.3. COMPARISON WITH CHARGING MODELS 93

which triggers the ion-dust streaming instability and excites dust acoustic waves ata certain threshold pressure around 50 Pa. The dust-acoustic waves tend to havevery amplitudes and are clearly non-linear. However, the onset of the low-frequencyself-excited waves can be well described by linear hydrodynamic dispersion relationswith collisionless Boltzmann electrons, static drifting collisional ions and counter-drifting collisional dust particles. The threshold pressure can be defined as thepressure for which the mode is driven from stable to unstable and will be depen-dent on the dust charge.

6.3 Comparison with charging models

The operating pressure range of the PK-4 discharge (10−500 Pa), spans the weaklycollisional and the intermediate collisional regimes for ions. Hence, the experimentaldetermination of the charge presented above can be used for the verification of thecharging models of these regimes.

In the weakly collisional regime both independent methods can be used for acomparison. The experimental data have shown a remarkable agreement with boththe charging model and molecular dynamics simulations for a variety of grain radii.

On the other hand, in the intermediate collisional regime only the force bal-ance method can be used. However, the existence of a pronounced minimum inthe charge well within the PK-4 operating pressures can facilitate a comparisonwith the interpolation formula. The experimental data have shown a very goodagreement with the exception of small systematic errors that can be contributed toan underestimation of the ion drag force.

Chapter 7

Summary

7.1 Paper I: Regimes for experimental tests of kineticeffects in dust acoustic waves

Dust acoustic waves (DAW) are waves similar to conventional ion acoustic waves,with the dust grains participating in the wave dynamics [Rao et al., 1990]. They areubiquitous features of complex plasmas produced in laboratory discharges, beingexcited by the ion-dust streaming instability [Merlino, 2009]. Involving the motionof massive dust grains, the DAW propagate in the low frequency regime defined bykvT d ≪ ω ≪ kvT i, typical frequencies are in the range of tens of Hz. They havereceived particular attention since they can be observed using laser light scatteringand charge coupled device (CCD) cameras, which provides the unique opportunityof investigations of wave-particle interactions at a particle level [e.g., Barkan etal., 1995; Thompson et al., 1997]. The physics of the mode are quite intuitive;the heavy dust species provides the inertia and the plasma pressure coupled to theelectric field provides the restoring force.

Linear weakly coupled dust acoustic waves have mostly been studied throughfluid models - from collisionless approaches to the inclusion of collisions with neu-trals [Pieper and Goree, 1996], particle drifts [Ivlev et al., 1999], plasma sources[D’Angelo, 1997], plasma absorption on dust, dust charge fluctuations [e.g., Me-landsø et al., 1993; Jana et al., 1993; Varma et al., 1993]- or through the multi-component [Rosenberg, 1993] and standard kinetic models [Shukla and Mamun,2002]. However, for the self-consistent inclusion of the effects of dust charge fluc-tuations and plasma absorption the use of the "full" kinetic model is necessary[Tsytovich et al., 2002].

In this paper the DAW waves are studied through the "full" kinetic model inthe fully ionized case. Aim of the work is to identify experimentally accessibleparameter regimes where charging effects / plasma absorption have a strong effecton the DAW dispersion, which will result in strong deviations from the traditionallyused collisionless hydrodynamic approach.

95

96 CHAPTER 7. SUMMARY

The theoretical advances of this paper include

1. The derivation of analytical asymptotic relations for the DAW phase velocitiesin wavelength regimes where (i) plasma absorption can be neglected, (ii) ionabsorption is important, (iii) both ion and electron absorption are strong.The results indicate a steep increase of the DAW phase velocity for largewavelengths and are also confirmed numerically.

2. The derivation of the first order in ωkvT i

imaginary parts of the low fre-quency plasma responses and their implementation for the determination ofthe damping rate of the waves under the small growth rate approximation.

3. The derivation of semi-analytical closed solutions of the dispersion equationthat are first order in kvT d

ω .

Results for concrete parameters (PK-4 facility, magnetic cusp) are presented andcriteria are formulated for the deviations of the DAW behavior from the collisionlessfluid approach to be maximum;

• possibility for the observation of large wavelengths (feasible in elongated dis-charge tubes)

• non-thermal plasmas with Te ≫ Ti (typical for low temperature discharges)

• dense dust clouds

• high plasma densities (feasible in magnetic confined low-temperature dis-charges)

• grains with large grain radius levitating in the bulk quasineutral dischargeplasma (feasible in microgravity).

7.2 Paper II: Kinetic models of partially ionized complexplasmas in the low frequency regime

Continuous absorption of plasma fluxes on the dust grains brings out the necessity ofan external source to replenish the plasma, so that the grains can sustain their quasi-equilibrium charges. The source can be either non-fluctuating or fluctuating. Anexample of fluctuating source, encountered in most laboratory discharges, is electronimpact ionization of neutrals, where the source fluctuations follow the fluctuationsof the exact electron distribution function. In partially ionized complex plasmas thesystem’s components are electrons, ions, dust grains and neutrals. The latter can betreated as a medium in thermodynamic equilibrium providing dissipation throughcollisions, the source of plasma particles and effects in grain charging [Tsytovich etal., 2005].

7.2. PAPER II: KINETIC MODELS OF PARTIALLY IONIZED COMPLEXPLASMAS IN THE LOW FREQUENCY REGIME 97

Figure 7.1: Typical plot of the real part of the permittivity.

In this paper kinetic models of partially ionized complex plasmas are formulatedin the low frequency regime, analyzed and compared in terms of their permittivityand static permittivity. Theoretical advances include

1. The low frequency integral plasma/dust responses, the static permittivity andthe permittivity are derived for the "full" kinetic model of partially ionizedcomplex plasmas.

2. The standard and "multi-component" kinetic models are formulated for par-tially ionized complex plasmas by employing the Bhatnagar-Gross-Krook re-laxation time approximation for the description of collisions with neutrals.

3. Absorption cross-sections are derived for ions taking into account the effectof neutrals in the weakly collisional regime, An analytical expression for thenew charging frequency is also derived.

4. The low frequency integral plasma/dust responses of all models are calculatedfor thermal distributions.

5. A simple condition is derived for the convergence of the multi-component withthe "full" kinetic model.

6. The ratio of induced to natural dust density fluctuations is derived for the"full" kinetic model, depending on its value great simplifications in the struc-ture of the kinetic equations are feasible.

On the numerical side, the permittivity and static permittivity of all models iscalculated for typical laboratory parameters. Novel results reveal


• The real part of the permittivity of the "full" kinetic model has multiple roots,which opens up the possibility for another low frequency electrostatic modein addition to the dust acoustic. This mode does not exist for the standardand multi-component kinetic models.

• The ratio of induced to natural dust density fluctuations is a decaying func-tion of the frequency. It is well below unity, already before the dust plasmafrequency ωpd, for all parameter ranges investigated. This implies that forkinetic models focusing on regimes typical of ion dynamic, induced dust fluc-tuations can be ignored. This will lead to major simplifications in the theory,since the induced part of the dust distribution function is the most cumber-some term in the mathematical treatment.

7.3 Paper III: Low frequency electrostatic modes inpartially ionized complex plasmas; a kinetic approach

In this paper the low frequency electrostatic modes in partially ionized complexplasmas are investigated employing the full kinetic model. This is the most self-consistent attempt to date. The results indicate that, apart from the dust acousticmode (DA), a novel long-wavelength (LW) mode exists. The characteristics of theLW mode are determined by the interplay between electron impact ionization ofneutrals and plasma absorption on dust. It propagates for long wavelengths and"induces" cut-offs in the DA mode.

The damping rates and dispersion relations of both modes are found from thesolution of the coupled system of equations ℑϵk,ωr,ωi = 0 , ℜϵk,ωr,ωi = 0 forrealistic laboratory parameters;

• A numerical scheme is devised, based on the strength of dissipation by neu-trals, that significantly simplifies the formidable numerical task with a negli-gible error.

• From physical and mathematical grounds, the multiple roots of the permit-tivity are classified to each mode.

• The sensitivity of the dispersion relations of both modes on basic plasma anddischarge parameters is investigated (gas pressure, dust density, dust size,electron temperature, operating gas).

• The DA mode kinetic dispersion is compared to hydrodynamic dispersionrelations.

Extensive numerical investigation of the long-wavelength mode has revealed itsmain characteristics:

7.3. PAPER III: LOW FREQUENCY ELECTROSTATIC MODES INPARTIALLY IONIZED COMPLEX PLASMAS; A KINETIC APPROACH 99

Figure 7.2: The dispersion relation for the long-wavelength mode as a function ofthe pressure for PK-4 density and temperature profiles (n(p) , Te(p)).

1. It is determined by the interplay between electron impact ionization andplasma absorption.

2. It does not exist for: low electron temperatures compared to the gas firstionization energy, relatively low pressures and relatively low dust densities.

3. Its dispersion relation appears to be adequately fitted by ωr(k) = A1 + A2k2+A2

3,

where the coefficients A1, A2 and A3 depend on the discharge, dust grain andplasma parameters, while A1 determines the cut-off frequency of the wave.

4. It is present for large wavelengths. However, as the ionization frequency/pressureincreases it can propagate at larger wavenumbers

5. The phase velocities are an order of magnitude larger than the DA phasevelocities.

6. The group velocity is negative, a characteristic common in ionization waves[Pekarek, 1968].

7. The presence of the mode in the large-wavelength part of the Fourier spec-trum leads to a cut-off of the DA mode in this range. The cut-off fre-quency/wavenumber of the DA mode is the same with the ones of the LWmode.

8. For very small wavenumbers it reaches time scales characteristic of ion dynam-ics and the low-frequency kinetic model does not suffice for a self-consistentsolution.


Discussion is provided regarding (i) the physical mechanism of the mode, (ii) itsrelation with the phenomenon of long range collective dust attraction [e.g., Tsy-tovich and Morfill, 2004; Castaldo et al., 2006; Ratynskaia et al., 2006; Tsytovichet al., 2006; de Angelis et al., 2010], (iii) its emergence in hydrodynamic models[Tsytovich and Watanabe, 2003], (iv) the possibility that it is driven unstable invery small wavenumbers [Else et al., 2009].

Finally, the necessary conditions, that experiments aiming to observe the LWmode should satisfy, are pointed out;

• possibility of observation of long wavelengths,

• relatively high operating pressures,

• elevated electron temperatures,

• operation with gases of low first ionization potential,

• sub-thermal drifts of ion/dust components.

7.4 Paper IV: Grain charging in an intermediatelycollisional plasma

The dust charge is the most fundamental parameter of complex plasma systems.For engineered complex plasmas, formed by injection of small spherical solid par-ticulates in laboratory discharges, different charging models exist depending on thevalue of plasma collisionality λD

ln,ithat is mostly determined by the gas pressure

[Khrapak and Morfill, 2009].Typical plasma discharges, fall either in the weakly collisional regime ln,i > λD

- where charging models have been extensively studied and verified both experi-mentally and by molecular dynamics simulations [Khrapak et al., 2005] - or in theintermediate collisional regime ln,i ∼ λD. In such a regime, neither a pair-particlenor a hydrodynamic approach is applicable to describe the ion transport to thegrain and therefore the use of interpolation formulas in inevitable. Interpolationformulas have been proposed based on molecular dynamics simulations, but theyhave never been tested experimentally.

In this paper we report the first experimental verification of an interpolationformula for the ion current in the intermediate collision regime. Dedicated experi-ments have been performed in the PK-4 dc discharge prototype in the Max-Planck-Institute for Extraterrestrial Physics in Garching, Germany. The experiments havebeen performed with Neon gas at elevated pressures 100 − 500 Pa. The dust grainflow has been illuminated by a laser sheet and the grain motion recorded by twofast video cameras, thus enabling measurements of the grain steady flow and thedust density. Plasma parameter measurements have been carried out by Langmuirprobes in absence of dust grains. The aforementioned experimental data can be

7.5. PAPER V: SPECTRA OF ION DENSITY AND POTENTIALFLUCTUATIONS IN WEAKLY IONIZED PLASMAS IN PRESENCE OFDUST GRAINS 101

Figure 7.3: Particle flow at p = 100 Pa. The picture has a field view of 22.2 ×16.6 mm2.

used for the determination of the dust charge through the force balance method,with the neutral drag, ion drag and electrostatic forces acting on the grain.

Comparison with the theoretical dust charge found through the flux balance

condition, with the ion flux given by IICi =

[(1

IW Ci

)γ

+(

1ISC

i

)γ]−1/γ

, has showna remarkable agreement within the experimental errors. The correct asymptoticbehavior of the interpolation formula in the weakly collisional regime has also beenconfirmed by comparison with previous experiments.

7.5 Paper V: Spectra of ion density and potentialfluctuations in weakly ionized plasmas in presence ofdust grains

Fluctuations are omnipresent at physical systems, even in thermodynamic equilib-rium, due to the discrete nature of matter. Initially, they were regarded as noiseand were undesirable in experiments. However, the fluctuation-dissipation theo-rem guarantees that the properties of any system (quantum or classical) can bedetermined from its intrinsic fluctuations [Kubo, 1966]. This has resulted in theimplementation of noise spectroscopic methods as diagnostics in a variety of phys-ical systems [Ichimaru, 1964], e.g for the determination of the kinetic parametersof a chemical reaction in an electrolytic solution [Feher and Weissman, 1973], forextraction of information about the quantum state of ultra-cold fermionic gases of


alkali atoms [Mihaila, 2006], for the determination of plasma parameters in quies-cent space environments [e.g., Meyer-Vernet, 1979; Meyer-Vernet and Perche, 1989;Moncuquet et al., 2006].

The presence of dust grains in plasma systems leads to enhancement of thespectral densities of ion density fluctuations and electrostatic field fluctuations, byorders of magnitude [Ratynskaia et al., 2007]. These important modifications stemfrom the large number of elementary charges residing on the dust surface and thedissipative nature of charging collisions between dust grains and plasma particles.They are mostly confined in the low frequency regime and are strongly dependent onthe dust density and the grain radius. Consequently, it has been recently proposedthat the modifications in the plasma spectra can be used as an in situ diagnostictool for sub-micron dust. Proof of principle of the diagnostic has already beenconfirmed and also verified by post-mortem analysis, yet for the interpretationof the experimental data the multi-component kinetic model has been employed[Ratynskaia et al., 2010].

The spectral densities have been derived for the "full" kinetic model only in thefully ionized case [de Angelis et al., 2006]. However, an important class of quiescentlaboratory discharges, the brush cathodes, operate in elevated pressure regimes[Persson, 1965]. In this paper, we derive the spectral densities of ion density andelectrostatic potential fluctuations in partially ionized complex plasmas in the lowfrequency regime. We study the effects of pressure and ionization numerically, forrealistic parameters and taking into account the effects of the finite probe size. Wealso compare with the results of the multi-component model to determine regimeswhere deviations are significant.

Theoretical advances include:

1. The derivation of the low frequency spectral densities of the electrostaticfield, electrostatic potential and ion density fluctuations for the "full" andmulti-component models of partially ionized complex plasmas. This enablesa comparison that shows the necessity of the "full" kinetic model.

2. The calculation of the low frequency spectral densities of the electrostaticfield, electrostatic potential and ion density fluctuations for weakly ionizedplasmas in absence of dust grains. This facilitates the estimation of the orderof magnitude enhancement of the spectral densities due to the presence ofdust.

3. The derivation of a criterion for the omission of plasma discreteness basedon the natural spectral densities for thermal distributions. Such a criterionproperly defines the low frequency regime and hence the range of validity ofthe kinetic model as ω/k < ΛαvT d, where Λα is a dimensionless quantity ofthe order of a few (that is a weak function of the dust and plasma parameters).

The numerical investigations have lead to the conclusions below:

7.6. PAPER VI: EFFECTS OF DUST PARTICLES IN PLASMA KINETICS;ION DYNAMICS TIME SCALES 103

• The results of both kinetic models reveal orders of magnitude enhancement(five to ten orders of magnitude) of the fluctuation level due to the presence ofdust, similar to previously reported spectral changes for fully ionized plasmas.

• Neutral gas density (pressure) can be responsible for significant modificationsof spectral density magnitudes, despite the opposite behavior of the equilib-rium dust charge number Zd. Hence the inclusion of the effect of neutrals isessential for a more realistic comparison with experiments.

• The "full" self-consistent model deviates from the multi-component modelsignificantly (above typical experimental errors) already for typical dust den-sities nd ≃ 105 cm−3 and dust radii a > 100 nm, values that are common forin situ produced dust.

• The effect of electron temperature variation in the spectral densities is nonmonotonic and can be attributed to the strength electron impact ionization.

7.6 Paper VI: Effects of dust particles in plasma kinetics;ion dynamics time scales

Application of the Klimontovich approach for complex plasma systems has so far ledto self-consistent kinetic models in the low frequency regime of dust dynamics, dueto treatment of electrons/ions as continuous Vlasov fluids [Tsytovich and de Angelis,1999]. This also implies some further restrictions. Namely, the dust densities shouldbe (i) large enough so that binary plasma collisions can be neglected when comparedto collisions with dust [Tsytovich, 1998], (ii) low enough so that the dust componentis its weakly coupled gaseous state. This limits the parameter regime of the abovemodels mostly in weakly coupled engineered complex plasma experiments [Fortovet al., 2005].

Aim of this paper is the development of a self-consistent kinetic model validin frequency regimes typical of ion dynamics. In such frequency ranges the dis-creteness of the ion and the electron component can no longer be neglected. Acentral assumption leading to great simplifications in the cumbersome algebra in-volved is that the induced fluctuating part of the dust distribution function canbe neglected when compared to the natural fluctuating part. In that case a fulldynamic description of dust can be avoided and dust charge fluctuations can befound independently by first order expansion of the charging equation.

Theoretical advances include:

1. The formulation of the basic assumptions of a new kinetic model of complexplasmas valid in frequency ranges characteristic of ion dynamics.

2. The derivation of the general forms of the integral responses, the effectivecharges and the permittivity of the system in such frequency ranges.


3. The derivation of the ion collision integrals, the ion kinetic equations and thedetailed analysis of their structure.

4. The derivation of the spectral densities of the ion density fluctuations andtheir physical interpretation.

The new kinetic model has been formulated in a generic fashion in order toaccount for both fluctuating and non-fluctuating plasma sources, hence it can beapplied for both space and engineered complex plasma systems. It is expected togive new interpretations to a variety of phenomena of both laboratory and astro-physical context

• ion stochastic acceleration [de Angelis et al., 2006],

• dust-ion acoustic waves [Shukla and Silin, 2002],

• spectra modification due to the presence of dust [Ratynskaia et al., 2010],

• scattering of electromagnetic radiation in dust clouds [de Angelis et al., 2002],

• self-consistent non-thermal distribution functions [Ricci et al., 2001],

• contribution of complex plasma populations to the distortion of the cosmicmicrowave background, the Sunyaev-Zeldovich effect [Sunyaev and Zeldovich,1970].

Chapter 8

Discussion and Outlook

8.1 Self-consistent treatment of electrostatic waves incomplex plasmas

Addition of drifting components in the treatment of dust acoustic wavesOne of the basic assumptions of the kinetic models is the absence of external elec-tric fields, ⟨E⟩ = 0. However, most laboratory discharge plasmas need an electricfield for their creation, which implies drifting dust and plasma components. ForDA waves the electrons can always be considered in Boltzmann equilibrium, whilethe ion drift can be sub-thermal, vi ≪ vT i, on the other hand the massive highlycharged grains are typically supra-thermal, vd > vT d. Therefore, the influence ofthe external electric field can be important.In fact, the relative drift between the ion and dust components can drive the DAwaves unstable [Rosenberg, 1993]. This is the typical excitation mechanism of theacoustic perturbations observed in the experiments [Merlino, 2009]. Even for rela-tively high pressures, ionization and ion drag force can be destabilizing factors thatare sufficient to overcome the strong collisional damping.Thus, the addition of a dc electric field in the self-consistent kinetic description isimportant, it will shed light in the competition between the stabilizing (collisionlessdamping, collisions with neutrals, dust charge fluctuations, plasma absorption) andthe destabilizing (ionization, relative drift) factors. We notice that this will notonly change the plasma / dust kinetic equations but also the expression for thepermittivity since the equations for the fluctuating parts will now contain the extraterm eγ⟨E⟩ · ∂δfγ

p,k,ω

∂p .

Treatment of the k → 0 limit of the long-wavelength modeThe long-wavelength mode, theoretically predicted in Paper III, is attributed tothe competition between electron impact ionization of neutrals and absorption ofions on the grains and is always damped in the low frequency regime. Plots of thedispersion relation ωr(k) of the LW mode, show that in the k → 0 limits the mode

105

106 CHAPTER 8. DISCUSSION AND OUTLOOK

eigenfrequencies are larger than the dust plasma frequency, which makes a low fre-quency kinetic model inadequate for its description in very small wavenumbers. Itis obvious that the mode is dependent on both ion and dust dynamics.However, the permittivity derived from the low frequency kinetic model is validfor all frequency regimes; The permittivity depends only on fluctuations that aredirectly proportional to the electric field fluctuations. None of these terms havebeen omitted, since all the components’ induced fluctuations are regarded, whereasthe omission of plasma discreetness will affect the effective charge but not the per-mittivity. Thus, we conclude that the same permittivity expression can be appliedfor the analysis of the LW mode in the whole frequency range, provided that thefrequency is no longer omitted in the ion integral responses.Such a treatment of the long-wavelength mode is very important. In case of anionization source and a ground state with no directed motion, analysis of the self-consistent hydrodynamic equations for partially ionized complex plasmas has re-vealed that the system is unstable for ion scale perturbations [Else et al., 2009;Khrapak and Morfill, 2010]. The instability length is very long corresponding tok → 0, which can have consequences on feasibility of the long-range attractionmechanism, a paradigm for the transition of the system to strongly coupled states[Tsytovich, 2006].

Self-consistent treatment of the dust-ion acoustic wavesDust-ion acoustic waves (DIA) are electrostatic waves similar to ion sound wavespropagating in the regime kvT i ≪ ω ≪ kvT e ; the ions provide the inertia, the elec-tron pressure provides the restoring force, while dust is essentially at rest [Shuklaand Silin, 1992]. In the fully ionized case, the collisionless hydrodynamic equationsfor k2λ2

De ≪ 1 yield the well-known dispersion relation ω = ωpiλDe k or alterna-tively vph =

√1+P

τ vT i. Therefore, even though dust does not participate in thewave dynamics, due to its effect in the ground state of the system through thequasi-neutrality condition, it leads to an increase of the phase velocity of the ionsound waves.DIA waves can be self-consistently treated either through the general permittivityexpression derived in Paper II, or through the permittivity of the kinetic model ofPaper VI. In that sense, DIA waves can lead to a first indirect verification of thevalidity of the new kinetic model.

8.2 Development of an in situ dust diagnostic based on thefluctuation spectra

Effect of chemical reactions in the fluctuation spectraIn situ dust formation takes place in a complicated environment involving chemicaltransformations [e.g., Bouchoule et al., 1991; Berndt et al., 2003; Benedikt, 2010],(i) breaking of molecular bonds of the precursor gases, (ii) formation of reactive

8.3. APPLICATIONS OF THE NEW KINETIC MODEL 107

radicals leading to larger molecules, (iii) avalanche condensation of molecules forthe formation of a solid core, (iv) surface attachment of radicals. It is expectedthat such processes will have an effect on the fluctuation spectra [Uddholm, 1983].But will the effect be strong enough to mask the order of magnitude enhancementof the ion density fluctuation spectra due to the effect of dust?The answer to this question is crucial to the development of the method as a di-agnostic. Apparently, a rigorous classical kinetic treatment is formidable, sincethe system involves strongly coupled subsystems and chemical bond formation ofquantum nature. However, estimates of the effect can be made by including ap-proximate collision operators in the Klimontovich equations, that in a relaxationtime approximation will just be sink terms with a frequency equal to the chem-ical reaction rate. In that sense they will contribute a chemical damping factor(ω − k · v + ı(νd,i + νn,i + νchem)) in the fluctuations, that bears a resemblance withthe effect of neutrals in the B.G.K. approximation.

Continuous and discrete size distributions for dustAfter their formation, dust grains continue to grow in size through coagulation(attachment collision of small dust particles forming a larger grain) and throughsurface growth (deposition of outer layers with dust acting as a spherical substrate).The radius of the dust grains is hence a dynamic variable, that can be approximatedeither as a discrete variable (when coagulation dominates) or as a continuous vari-able (when surface growth dominates).Each treatment will have an effect both in the results and the tolerance errors of theoutputs (dust density and grain size) of the diagnostic. So far, experimental resultshave been compared with theoretical models assuming discrete dust components,that were treated in a multi-component fashion.

Multi-component or "full" kinetic model?So far only multi-component kinetic models have been used for the theoretical inter-pretation of the results. Treatment of multiple dust species within the frameworkof the "full" kinetic model will definitely result in cumbersome results, due to thestrong coupling of the kinetic equations. For dust radii lower than 100 nm, theeffects of absorption and dust charge fluctuations in the spectra will be negligible,however, dust grains will diameters approaching the micron range are also possibleto exist.

8.3 Applications of the new kinetic model

Self consistent solutions of the ion kinetic equationThe numerical solution of the ion kinetic equation will provide self-consistent reg-ular ion distribution functions and describe their temporal evolution. These areexpected to be non-thermal and with a strong dependence on the source of plasmaparticles [Ricci et al., 2001].

108 CHAPTER 8. DISCUSSION AND OUTLOOK

The solution of the integro-differential equation for Φip is a formidable numerical

task, not only due to the products of the distribution functions of the collidingparticles present in the collision integral, but also due to the dependence of thepermittivity on Φi

p through the integral responses χik,ω , qi

k,ω , Sk,ω , Sk,ω .

Stochastic ion accelerationThe collision integrals can be used for the study of stochastic acceleration of ionsdue to their interaction with dust grains [de Angelis et al., 2005]. For dust particlesit has been shown that charge fluctuations in dust-dust interactions produce aninstability which can lead to stochastic heating of the dust grains, under astrophys-ical [Ivlev et al., 2010], laboratory [Marmolino et al., 2009] and fusion conditions[Marmolino et al., 2008]. The linear stage of this instability can be well describedby the dust kinetic equation resulting from fluctuation theory. It is of great interestwhether a similar effect could be predicted for ions, in that case the second momentof the ion kinetic equation would be cast into a form dϵ

dt = νϵϵ where ϵ is the meanlocal energy of ions and νϵ the energy growth rate.The new kinetic model, that is applicable for both astrophysical and laboratory con-ditions, can be employed to investigate the conditions for νϵ > 0, whether the dustcharge should be positive or negative for the instability threshold to be reached,or even how dissipation by neutrals can quench this instability. Already, by simpleinspection of the ion kinetic equation, one can see that the term describing dissi-pation due to absorption on dust can be reduced due to collective effects, and thatthe fluctuating ionization source also contributes to the positivity of νϵ.

Extension to frequencies characteristic of electron dynamicsIn the new kinetic model, the adiabatic assumption has been employed for theelectrons. This confines the frequency range of validity in ωpd < ω ≪ kvT e. Anextension to frequencies typical of electron dynamics can easily be implemented byusing a full dynamic description of electrons similar to the ion description.This will enable the self-consistent description of the scattering of electromagneticradiation through complex plasma configurations and will provide the modificationsof the relevant cross-sections as functions of the dust parameters. The latter couldhave major astrophysical implications, i.e the effects of space dust populations inthe distortion of the cosmic microwave background (Sunyaev - Zeldovich effect)[Sunyaev and Zeldovich, 1970].

Appendices

109

Appendix A

Generalized Approach in theComputation of the IntegralResponses

The approach in the computation of the integral responses is based on the fact thatfor isotropic distribution functions, spherical coordinates and with the appropriatechoice of the rotation of the coordinate system; the integrations over the azimuthaland elevation angles can be computed analytically, leaving only the speed integral tobe evaluated numerically (preferably in a dimensionless form). The methodology isvalid for all frequencies and is performed for Maxwellian distributions. Nevertheless,it can be followed for any isotropic distribution.

A.1 Calculation of the ion responses

The calculation of the ion responses for thermal distributions is performed followingthe methodology below;

• Since we are interested in the temporal attenuation of the waves, we useω = ωr + ıωi (initial value problem). In case we were interested in the spatialattenuation, we would have k = kr + ıki (boundary value problem).

• We decompose in real and imaginary parts. Note that only in the low fre-quency regime ωr ≪ kvT i the ion responses will be either purely real or purelyimaginary, due to the omission of ωr.

• We choose a coordinate system such that k//z, hence k · v = kvz.

• For the computation of the integrals we choose spherical coordinates, d3v =sin θv2dϕdθdv and kvz = kv cos θ.

• The integration over the azimuthal angle ϕ is trivial, d3v = 2π sin θv2dθ dv.

111

112APPENDIX A. GENERALIZED APPROACH IN THE COMPUTATION OF

THE INTEGRAL RESPONSES

• The integration over the elevation angle θ is always analytical and can beperformed using a number of auxiliary integrals, defined below. The resultswill be complicated functions of the speed either of logarithmic or inversetangent nature.

• The integration over the speed v is performed via the transformation y =miv2

2Ti. It is convenient define a number of auxiliary functions in the dimen-

sionless y-space in order to simplify the cumbersome results. These are thelogarithmic and inverse tangent kernels, the total imaginary part of denomina-tor of the responses, the ion capture frequencies, the ion capture cross-sectionsand their charge derivative.

Auxiliary IntegralsThe angular integrals can be classified in one of the following forms, with A = ωr ,B = kv and C = νtot,

I1 =∫ π

0

sin θ

(A − B cos θ)2 + C2 dθ =1

BC

[arctan

(A + B

C

)− arctan

(A − B

C

)]I2 =

∫ π

0

cos θ sin θ

(A − B cos θ)2 + C2 dθ = −1

2B2 ln[

(A + B)2 + C2

(A − B)2 + C2

]+

A

B2C

[arctan

(A + B

C

)− arctan

(A − B

C

)]I3 =

∫ π

0

cos2 θ sin θ

(A − B cos θ)2 + C2 dθ =2

B2 −A

B3 ln[

(A + B)2 + C2

(A − B)2 + C2

]+

A2 − C2

B3C

[arctan

(A + B

C

)− arctan

(A − B

C

)]I4 =

∫ π

0

B sin θ cos θ(A − B cos θ)(A − B cos θ)2 + C2 dθ = −2 +

C

B

[arctan

(A + B

C

)− arctan

(A − B

C

)]+

A

2Bln[

(A + B)2 + C2

(A − B)2 + C2

]I5 =

∫ π

0

(A − B cos θ) sin θ

(A − B cos θ)2 + C2 dθ =1

2Bln[

(A + B)2 + C2

(A − B)2 + C2

].

The integrals can be easily computed with the transformation cos θ = x. Otheruseful integrals are the following

I6 =∫ ∞

0(A + By)e−ydy = A + B ,

I7 =∫ ∞

0y2e−y2

dy =√

π

4,

both easily computed by employing integration by parts.

A.1. CALCULATION OF THE ION RESPONSES 113

Ion capture cross-sections in the weakly collisional regimeFor the collision enhanced collection model a generalization of our discussion aboutthe charging currents in terms of cross-sections yields; that there is a collisional ionpopulation occurring with probability R0

ln,ithat has the geometrical cross-section

πR20, there is an independent collisionless ion population with probability

(1 − R0

ln,i

)that has the O.M.L cross-section πa2

(1 − 2qe

amiv2

),

σi(q, v) = πR20

R0

ln,i+(

1 − R0

ln,i

)πa2

(1 − 2qe

amiv2

). (A.1)

In order to compute the radius of the perturbed plasma region R0 self-consistently,the electrostatic potential around a test grain should be known. It can be foundthrough the Poisson equation, which in the general case (taking into account ab-sorption of particles on the grain, trapped ions, potential barriers, collisions withneutrals and other non-linearities) will be an integro-differential equation to besolved numerically. Such complexity can be avoided by assuming that the inter-action potential is of the Yukawa type, i.e ϕ(r) = q

r exp(− r

λ

), with the effective

screening length λ regarded as an unknown parameter. Curve fitting of numeri-cal solutions to the Yukawa form, in various scenarios, has shown that λ is slightlylarger than the ion Debye length λDi, hence λ ≃ λDi is still a viable approximation.

From the definition of R0 we have

|U(R0)| = Ti ⇒ − qe

R0exp

(−R0

λ

)= Ti , (A.2)

since in absence of emission from the grain, the dust charge is always negative.The above equation is a transcendental equation for R0 that should be solvednumerically. It is obvious that R0 is charge dependent (R0(q)). In equilibrium, forq = −Zde,

Zde2

R0exp

(−R0

λ

)= Ti ⇒ Zde2

Tiexp

(−R0

λ

)= R0

Zde2

aTi

a

λexp

(−R0

λ

)= R0

λ⇒ Zde2

aTe

Te

Ti

a

λexp

(−R0

λ

)= R0

λ

z

τ

a

λexp

(−R0

λ

)= R0

λ⇒ βT exp

(−R0

λ

)= R0

λ, (A.3)

where βT = zτ

aλ is the scattering parameter at the ion thermal velocity. Using nor-

malization over the effective screening length x = R0λ , the transcendental equation

for the radius R0 becomes βT exp (x) = x. In the case that βT ≪ 1, we have that

x ≪ 1 ⇒ exp (x) ≃ 1 ⇒ x ≃ βT ⇒ R0

λ= z

τ

a

λ⇒ R0 = za

τ



and R0 ≪ λ.The dependence of R0 on the charge will introduce an extra factor in the charg-

ing frequency. We compute the first derivative of R0(q) by direct differentiation ofthe transcendental equation,

∂

∂q

[− qe

R0(q)exp

(−R0(q)

λ

)]= ∂Ti

∂q⇒ ∂

∂q

[q

R0(q)exp

(−R0(q)

λ

)]= 0

1R0

exp(

−R0(q)λ

)− q

R20

∂R0

∂qexp

(−R0(q)

λ

)− q

λR0

∂R0

∂qexp

(−R0(q)

λ

)= 0

1 − q

R0

∂R0

∂q− q

λ

∂R0

∂q= 0 ⇒ q

∂R0

∂q

(1

R0+ 1

λ

)= 1 ⇒ ∂R0

∂q= R0

q

(1

1 + R0/λ

).

Finally, the derivative at the equilibrium charge will be

∂R0

∂q|q=qeq = − R0

Zde

(1

1 + R0/λ

). (A.4)

The new flux balance condition will be given by the equality of electron andion currents, we also use the approximation

(1 − R0

ln,i

)≃ 1 valid in the weakly

collisional regime,√

8πeneuT ea2 exp(

qe

aTe

)=

√8πeniuT i

R30

ln,i+

√8πeniuT ia

2(

1 − qe

aTi

)neuT e exp

(qe

aTe

)= niuT i

[1 − qe

aTi+ R3

0a2ln,i

]. (A.5)

For the computation of the charging frequency one should not forget to takeinto account the dependence of R0 on the equilibrium charge,

νch = − ∂

∂q

∑α

⟨Iα(q)⟩ = −∂Ie(q)∂q

− ∂Ii(q)∂q

=√

8πeneuT ea2 e

aTeexp

(qe

aTe

)+

√8πeniuT ia

2(

e

aTi− 3R2

0a2ln,i

∂R0

∂q

)=

√8πeniuT ia

2 e

aTe

[1 − qe

aTi+ R3

0a2ln,i

]+

√8πeniuT ia

2(

e

aTi− 3R2

0a2ln,i

∂R0

∂q

)=

√8πeniuT ia

2

e

aTe

[1 − qe

aTi+ R3

0a2ln,i

]+(

e

aTi− 3R2

0a2ln,i

∂R0

∂q

)=

√8πe2niuT ia

Ti

Ti

Te

[1 − qe

aTi+ R3

0a2ln,i

]+(

1 − aTi

e

3R20

a2ln,i

∂R0

∂q

)=

√8πe2niuT ia

Ti

τ

[1 + z

τ+ R3

0a2ln,i

]+(

1 − aTi

e

3R20

a2ln,i

∂R0

∂q

)=

√8πe2niuT ia

Ti

1 + τ + z + R3

0τ

a2ln,i− aTi

e

3R20

a2ln,i

∂R0

∂q

,


where all the quantities are evaluated in equilibrium charge and we substituted forthe exponential from the equilibrium condition. Moreover, we use

√8πe2niuT ia

Ti= 4πnie

2

Ti

uT ia√2π

= uT ia√2πλ2

Di

= a√2π

ωpi

λDi,

−aTi

e

3R20

a2ln,i

∂R0

∂q= aTi

e

3R20

a2ln,i

R0

Zde

(1

1 + R0/λ

)= aTi

ZDe23R3

0a2ln,i

(1

1 + R0/λ

)⇒

−aTi

e

3R20

a2ln,i

∂R0

∂q= aTe

ZDe2Ti

Te

3R30

a2ln,i

(1

1 + R0/λ

)= τ

z

3R30

a2ln,i

(1

1 + R0/λ

),

we substitute and acquire

νch = a√2π

ωpi

λDi

1 + τ + z + R3

0τ

a2ln,i+ τ

z

3R30

a2ln,i

(1

1 + R0/λ

)= a√

2π

ωpi

λDi

1 + τ + z + R2

0a2

R0

ln,i

[τ + 3τ

z

(1

1 + R0/λ

)]. (A.6)

Comparing with the charging frequency in absence of ion-neutral collisions (ln,i →∞), νch = a√

2π

ωpi

λDi1 + τ + z, we notice that the charging frequency increases.

There are two additional terms: one due to the altered equilibrium condition andthe other due to the dependence of the radius R0 on the charge.

Finally, an exact computation can be made in the case βT ≪ 1 ⇒ zaτ ≪ λ, then

R0 = zaτ and R0 ≪ λ ⇒ 1 + R0/λ ≃ 1,

νch = a√2π

ωpi

λDi

1 + τ + z + z3a3

τ3ln,ia2

[τ + 3τ

z

(1

1 + R0/λ

)]= a√

2π

ωpi

λDi

1 + τ + z + z3

τ2a

ln,i

(1 + 3

z

). (A.7)

Auxiliary functions for the ion capture cross-sections

Using the narrowness of the dust distribution around the equilibrium charge weacquire νd,i =

∫vσi(q, v)Φd

p′(q) dqd3p′

(2π)3 ≃ vσi(qeq, v)∫

Φdp′(q) dqd3p′

(2π)3 ≃ ndvσi(qeq, v).For the capture cross-sections of ions on dust, we can either use the O.M.L. modelor the collision enhanced collection model (C.E.C).



We now express both cross-sections in the y-space,

σi(qeq, v) = πR20

R0

ln,i+(

1 − R0

ln,i

)πa2

(1 + 2Zde2

amiv2

)= πa2 R2

0a2

R0

ln,i+(

1 − R0

ln,i

)πa2

(1 + Zde2

aTe

Te

Ti

1y

)= πa2 R2

0a2

R0

ln,i+(

1 − R0

ln,i

)πa2 z

τy

(1 + τ

zy)

= πa2 z

τy

τ

z

R20

a2R0

ln,iy +

(1 − R0

ln,i

)(1 + τ

zy)

while for the O.M.L. cross-sections we use the limit ln,i → ∞ and acquire σi(qeq, y) =πa2 z

τy

(1 + τ

z y). We can combine both expressions in a form convenient for the

computation of the integral responses, σi(qeq, y) = πa2 zτy σ0(y), where σ0(y) is the

dimensionless y-transformed cross-section function defined by

σ0(y) =

1 + τ

z y, O.M.Lτz

R20

a2R0ln,i

y + (1 − R0ln,i

)(1 + τz y), C.E.C.

(A.8)

In some responses the charge derivative of the cross-section, evaluated at the equi-librium charge, is present. We compute for both charging models and transform iny-space,

∂σi(q, v)∂q

=∂

∂q

[πR3

0ln,i

]+

∂

∂q

[(1 −

R0

ln,i

)πa

2(

1 −2qe

amiv2

)]= 3π

R20

ln,i

∂R0

∂q|q=qeq −

∂R0

∂q|q=qeq

1ln,i

πa2(

1 −2qeqe

amiv2

)−(

1 −R0

ln,i

)πa

2 2e

amiu2

= −3πR2

0ln,i

R0

Zde(

11 + R0/λ

) − (1 −R0

ln,i

)πa2 2e

amiu2 +R0

Zde(

11 + R0/λ

)1

ln,i

πa2(1 +

2Zde2

amiv2 )

= −3πR2

0ln,i

R0

Zde(

11 + R0/λ

) − (1 −R0

ln,i

)πa2 e

aTiy+

R0

Zde(

11 + R0/λ

)1

ln,i

πa2(1 +

Zde2

aTiy)

= −3πa2

Zde

R20

a2R0

ln,i

(1

1 + R0/λ) − (1 −

R0

ln,i

)πa2

Zde

z

τy+

πa2

Zde(

11 + R0/λ

)R0

ln,i

z

τy(1 +

τ

zy)

= −πa2

Zde

z

τy

3

R20

a2R0

ln,i

τ

z

(1

1 + R0/λ

)y +(

1 −R0

ln,i

)−(

11 + R0/λ

)R0

ln,i

(1 +

τ

zy

),

while for O.M.L. cross-sections we use the limit ln,i → ∞ and acquire σ′i(qeq, y) =

− πa2

Zdez

τy . We can combine both expressions in a form convenient for the com-putation of the integral responses, σ′

i(qeq, y) = − πa2

Zdez

τy σ′0(y), where σ′

0(y) is thedimensionless y-transformed cross-section derivative function defined by

σ′0(y) =

1, O.M.L

3 R20

a2R0ln,i

τz

(1

1+R0/λ

)y +

(1 − R0

ln,i

)−(

11+R0/λ

)R0ln,i

(1 + τ

zy)

, C.E.C.

(A.9)


Auxiliary FunctionsWe now define a number of auxiliary functions that appear in the ion integralresponses and give their expressions in the -y- space.

For the inverse tangent kernel we have

Ψ±(y) = arctan[

ωr ± kv

ωi + νn,i + νd,i(v)

]= arctan

[ωr ± kv

ωi + nnσn,ivT i + ndvσi(qeq, v)

]

= arctan

ωr ± k√

2Ti

mi

√y

ωi + nnσn,ivT i + nd

√2Ti

mi

√yσi(qeq, y)

= arctan

[ωr ± k

√2vT i

√y

ωi + nnσn,ivT i + nd

√2vT i

√yσi(qeq, y)

]

= arctan

[ωr ± k

√2vT i

√y

ωi + nnσn,ivT i + ndπa2 zyτ

√2vT i

√yσ0(y)

]

= arctan

ωr√

y√2vT i

± ky

ωi√

y√2vT i

+ nnσn,i√

y√2 + ndπa2 z

τ σ0(y)

.

The above expression appears in the responses as Ψtot = Ψ+ − Ψ−. In the lowfrequency limit, ωr ≪ kvT i, which yields

ΨLFtot = arctan

[ky

ωi√

y√2vT i

+nnσn,i

√y

√2

+ ndπa2 zτ σ0(y)

]− arctan

[−ky

ωi√

y√2vT i

+nnσn,i

√y

√2

+ ndπa2 zτ σ0(y)

]

= 2 arctan

[ky

ωi√

y√2vT i

+nnσn,i

√y

√2

+ ndπa2 zτ σ0(y)

].

In the fully ionized case, in absence of collisions with neutrals we additionally haveσn,i → 0 , ωi → 0 and σ0(y) = 1 + τ

z y, which yields

ΨLF,ftot = 2 arctan

[ky

ndπa2 zτ σ0(y)

]= 2 arctan

[ky

ndπa2 zτ

(1 + τ

z y)]

= 2 arctan[

κiy

1 + τz y

].

In the latter, we have defined the dimensionless wavenumber κi = kndπa2 z

τ, where

lid =(ndπa2 z

τ

)−1 is approximately equal to the characteristic ion absorption length


THE INTEGRAL RESPONSESvT i

νdi. Apparently, k ≫ (lid)−1 or κi ≫ 1 implies spatial scales where absorption of

ions on dust is not important.For the logarithmic kernel we have

L(y) = ln[

(ωr + kv)2 + (ωi + νn,i + νd,i(v))2

(ωr − kv)2 + (ωi + νn,i + νd,i(v))2

]= ln

[(ωr + kv)2 + (ωi + νn,i + νd,i(v))2

(ωr − kv)2 + (ωi + νn,i + νd,i(v))2

]= ln

[(ωr + kv)2 + (ωi + nnσn,ivT i + ndvσi(qeq, v))2

(ωr − kv)2 + (ωi + nnσn,ivT i + ndvσi(qeq, v))2

]= ln

[(ωr + k

√2vT i

√y)2 + (ωi + nnσn,ivT i + nd

√2vT i

√yσi(qeq, y))2

(ωr − k√

2vT i√

y)2 + (ωi + nnσn,ivT i + nd

√2vT i

√yσi(qeq, y))2

]

= ln

[(ωr + k

√2vT i

√y)2 + (ωi + nnσn,ivT i + ndπa2√

2vT i√

y zyτ σ0(y))2

(ωr − k√

2vT i√

y)2 + (ωi + nnσn,ivT i + ndπa2√

2vT i√

y zyτ σ0(y))2

]

= ln

(

ωr√

y√2vT i

+ ky)2

+(

ωi√

y√2vT i

+ nnσn,i√

y√2 + ndπa2 z

τ σ0(y))2

(ωr

√y√

2vT i− ky

)2+(

ωi√

y√2vT i

+ nnσn,i√

y√2 + ndπa2 z

τ σ0(y))2

.

which clearly vanishes in the low frequency limit.For the total imaginary part of the expression ω − k · v + ıνi(v) we have

C(y) = ωi + νn,i + νd,i(v)= ωi + nnσn,ivT i + ndvσi(qeq, v)

= ωi + nnσn,ivT i + nd

√2vT i

√yσi(qeq, y)

= ωi + nnσn,ivT i + ndπa2√2vT i

z

yτ

√yσ0(y)

=√

2vT i√y

(ωi

√y

√2vT i

+nnσn,i

√y

√2

+ ndπa2 z

τσ0(y)

).

For convergence to the expressions of the low frequency responses it is convenientto define C(y) =

√2kvT iCk(y) or equivalently

Ck(y) =

(ωi

√y√

2vT i+ nnσn,i

√y√

2 + ndπa2 zτ σ0(y)

)k√

y.

In the low frequency limit the expression will remain the same being independent


of the frequency, while in the fully ionized case

Ck(y) =ndπa2 z

τ σ0(y)k√

y

=ndπa2 z

τ (1 + τz y)

k√

y

=1 + τ

z y

κi√

y.

The integral response χik,ω

The response is the generalization of the ion susceptibility in collisional plasmas, itis defined by

χik,ω = 4π e2

k2

∫1

ω − k · v + ıνi(v)

(k ·

∂Φip

∂p

)d3p

(2π)3 . (A.10)

We use the Maxwellian derivative property, ω = ωr + ıωi, set C(v) = ωi + νn,i +νd,i(v) and decompose in real and imaginary parts.

χik,ω = 4π e2

k2

∫1

ω − k · v + ıνd,i(v) + ıνn,i

(k ·

∂Φip

∂p

)d3p

(2π)3

= 4π e2

k2

∫1

ωr − k · v + ı(ωi + νd,i(v) + νn,i)

(k ·

∂Φip

∂p

)d3p

(2π)3

= −4π e2

Tik2

∫k · v

ωr − k · v + ı(ωi + νd,i(v) + νn,i)Φi(v)d3v

= −4π e2

Tik2

∫k · v

(ωr − k · v) + ı C(v)Φi(v)d3v

= −4π e2

Tik2

∫(k · v) [(ωr − k · v) − ı C(v)]

(ωr − k · v)2 + C2(v)Φi(v)d3v

= −4π e2

Tik2

∫(k · v)(ωr − k · v)

(ωr − k · v)2 + C2(v)Φi(v)d3v

+ ı4π e2

Tik2

∫(k · v) C(v)

(ωr − k · v)2 + C2(v)Φi(v)d3v .



For the real part we use spherical coordinates and evaluate the integral through I4and I7

ℜχik,ω = −

4π e2

Tik2

∫(k · v)(ωr − k · v)

(ωr − k · v)2 + C2(v)Φi(v)d3v

= −4π e2

Tik2

∫ ∞

0

(∫ π

0

kv sin θ cos θ(ωr − kv cos θ)(ωr − kv cos θ)2 + C2(v)

dθ

)2π v2Φi(v)dv

=(4π)2 e2

Tik2

∫ ∞

0v2Φi(v)dv −

8π2e2

Tik3

∫ ∞

0vC(v) (Ψ+(v) − Ψ−(v)) Φi(v)dv

−4π e2

Tik2ωr

k

∫ ∞

0vL(v)Φi(v)dv

=(4π)2nie

2

Tik2

(mi

2π Ti

)3/2∫ ∞

0v2 exp

(−

miv2

2Ti

)dv −

8π2e2

Tik3

∫ ∞

0vC(v)[Ψ+(v)

− Ψ−(v)]Φi(v)dv −4π2 e2

Tik2ωr

k

∫ ∞

0vL(v)Φi(v)dv

=(4π)2nie

2√

πTik2

∫ ∞

0

√ye−ydy −

8π2e2

Tik3

∫ ∞

0vC(v) (Ψ+(v) − Ψ−(v)) Φi(v)dv

−4π2 e2

Tik2ωr

k

∫ ∞

0vL(v)Φi(v)dv

=1

k2λ2Di

−8π2nie

2

Tik3

(mi

2π Ti

)3/2∫ ∞

0vC(v) (Ψ+(v) − Ψ−(v)) exp

(−

miv2

2Ti

)dv

−4π2 e2

Tik2ωr

k

∫ ∞

0vL(v)Φi(v)dv

=1

k2λ2Di

−4π nie

2

Tik3

√mi

2π Ti

∫ ∞

0C(y) (Ψ+(y) − Ψ−(y)) e−ydy

−4π2 e2

Tik2ωr

k

∫ ∞

0vL(v)Φi(v)dv

=1

k2λ2Di

−1

λ2Dik

21

√2πkvT i

∫ ∞

0C(y) (Ψ+(y) − Ψ−(y)) e−ydy

−4π2 e2

Tik2ωr

k

∫ ∞

0vL(v)Φi(v)dv

=1

k2λ2Di

−1

λ2Dik

21

√π

∫ ∞

0e−yCk(y) (Ψ+(y) − Ψ−(y)) dy

−4π2 e2

Tik2ωr

k

∫ ∞

0vL(v)Φi(v)dv

=1

k2λ2Di

−1

k2λ2Di

1√

π

∫ ∞

0e−yCk(y) (Ψ+(y) − Ψ−(y)) dy

−1

k2λ2Di

πωr

k

(mi

2π Ti

)3/2∫ ∞

0vL(v) exp

(−

miv2

2Ti

)dv

=1

k2λ2Di

1 −

1√

π

∫ ∞

0e−yCk(y) (Ψ+(y) − Ψ−(y)) dy −

ωr

2√

2π kvT i

∫ ∞

0e−yL(y)dy

.


For the imaginary part we evaluate the angular integral through I2

ℑχk,ω = +4π e2

Tik2

∫(k · v) C(v)

(ωr − k · v)2 + C2(v)Φi(v)d3v

=4π e2

Tik2

∫ ∞

0

(∫ π

0

cos θ sin θ

(ωr − kv cos θ)2 + C2(v)

)2π kv3C(v)Φi(v)dv

=4π e2

Tik2

∫ ∞

0

[−

12k2v2 L(v) +

ωr

k2v2C(v)(Ψ+(v) − Ψ−(v))

]2π kv3C(v)Φi(v)dv

= −4π e2

Tik2π

k

∫ ∞

0vL(v)C(v)Φi(v)dv +

4π e2

Tik22πωr

k

∫ ∞

0v (Ψ+(v) − Ψ−(v)) Φi(v)dv

= −1

k2λ2Di

π

k

(mi

2π Ti

)3/2∫ ∞

0vL(v)C(v) exp

(−

miv2

2Ti

)dv

+4π e2

Tik22πωr

k

∫ ∞

0v (Ψ+(v) − Ψ−(v)) Φi(v)dv

= −1

k2λ2Di

12k

√mi

2π Ti

∫ ∞

0L(y)C(y)e−ydy

+4π e2

Tik22πωr

k

∫ ∞

0v (Ψ+(v) − Ψ−(v)) Φi(v)dv

= −1

k2λ2Di

12√

π

∫ ∞

0e−yCk(y)L(y)dy +

4π e2

Tik22πωr

k

∫ ∞

0v (Ψ+(v) − Ψ−(v)) Φi(v)dv

= −1

k2λ2Di

12√

π

∫ ∞

0e−yCk(y)L(y)dy

+1

k2λ2Di

2πωr

k

(mi

2π Ti

)3/2∫ ∞

0v (Ψ+(v) − Ψ−(v)) exp

(−

miv2

2Ti

)dv

= −1

k2λ2Di

12√

π

∫ ∞

0e−yCk(y)L(y)dy +

1k2λ2

Di

ωr√2πkvT i

∫ ∞

0e−y (Ψ+(y) − Ψ−(y)) dy

=1

k2λ2Di

ωr√

2πkvT i

∫ ∞

0e−y (Ψ+(y) − Ψ−(y)) dy −

12√

π

∫ ∞

0e−yCk(y)L(y)dy

.

Overall, we have

χik,ω = 1

k2λ2Di

1 − 1√

π

∫ ∞

0e−yCk(y) (Ψ+(y) − Ψ−(y)) dy − ωr

2√

2π kvT i

∫ ∞

0e−yL(y)dy

+ ı

k2λ2Di

ωr√

2πkvT i

∫ ∞

0e−y (Ψ+(y) − Ψ−(y)) dy − 1

2√

π

∫ ∞

0e−yCk(y)L(y)dy

.

(A.11)



In the low frequency limit the imaginary part of the response will vanish, while forthe real part we get

χik,ω =

1k2λ2

Di

1 −

1√

π

∫ ∞

0e−yCk(y)ΨLF

tot dy

=1

k2λ2Di

1 −2

√π

∫ ∞

0e−y

(ωi

√y√

2vT i+ nnσn,i

√y√

2+ ndπa2 z

τσ0(y)

)k

√y

× arctan

[ky

ωi√

y√2vT i

+ nnσn,i√

y√2

+ ndπa2 zτ

σ0(y)

]dy , (A.12)

where the second part of the response expresses deviations from Debye screeningdue to absorption on dust and collisions with neutrals.In the fully ionized case, the relation will be further simplified

χik,ω = 1

k2λ2Di

1 − 2√

π

∫ ∞

0e−y 1 + τ

z y

κi√

yarctan

[κiy

1 + τz y

]dy

. (A.13)

The integral response Gk,ω

The response is defined by

Gk,ω =∫

1ı(ω − k · v + ıνi(v))

ΦMp

d3p

(2π)3 . (A.14)

We use ω = ωr + ıωi, set C(v) = ωi + νn,i + νd,i(v) and decompose in real andimaginary parts,

Gk,ω =∫

1ı(ω − k · v + ıνi(v))

ΦMp

d3p

(2π)3

= −∫

1(ωi + νd,i(v) + νn,i) − ı(ωr − k · v)

ΦMp

d3p

(2π)3

= −∫

(ωi + νd,i(v) + νn,i) + ı(ωr − k · v)(ωi + νd,i(v) + νn,i)2 + (ωr − k · v)2 ΦM

p

d3p

(2π)3

= −∫

C(v) + ı(ωr − k · v)(ωr − k · v)2 + C2(v)

ΦMp

d3p

(2π)3

= −∫

C(v)(ωr − k · v)2 + C2(v)

ΦMp

d3p

(2π)3 − ı

∫ωr − k · v

(ωr − k · v)2 + C2(v)ΦM

p

d3p

(2π)3 .


For the real part we use spherical coordinates and evaluate the angular integralthrough I1

ℜGk,ω = −∫

C(v)(ωr − k · v)2 + C2(v)

ΦMp

d3p

(2π)3

= −∫ ∞

0

(∫ π

0

sin θ

(ωr − kv cos θ)2 + C2(v)dθ

)2π v2C(v)ΦM (v)dv

= −∫ ∞

0

2π v2C(v)kvC(v)

(Ψ+(v) − Ψ−(v)) ΦM (v)dv

= −2π

k

∫ ∞

0v (Ψ+(v) − Ψ−(v)) ΦM (v)dv

= −2π

k

(mi

2π Ti

)3/2 ∫ ∞

0v (Ψ+(v) − Ψ−(v)) exp

(−miv

2

2Ti

)dv

= −2π

k

(mi

2π Ti

)3/2Ti

mi

∫ ∞

0(Ψ+(y) − Ψ−(y)) e−y dy

= − 1k

√mi

2π Ti

∫ ∞

0(Ψ+(y) − Ψ−(y)) e−y dy

= − 1√2πkvT i

∫ ∞

0(Ψ+(y) − Ψ−(y)) e−y dy


ℑGk,ω = −∫

ωr − k · v

(ωr − k · v)2 + C2(v)ΦM

p

d3p

(2π)3

= −∫ ∞

0

(∫ π

0

(ωr − kv cos θ) sin θ


)2π v2ΦM (v)dv

= −∫ ∞

0

2π v2

2kvL(v)ΦM (v)dv

= −π

k

∫ ∞

0v L(v)ΦM (v)dv

= −π

k

(mi

2π Ti

)3/2 ∫ ∞

0v L(v) exp

(−miv

2

2Ti

)dv

= −π

k

(mi

2π Ti

)3/2Ti

mi

∫ ∞

0L(y)e−ydy

= − 12k

√mi

2π Ti

∫ ∞

0L(y)e−ydy

= − 12√

2πkvT i

∫ ∞

0L(y)e−ydy .



Overall, we get

Gk,ω = −

1√2πkvT i

∫ ∞

0e−y (Ψ+(y) − Ψ−(y)) dy

−ı

1

2√

2πkvT i

∫ ∞

0e−yL(y)dy

.

(A.15)In the low frequency regime the imaginary part will vanish, while for the real partwe have

Gk,ω = − 1√2πkvT i

∫ ∞

0e−yΨLF

tot dy

= − 2√2πkvT i

∫ ∞

0e−y arctan

kyωi

√y√

2vT i+ nnσn,i

√y√


dy

= − 1kvT i

√2π

∫ ∞

0e−y arctan

kyωi

√y√

2vT i+ nnσn,i

√y√


dy .

(A.16)

For the fully ionized case, the response does not exist, being dependent on the dis-tribution function of the neutrals ΦM

p . The response is directly related to collisionswith neutrals via the B.G.K approximation and electron impact ionization, beingalways multiplied by νn,i or νe.

The integral response qik,ω(q)


qik,ω(q) =

∫evσi(q, v)

ı(ω − k · v + ıνi(v))Φi

p

d3p

(2π)3 . (A.17)

It is evaluated for q = qeq. Initially, we use ω = ωr+ıωi, set C(v) = ωi+νn,i+νd,i(v)and decompose in real and imaginary parts,

qik,ω(q) = −

∫ıevσi(q, v)

(ωr − k · v) + ı(ωi + νn,i + νd,i(v)) Φip

d3p

(2π)3

= −∫

ıevσi(q, v)(ωr − k · v) + ıC(v) Φi

pd3p

(2π)3

= −∫

ıevσi(q, v) [(ωr − k · v) − ıC(v)](ωr − k · v)2 + C2(v) Φi

pd3p

(2π)3

= −∫

evσi(q, v)C(v)(ωr − k · v)2 + C2(v) Φi

pd3p

(2π)3 − ı

∫evσi(q, v)(ωr − k · v)(ωr − k · v)2 + C2(v) Φi

pd3p

(2π)3 .

For the real part we use spherical coordinates and evaluate the angular integralthrough I1. We also use the dimensionless wavenumber κi = k

ndπa2z/τ and the


relation − eni

nd= qeq

1+PP

ℜqik,ω = −

∫evσi(q, v)C(v)

(ωr − k · v)2 + C2(v)Φi

p

d3p

(2π)3

= −∫ ∞

0

(∫ π

0

sin θ


)2πev3σi(q, v)C(v)Φi(v)dv

= −∫ ∞

0

2πev3σi(q, v)C(v)kvC(v)

(Ψ+(v) − Ψ−(v)) Φi(v)dv

= −2πe

k

∫ ∞

0v2σi(q, v) (Ψ+(v) − Ψ−(v)) Φi(v)dv

= −2πeni

k

(mi

2πTi

)3/2 ∫ ∞

0vσi(q, v) (Ψ+(v) − Ψ−(v)) exp

(−miv

2

2Ti

)vdv

= −2πeni

k

√2

2π√

2π

∫ ∞

0

√yσi(qeq, y) (Ψ+(y) − Ψ−(y)) e−ydy

= − eni

ndk

1√π

ndπa2 z

τ

∫ ∞

0

σ0(y)√

y(Ψ+(y) − Ψ−(y)) e−ydy

= qeq√π

1 + P

P

∫ ∞

0

σ0(y)e−y

κi√

y[Ψ+(y) − Ψ−(y)] dy .


ℑqik,ω = −

∫evσi(q, v)(ωr − k · v)(ωr − k · v)2 + C2(v)

Φip

d3p

(2π)3

= −∫ ∞

0

(∫ π

0



)2πev3σi(q, v)Φi(v)dv

= −∫ ∞

0

2πev3σi(q, v)2kv

L(v)Φi(v)dv

= −πe

k

∫ ∞

0vσi(q, v)L(v)Φi(v)vdv

= −πeni

k

(mi

2πTi

)3/2 ∫ ∞

0vσi(q, v)L(v) exp

(−miv

2

2Ti

)vdv

= −eni

k

12√

π

∫ ∞

0

√yσi(y)L(y)e−ydy

= −eni

nd

12√

π

∫ ∞

0

ndπa2 zτ

k

σ0(y)e−y

√y

L(y)dy

= qeq

2√

π

1 + P

P

∫ ∞

0

σ0(y)e−y

κi√

yL(y)dy .



Overall, we get

qik,ω =

qeq√

π

1 + P

P

∫ ∞

0

σ0(y)e−y

κi√

y[Ψ+(y) − Ψ−(y)] dy

+ ı

qeq

2√

π

1 + P

P

∫ ∞

0

σ0(y)e−y

κi√

yL(y)dy

. (A.18)

In the low frequency regime the imaginary part will vanish, while for the real partwe have

qik,ω = qeq√

π

1 + P

P

∫ ∞

0

σ0(y)e−y

κi√

yΨLF

tot dy

= 2qeq√π

1 + P

P

∫ ∞

0

σ0(y)e−y

κi√

yarctan

kyωi

√y√

2vT i+ nnσn,i

√y√


dy .

(A.19)

In the fully ionized case the response will become

qik,ω = 2qeq√

π

1 + P

P

∫ ∞

0e−y 1 + τ

z y

κi√

yarctan

[κiy

1 + τz y

]dy . (A.20)

The integral response λik,ω(q)


λik,ω(q) =

∫evσi(q, v)

ı(ω − k · v + ıνi(v))ΦM

p

d3p

(2π)3 . (A.21)

It is evaluated for q = qeq. From a simple inspection we notice that it is exactlythe same with qi

k,ω(q), but instead of the ion distribution we have the normalized

distribution of the neutrals. Since, ΦMp = Φi

p

niwe will also have λi

k,ω = qik,ω

ni, i.e

λik,ω =

qeq

ni√

π

1 + P

P

∫ ∞

0

σ0(y)e−y

κi√

y[Ψ+(y) − Ψ−(y)] dy

+ ı

qeq

2ni√

π

1 + P

P

∫ ∞

0

σ0(y)e−y

κi√

yL(y)dy

. (A.22)


λik,ω = qeq

ni√

π

1 + P

P

∫ ∞

0

σ0(y)e−y

κi√

yΨLF

tot dy

= 2qeq

ni√

π

1 + P

P

∫ ∞

0

σ0(y)e−y

κi√

yarctan

kyωi

√y√

2vT i+ nnσn,i

√y√


dy .

(A.23)


For the fully ionized case, the responses does not exist, being dependent on the dis-tribution function of the neutrals ΦM

p . The response is directly related to collisionswith neutrals via the B.G.K approximation and electron impact ionization, beingalways multiplied by νn,i or νe.

The integral response βik,ω(q)


βik,ω(q) =

∫evσ′

i(q, v)ı(ω − k · v + ıνi(v))

Φip

d3p

(2π)3 . (A.24)

It is evaluated for q = qeq. We decompose in real and imaginary parts.

βik,ω(q) =

∫evσ′

i(qeq, v)ı(ω − k · v + ıνi(v))

Φip

d3p

(2π)3

=∫

evσ′i(qeq, v)

ı((ωr − k · v) + ı(ωi + νd,i(v) + νn,i))Φi

p

d3p

(2π)3

=∫

−ıevσ′i(qeq, v)

(ωr − k · v) + ıC(v)Φi

p

d3p

(2π)3

= −∫

evσ′i(qeq, v)(C(v) + ı(ωr − k · v))

(ωr − k · v)2 + C2(v)Φi

p

d3p

(2π)3 .

For the real part we evaluate the angular integral through I1

ℜβik,ω(q) = −

∫evσ′

i(qeq, v)C(v)(ωr − k · v)2 + C2(v) Φi

pd3p

(2π)3

= −e

∫ ∞

0

∫ π

0

sin θ

(ωr − kv cos θ)2 + C2(v) dθ

2πv3σ′

i(qeq, v)C(v)Φi(v)dv

= −e

∫ ∞

0

1kvC(v) (Ψ+(v) − Ψ−(v)) 2πv3σ′

i(qeq, v)C(v)Φi(v)dv

= −2πe

k

∫ ∞

0(Ψ+(v) − Ψ−(v)) v2σ′

i(qeq, v)Φi(v)dv

= −2πnie

k

(mi

2πTi

)3/2∫ ∞

0(Ψ+(v) − Ψ−(v)) vσ′

i(qeq, v) exp(

−miv2

2Ti

)vdv

= −2πnie

k

12π

√π

∫ ∞

0(Ψ+(y) − Ψ−(y)) √

yσ′i(qeq, y)e−ydy

= ni

Zdndk

ndπa2 zτ√

π

∫ ∞

0(Ψ+(y) − Ψ−(y)) σ′

0(y)e−y

√y

dy

= ni

ne

ne

Zdnd

1√π

∫ ∞

0

σ′0(y)e−y

κi√

y[Ψ+(y) − Ψ−(y)] dy

= 1 + P

P

1√π

∫ ∞

0

σ′0(y)e−y

κi√

y[Ψ+(y) − Ψ−(y)] dy .




ℑβik,ω(q) = −

∫evσ′

i(qeq, v)(ωr − k · v)(ωr − k · v)2 + C2(v)

Φip

d3p

(2π)3

= −2πe

∫ ∞

0

∫ π

0

sin θ(ωr − kv cos θ)(ωr − kv cos θ)2 + C2(v)

v3σ′

i(qeq, v)Φi(v)dv

= −2πe

∫ ∞

0

12kv

L(v) v3σ′i(qeq, v)Φi(v)dv

= −πe

k

∫ ∞

0vL(v)σ′

i(qeq, v)Φi(v)vdv

= −πnie

k

(mi

2πTi

)3/2 ∫ ∞

0vL(v)σ′

i(qeq, v) exp(

−miv2

2Ti

)vdv

= −πnie

k

12π

√π

∫ ∞

0

√yσ′

i(qeq, y)e−yL(y)dy

= nie

eZdk

πa2 zτ

2√

π

∫ ∞

0

σ′0(y)e−y

√y

L(y)dy

= ni

ndZd

12√

π

∫ ∞

0

σ′0(y)e−y

κi√

yL(y)dy

= 1 + P

P

12√

π

∫ ∞

0

σ′0(y)e−y

κi√

yL(y)dy .

Overall, we have

βik,ω(q) = 1 + P

P

1√π

(∫ ∞

0

σ′0(y)e−y

κi√

y[Ψ+(y) − Ψ−(y)] dy

)+ ı

(12

∫ ∞

0

σ′0(y)e−y

κi√

yL(y)dy

) . (A.25)


βik,ω(q) = 1 + P

P

1√π

∫ ∞

0

σ′0(y)e−y

κi√

yΨLF

tot dy

= 1 + P

P

2√π

∫ ∞

0

σ′0(y)e−y

κi√

yarctan

kyωi

√y√

2vT i+ nnσn,i

√y√


dy .

(A.26)

In the fully ionized case the response can be further simplified

βik,ω(q) = 1 + P

P

2√π

∫ ∞

0

e−y

κi√

yarctan

[κiy

1 + τz y

]dy . (A.27)


The integral response SIk,ω(q)


SIk,ω(q) = e2

∫vσi(q, v)

ı(ω − k · v + ıνi(v))k

k·

∂Φip

∂p

d3p

(2π)3 . (A.28)

It is evaluated for q = qeq. We use the Maxwellian derivative property and decom-pose in real and imaginary parts.

SIk,ω(q) = − e2

kTi

∫vσi(qeq, v)(k · v)

ı(ω − k · v + ı(νd,i(v) + νn,i))Φi(v)d3v

= − e2

kTi

∫vσi(qeq, v)(k · v)

ı(ωr − k · v) − C(v)Φi(v)d3v

= + e2

kTi

∫vσi(qeq, v)(k · v)(C(v) + ı(ωr − k · v))

(ωr − k · v)2 + C2(v)Φi(v)d3v .

For the imaginary part we evaluate the angular integral through I4 and I6

ℑSIk,ω(q) =

e2

kTi

∫vσi(qeq , v)(k · v)(ωr − k · v)

(ωr − k · v)2 + C2(v)Φi(v)d3v

=e2

kTi

∫ ∞

0

∫ π

0

kv cos θ sin θ(ωr − kv cos θ)(ωr − kv cos θ)2 + C2(v)

dθ

2πv3σi(qeq , v)Φi(v)dv

=e2

kTi

∫ ∞

0

−2 +

C(v)kv

(Ψ+(v) − Ψ−(v)) +ωr

2kvL(v)


= −e2

kTi

∫ ∞


+e2

kTi

∫ ∞

0

C(v)kv

(Ψ+(v) − Ψ−(v)) +ωr

2kvL(v)


= −4πe2ni

kTi

(mi

2πTi

)3/2∫ ∞

0v2σi(qeq , v) exp

(−

miv2

2Ti

)vdv

+e2

kTi

∫ ∞

0

C(v)kv

(Ψ+(v) − Ψ−(v)) +ωr

2kvL(v)


= −1

kλ2Di

√Ti

mi

22π

√2π

∫ ∞

0yσi(qeq , y)e−ydy

+e2

kTi

∫ ∞

0

C(v)kv

(Ψ+(v) − Ψ−(v)) +ωr

2kvL(v)




ℑSIk,ω(q) = −

1kλ2

Di

√Ti

mi

22π

√2π

πa2 z

τ

∫ ∞

0σ0(y)e−ydy

+e2

kTi

∫ ∞

0

C(v)kv

(Ψ+(v) − Ψ−(v)) +ωr

2kvL(v)


= −1

kλ2Di

a2√

2πvT i

z

τσ0(1) +

2πe2

k2Ti

∫ ∞

0C(v) (Ψ+(v) − Ψ−(v)) v2σi(qeq , v)Φi(v)dv

+πωre2

k2Ti

∫ ∞

0L(v)v2σi(qeq , v)Φi(v)dv

= −1

kλ2Di

a2√

2πvT i

z

τσ0(1) +

2πnie2

k2Ti

(mi

2πTi

)3/2∫ ∞

0C(v)(Ψ+(v)

− Ψ−(v))vσi(qeq , v) exp(

−miv

2

2Ti

)vdv +

πωre2

k2Ti

∫ ∞


= −1

kλ2Di

a2√

2πvT i

z

τσ0(1) +

12k2λ2

Di

12π3/2

∫ ∞

0C(y)(Ψ+(y)

− Ψ−(y))√yσi(qeq , y)e−ydy +πωre2

k2Ti

∫ ∞


= −1

kλ2Di

a2√

2πvT i

z

τσ0(1) +

12k2λ2

Di

12π3/2

πa2z

τ

∫ ∞

0

C(y)√

y(Ψ+(y)

− Ψ−(y))σ0(y)e−ydy +πωre2

k2Ti

∫ ∞


= −1

kλ2Di

a2√

2πvT i

z

τσ0(1) +

12kλ2

Di

a2√

2πvT i

z

τ

∫ ∞

0

σ0(y)e−y

√y

Ck(y)(Ψ+(y)

− Ψ−(y))dy +ωr

4k2λ2Di

(mi

2πTi

)3/2∫ ∞

0L(y)vσi(qeq , v) exp

(−

miv2

2Ti

)vdv

= −1

kλ2Di

a2√

2πvT i

z

τσ0(1) +

12kλ2

Di

a2√

2πvT i

z

τ

∫ ∞

0

σ0(y)e−y

√y

Ck(y)(Ψ+(y)


4k2λ2Di

12π

√π

∫ ∞

0L(y)√yσi(qeq , y)e−ydy

= −1

kλ2Di

a2√

2πvT i

z

τσ0(1) +

12kλ2

Di

a2√

2πvT i

z

τ

∫ ∞

0

σ0(y)e−y

√y

Ck(y)(Ψ+(y)


4k2λ2Di

a2

2√

π

z

τ

∫ ∞

0

σ0(y)e−y

√y

L(y)dy

= −1

kλ2Di

a2√

2πvT i

z

τσ0(1) +

12kλ2

Di

a2√

2πvT i

z

τ

∫ ∞

0

σ0(y)e−y

√y

Ck(y)(Ψ+(y)

− Ψ−(y))dy +1

4kλ2Di

a2√

2πvT i

z

τ

ωr√2kvT i

∫ ∞

0

σ0(y)e−y

√y

L(y)dy

= −1

kλ2Di

a2√

2πvT i

z

τσ0(1) −

12

∫ ∞

0

σ0(y)e−y

√y

Ck(y) (Ψ+(y) − Ψ−(y)) dy

−14

ωr√2kvT i

∫ ∞

0

σ0(y)e−y

√y

L(y)dy .



ℜSIk,ω(q) = +

e2

kTi

∫vσi(qeq , v)(k · v)C(v)(ωr − k · v)2 + C2(v)

Φi(v)d3v

=e2

kTi

∫ ∞

0

∫ π

0

cos θ sin θ


2πkv4σi(qeq , v)C(v)Φi(v)dv

=e2

k2Ti

∫ ∞

0

−

12

L(v) +ωr

C(v)(Ψ+(v) − Ψ−(v))

2πv2σi(qeq , v)C(v)Φi(v)dv

= −πe2

k2Ti

∫ ∞

0L(v)v2σi(qeq , v)C(v)Φi(v)dv

+2πωre2

k2Ti

∫ ∞

0(Ψ+(v) − Ψ−(v)) v2σi(qeq , v)Φi(v)dv

= −4πnie

2

4k2Ti

(mi

2πTi

)3/2∫ ∞

0L(v)vσi(qeq , v)C(v) exp

(−

miv2

2Ti

)vdv

+2πωre2

k2Ti

∫ ∞


= −1

4k2λ2Di

a2

2√

π

z

τ

∫ ∞

0L(y)C(y)

σ0(y)e−y

√y

dy

+2πωre2

k2Ti

∫ ∞


= −1

4kλ2Di

a2√

2πvT i

z

τ

∫ ∞

0

σ0(y)e−y

√y

Ck(y)L(y)dy

+4πωrnie

2

2k2Ti

(mi

2πTi

)3/2∫ ∞

0(Ψ+(v) − Ψ−(v)) vσi(qeq , v) exp

(−

miv2

2Ti

)vdv

= −1

4kλ2Di

a2√

2πvT i

z

τ

∫ ∞

0

σ0(y)e−y

√y

Ck(y)L(y)dy

+ωr

2k2λ2Di

12π

√π

∫ ∞

0

√yσi(qeq , y) (Ψ+(y) − Ψ−(y)) e−ydy

= −1

4kλ2Di

a2√

2πvT i

z

τ

∫ ∞

0

σ0(y)e−y

√y

Ck(y)L(y)dy

+ωr

2k2λ2Di

a2

2√

π

z

τ

∫ ∞

0

σ0(y)e−y

√y

(Ψ+(y) − Ψ−(y)) dy

= −1

4kλ2Di

a2√

2πvT i

z

τ

∫ ∞

0

σ0(y)e−y

√y

Ck(y)L(y)dy

+1

2kλ2Di

a2√

2πvT i

z

τ

ωr√2kvT i

∫ ∞

0

σ0(y)e−y

√y

(Ψ+(y) − Ψ−(y)) dy

=1

2kλ2Di

a2√

2πvT i

z

τ−

12

∫ ∞

0

σ0(y)e−y

√y

Ck(y)L(y)dy

+ωr√

2kvT i

∫ ∞

0

σ0(y)e−y

√y

(Ψ+(y) − Ψ−(y)) dy .



Overall, we get

SIk,ω(q) =

12kλ2

Di

a2√

2πvT i

z

τ−

12

∫ ∞

0

σ0(y)e−y

√y

Ck(y)L(y)dy

+ωr√

2kvT i

∫ ∞

0

σ0(y)e−y

√y

(Ψ+(y) − Ψ−(y)) dy − ı1

kλ2Di

a2√

2πvT i

z

τσ0(1)

−12

∫ ∞

0

σ0(y)e−y

√y

Ck(y) (Ψ+(y) − Ψ−(y)) dy −14

ωr√2kvT i

∫ ∞

0

σ0(y)e−y

√y

L(y)dy .

(A.29)In the low frequency regime the real part will vanish, while for the imaginary partwe have

SIk,ω(q) = −ı

1

kλ2Di

a2√

2πvT i

z

τ

σ0(1) −

12

∫ ∞

0

σ0(y)e−y

√y

Ck(y)ΨLFtot dy

= −ı1

kλ2Di

a2√

2πvT i

z

τσ0(1) −

∫ ∞

0

σ0(y)e−y

√y

(ωi

√y√

2vT i+ nnσn,i

√y√

2+ ndπa2 z

τσ0(y)

)k

√y

× arctan

[ky

ωi√

y√2vT i

+ nnσn,i√

y√2

+ ndπa2 zτ

σ0(y)

]dy

= −ı1

kλ2Di

a2√

2πvT i

z

τσ0(1) −

∫ ∞

0σ0(y)e−y

(ωi

√y√

2vT i+ nnσn,i

√y√

2+ ndπa2 z

τσ0(y)

)ky

× arctan

[ky

ωi√

y√2vT i

+ nnσn,i√

y√2

+ ndπa2 zτ

σ0(y)

]dy . (A.30)

In the fully ionized case the response will be further simplified

SIk,ω(q) = −ı

1

kλ2Di

a2√

2πvT i

z

τ

1 + τ

z−∫ ∞

0e−y

(1 + z

τy)2

κiyarctan

[κiy

1 + τz

y

]dy

.

(A.31)

The integral response SIk,ω(q, q′)


SIk,ω(q, q′) = e

∫v2σi(q, v)σi(q′, v)

ı(ω − k · v + ıνi(v))Φi

p

d3p

(2π)3 . (A.32)

It is evaluated for q = q′ = qeq. We decompose in real and imaginary parts.

SIk,ω(q, q′) = −e

∫ıv2σ2

i (qeq, v)(ωr − k · v) + ı(ωi + νn,i + νd,i(v)) Φi

pd3p

(2π)3

= −e

∫ıv2σ2

i (qeq, v) ((ωr − k · v) − ıC(v))(ωr − k · v)2 + C2(v) Φi

pd3p

(2π)3

= −e

∫v2σ2

i (qeq, v)C(v)Φip

(ωr − k · v)2 + C2(v)d3p

(2π)3 − ıe

∫v2σ2

i (qeq, v)(ωr − k · v)(ωr − k · v)2 + C2(v) Φi

pd3p

(2π)3 .



ℜSIk,ω(q, q′) = −e

∫v2σ2

i (qeq, v)C(v)(ωr − k · v)2 + C2(v) Φi

pd3p

(2π)3

= −e

∫ ∞

0

(∫ π

0

sin θ


)2πv4σ2

i (qeq, v)C(v)Φi(v)dv

= −e

∫ ∞

0

2πv4σ2i (qeq, v)C(v)kvC(v) Φi(v) [Ψ+(v) − Ψ−(v)] dv

= −e

∫ ∞

0

2πv3σ2i (qeq, v)k

Φi(v) [Ψ+(v) − Ψ−(v)] dv

= −2πnie

k

(mi

2πTi

) 32∫ ∞

0v2σ2

i (qeq, v) exp(

−miv2

2Ti

)[Ψ+(v) − Ψ−(v)] vdv

= −2πnie

k

22π

√2π

√Timi

∫ ∞

0yσ2

i (qeq, y)e−y [Ψ+(y) − Ψ−(y)] dy

= −eni

nd

√2π

vT i(πa2 z

τ)∫ ∞

0

ndπa2 zτ

k

σ20(y)e−y

y[Ψ+(y) − Ψ−(y)] dy

= −eni

nd

√2π

vT i(πa2 z

τ)∫ ∞

0

σ20(y)e−y

κiy[Ψ+(y) − Ψ−(y)] dy

= qeq1 + P

P

2πa2vT izτ√

2π

∫ ∞

0

e−yσ20(y)

κiy[Ψ+(y) − Ψ−(y)] dy .


ℑSIk,ω(q, q′) = −e

∫v2σ2

i (qeq, v)(ωr − k · v)(ωr − k · v)2 + C2(v) Φi

pd3p

(2π)3

= −e

∫ ∞

0

(∫ π

0



)2πv4σ2

i (qeq, v)Φi(v)dv

= −e

∫ ∞

0

2πv4σ2i (qeq, v)

2kvL(v)Φi(v)dv

= −πnie

k

(mi

2πTi

)3/2∫ ∞

0v2σ2

i (qeq, v)L(v) exp(

−miv2

2Ti

)vdv

= −πnie

k

22π

√2π

vT i

∫ ∞

0yσ2

i (qeq, y)L(y)e−ydy

= −nie

nd

πa2 zτ√

2πvT i

∫ ∞

0

e−yσ20(y)

κiyL(y)dy

= qeq1 + P

P

πa2vT izτ√

2π

∫ ∞

0

e−yσ20(y)

κiyL(y)dy .



Overall, we get

SIk,ω(q, q′) =

qeq

1 + P

P

2πa2vT izτ√

2π

∫ ∞

0

e−yσ20(y)


+ ı

qeq

1 + P

P

πa2vT izτ√

2π

∫ ∞

0

e−yσ20(y)

κiyL(y)dy

. (A.33)


SIk,ω(q, q′) =

qeq

1 + P

P

2πa2vT izτ√

2π

∫ ∞

0

e−yσ20(y)

κiyΨLF

tot dy

= qeq

1 + P

P

4πa2vT izτ√

2π

∫ ∞

0

e−yσ20(y)

κiy

× arctan

kyωi

√y√

2vT i+ nnσn,i

√y√


dy . (A.34)

In the fully ionized case the above expression will be further simplified to

SIk,ω(q, q′) = qeq

1 + P

P

4πa2vT izτ√

2π

∫ ∞

0e−y

(1 + τ

z y)2

κiyarctan

[κiy

1 + τz y

]dy . (A.35)

The integral response S′Ik,ω(q, q′)


S′Ik,ω(q, q′) = e

∫v2σi(q, v)σ′

i(q′, v)ı(ω − k · v + ıνi(v))

Φip

d3p

(2π)3 . (A.36)

It is evaluated for q = q′ = qeq. We decompose in real and imaginary parts.

S′Ik,ω(q, q′) = −e

∫ıv2σi(qeq, v)σ′

i(qeq, v)(ωr − k · v) + ı(ωi + νn,i + νd,i(v))

Φip

d3p

(2π)3

= −e

∫ıv2σi(qeq, v)σ′

i(qeq, v) ((ωr − k · v) − ıC(v))(ωr − k · v)2 + C2(v)

Φip

d3p

(2π)3

= −e

∫v2σi(qeq, v)σ′

i(qeq, v)C(v)(ωr − k · v)2 + C2(v)

Φip

d3p

(2π)3

− ıe

∫v2σi(qeq, v)σ′

i(qeq, v)(ωr − k · v)(ωr − k · v)2 + C2(v)

Φip

d3p

(2π)3 .


For the real part we evaluate the angular integral through I1 and also use ni

ndZd2πa2vT i

zτ =

2πe2ni

mivT i

and

.

ℜS′Ik,ω(q, q′) = −e

∫v2σi(qeq , v)σ′

i(qeq , v)C(v)(ωr − k · v)2 + C2(v)

Φip

d3p

(2π)3

= −e

∫ ∞

0

(∫ π

0

sin θ


)2πv4σi(qeq , v)σ′

i(qeq , v)C(v)Φi(v)dv

= −e

∫ ∞

0

2πv3σi(qeq , v)σ′i(qeq , v)

kΦi(v) [Ψ+(v) − Ψ−(v)] dv

= −2πnie

k

(mi

2πTi

)3/2∫ ∞

0v2σi(qeq , v)σ′

i(qeq , v) exp(

−miv

2

2Ti

)[Ψ+(v) − Ψ−(v)] vdv

= −2πnie

k

22π

√2π

√Timi

∫ ∞

0yσi(qeq , y)σ′

i(qeq , y)e−y [Ψ+(y) − Ψ−(y)] dy

=ni

Zdnd

√2π

(πa2vT iz

τ)∫ ∞

0

ndπa2 zτ

k

σ0(y)σ′0(y)e−y

y[Ψ+(y) − Ψ−(y)] dy

=ni

Zdnd

2πa2vT izτ√

2π

∫ ∞

0

e−yσ0(y)σ′0(y)


=2πe2ni

mivT i

a

nd

1√

2π

∫ ∞

0


κiy[Ψ+(y) − Ψ−(y)] dy .


ℑS′Ik,ω(q, q′) = −e

∫v2σi(qeq , v)σ′

i(qeq , v)(ωr − k · v)(ωr − k · v)2 + C2(v)

Φip

d3p

(2π)3

= −e

∫ ∞

0

(∫ π

0



)2πv4σi(qeq , v)σ′

i(qeq , v)Φi(v)dv

= −e

∫ ∞

0

2πv4σi(qeq , v)σ′i(qeq , v)

2kvL(v)Φi(v)dv

= −πnie

k

(mi

2πTi

)3/2∫ ∞

0v2σi(qeq , v)σ′

i(qeq , v)L(v) exp(

−miv

2

2Ti

)vdv

= −πnie

k

22π

√2π

vT i

∫ ∞

0yσi(qeq , y)σ′

i(qeq , y)L(y)e−ydy

=nie

nd

1√

2π

πa2vT izτ

Zde

∫ ∞

0


κiyL(y)dy

=ni

Zdnd

1√

2π(πa2vT i

z

τ)∫ ∞

0


κiyL(y)dy

=πe2ni

mivT i

a

nd

1√

2π

∫ ∞

0


κiyL(y)dy .



Overall, we get

S′Ik,ω(q, q′) =

2πe2ni

mivT i

a

nd

1√2π

∫ ∞

0



+

ı

πe2ni

mivT i

a

nd

1√2π

∫ ∞

0


κiyL(y)dy

.

(A.37)


S′Ik,ω(q, q′) = 2πe2ni

mivT i

a

nd

1√2π

∫ ∞

0


κiyΨLF

tot dy

= 4πe2ni

mivT i

a

nd

1√2π

∫ ∞

0


κiy

× arctan

kyωi

√y√

2vT i+ nnσn,i

√y√


dy . (A.38)

Whereas in the fully ionized case we get

S′Ik,ω(q, q′) = 4πe2ni

mivT i

a

nd

1√2π

∫ ∞

0e−y 1 + τ

z y

κiyarctan

[κiy

1 + τz y

]dy . (A.39)

A.2 Calculation of the dust responses

The methodology employed in the calculation of the dust responses is essentiallythe same as in the ion responses, apart from the transformation used in the compu-tation of the speed integral. Here we use the transformation y =

√mdv2

2Td, in order to

collapse to the plasma dispersion function for the response χd,eqk,ω in absence of neu-

trals. Other useful quantities are the dimensionless phase velocity ζ = ωr√2kvT d

, thedimensionless total effective damping of the dust susceptibility νt1 = ωi+νn,d√

2kvT d, the

dimensionless total effective damping of the charging response νt2 = ωi+νch+νn,d√2kvT d

.

The integral response χd,eqk,ω

The response is the generalization of the dust susceptibility in the presence ofneutrals and is defined by

χd,eqk,ω =

4πq2eq

k2

∫1

ω − k · v + ıνn,dk ·

∂Φdp

∂ pd3p

(2π)3 . (A.40)

A.2. CALCULATION OF THE DUST RESPONSES 137

We decompose in real and imaginary parts and we define νtot1 = ωi + νn,d, asthe total effective damping term present in the response. Using the property k ·∂Φd

p′

∂p′d3p′

(2π)3 = − k·v′

TdΦdd3v of the Maxwellian distribution we end up with

χd,eqk,ω = −

4π q2eq

Tdk2

∫(k · v)

ωr − k · v + ıνtot1Φd(v)d3v

= −4π q2

eq

Tdk2

∫(k · v)(ωr − k · v − ıνtot1)

(ωr − k · v)2 + ν2tot1

Φd(v)d3v

= −4π q2

eq

Tdk2

∫(k · v)(ωr − k · v)

(ωr − k · v)2 + ν2tot1

Φd(v)d3v

+ ı4π q2

eq

k2

∫(k · v)νtot1

(ωr − k · v)2 + ν2tot1

Φd(v)d3v .

For the real part we use the integral I4

ℜχd,eqk,ω

= −4π q2

eq

Tdk2

∫(k · v)(ωr − k · v)

(ωr − k · v)2 + ν2tot1

Φd(v)d3v

= −4π q2

eq

Tdk2

∫ ∞

0

(∫ π

0

kv sin θ cos θ(ωr − kv cos θ)(ωr − kv cos θ)2 + ν2

tot1dθ

)2π v2Φd(v)dv

= −4π q2

eq

Tdk2

∫ ∞

0[−2 +

νtot1

kv

(arctan

(ωr + kv

νtot1

)− arctan

(ωr − kv

νtot1

))+

ωr

2kvln(

(ωr + kv)2 + ν2tot1

(ωr − kv)2 + ν2tot1

)] 2π v2Φd(v)dv

=4π q2

eq

Tdk2

∫ ∞

04π v2Φd(v)dv

−4π q2

eq

Tdk2

∫ ∞

0

νtot1

kv

(arctan

(ωr + kv

νtot1

)− arctan

(ωr − kv

νtot1

))2π v2Φd(v)dv

−4π q2

eq

Tdk2

∫ ∞

0

ωr

2kvln(



)2π v2Φd(v)dv

=4π ndq2

eq

Tdk2

(md

2π Td

)3/2∫ ∞

04π v2 exp

(−

mdv2

2Td

)dv

−4π q2

eq

Tdk2

∫ ∞

0

νtot1

kv

(arctan

(ωr + kv

νtot1

)− arctan

(ωr − kv

νtot1

))2π v2Φd(v)dv

−4π q2

eq

Tdk2

∫ ∞

0

ωr

2kvln(



)2π v2Φd(v)dv



ℜχd,eqk,ω

=1

k2λ2Dd

2√

π

∫ ∞

0

√xe−xdx

−4π q2

eq

Tdk2

∫ ∞

0

νtot1

kv

(arctan

(ωr + kv

νtot1

)− arctan

(ωr − kv

νtot1

))2π v2Φd(v)dv

−4π q2

eq

Tdk2

∫ ∞

0

ωr

2kvln(



)2π v2Φd(v)dv

=1

k2λ2Dd

−4π ndq2

eq

Tdk22πνtot1

k

(md

2π Td

) 32∫ ∞

0

[arctan

(ωr + kv

νtot1

)− arctan

(ωr − kv

νtot1

)]× v exp

(−

mdv2

2Td

)dv −

4π q2eq

Tdk2

∫ ∞

0

ωr

2kvln(



)2π v2Φd(v)dv

=1

k2λ2Dd

−1

k2λ2Dd

2√

π

νtot1

k

√md

2Td

×∫ ∞

0

arctan

ωr + k

√2Tdmd

y

νtot1

− arctan

ωr − k

√2Tdmd

y

νtot1

ye−y2dy

−4π q2

eq

Tdk2

∫ ∞

0

ωr

2kvln(



)2π v2Φd(v)dv

=1

k2λ2Dd

−1

k2λ2Dd

2√

πνt1

∫ ∞

0ye−y2

(arctan

(ζ + y

νt1

)− arctan

(ζ − y

νt1

))dy

−4π q2

eq

Tdk2

∫ ∞

0

ωr

2kvln(



)2π v2Φd(v)dv

=1

k2λ2Dd

−1

k2λ2Dd

2√

πνt1

∫ ∞

0ye−y2

(arctan

(ζ + y

νt1

)− arctan

(ζ − y

νt1

))dy

−4π ndq2

eq

Tdk2πωr

k

(md

2π Td

)3/2∫ ∞

0ln(



)v exp

(−

mdv2

2Td

)dv

=1

k2λ2Dd

−1

k2λ2Dd

2√

πνt1

∫ ∞

0ye−y2

(arctan

(ζ + y

νt1

)− arctan

(ζ − y

νt1

))dy

−1

k2λ2Dd

ωr√πk

√md

2Td

∫ ∞

0ln

(ωr + k

√2Tdmd

y)2 + ν2tot1

(ωr − k

√2Tdmd

y)2 + ν2tot1

ye−y2dy

=1

k2λ2Dd

−1

k2λ2Dd

2√

πνt1

∫ ∞

0ye−y2

(arctan

(ζ + y

νt1

)− arctan

(ζ − y

νt1

))dy

−1

k2λ2Dd

1√

πζ

∫ ∞

0ye−y2

ln(

(ζ + y)2 + ν2t1

(ζ − y)2 + ν2t1

)dy

=1

k2λ2Dd

1 −2

√π

∫ ∞

0νt1 ye−y2

(arctan

(ζ + y

νt1

)− arctan

(ζ − y

νt1

))dy

−1

√π

∫ ∞

0ζ ye−y2

ln(

(ζ + y)2 + ν2t1

(ζ − y)2 + ν2t1

)dy .


For the imaginary part we use the integral I2

ℑχd,eqk,ω

=4π q2

eq

Tdk2

∫(k · v)νtot1

(ωr − k · v)2 + ν2tot1

Φd(v)d3v

=4π q2

eq

Tdk2

∫ ∞

0

(∫ π

0

kv cos θ sin θ

(ωr − kv cos θ)2 + ν2tot1

dθ

)2πνtot1v2Φd(v)dv

=4π q2

eq

Tdk22πωr

k

∫ ∞

0

(arctan

(ωr + kv

νtot1

)− arctan

(ωr − kv

νtot1

))vΦd(v)dv

−4π q2

eq

Tdk2πνtot1

k

∫ ∞

0ln(



)vΦd(v)dv

=4π ndq2

eq

Tdk22πωr

k

(md

2π Td

)3/2∫ ∞

0

(arctan

(ωr + kv

νtot1

)− arctan

(ωr − kv

νtot1

))× v exp

(−

mdv2

2Td

)dv −

4π q2eq

Tdk2πνtot1

k

∫ ∞

0ln(



)vΦd(v)dv

=1

k2λ2Dd

2ωr

k

√md

2π Td

×∫ ∞

0

arctan

ωr + k

√2Tdmd

y

νtot1

− arctan

ωr − k

√2Tdmd

y

νtot1

ye−y2dy

−4π q2

eq

Tdk2πνtot1

k

∫ ∞

0ln(



)vΦd(v)dv

=1

k2λ2Dd

2√

π

∫ ∞

0ζ ye−y2

(arctan

(ζ + y

νt1

)− arctan

(ζ − y

νt1

))dy

−4π q2

eq

Tdk2πνtot1

k

∫ ∞

0ln(



)vΦd(v)dv

=1

k2λ2Dd

2√

π

∫ ∞

0ζ ye−y2

(arctan

(ζ + y

νt1

)− arctan

(ζ − y

νt1

))dy

−4π ndq2

eq

Tdk2πνtot1

k

(md

2π Td

)3/2∫ ∞

0ln(



)v exp

(−

mdv2

2Td

)dv

=1

k2λ2Dd

2√

π

∫ ∞

0ζ ye−y2

(arctan

(ζ + y

νt1

)− arctan

(ζ − y

νt1

))dy

−1

k2λ2Dd

νtot1

k

√md

2π Td

∫ ∞

0ye−y2

ln

(ωr + k

√2Tdmd

y)2 + ν2tot1

(ωr − k

√2Tdmd

y)2 + ν2tot1

dy

=1

k2λ2Dd

2

√π

∫ ∞

0ζ ye−y2

(arctan

(ζ + y

νt1

)− arctan

(ζ − y

νt1

))dy

−1

√π

∫ ∞

0νt1 ye−y2

ln(

(ζ + y)2 + ν2t1

(ζ − y)2 + ν2t1

)dy .



Overall, we have

χd,eqk,ω = 1

k2λ2Dd

1 − 2√π

∫ ∞

0νt1 ye−y2

(arctan

(ζ + y

νt1

)− arctan

(ζ − y

νt1

))dy

− 1√π

∫ ∞

0ζ ye−y2

ln(

(ζ + y)2 + ν2t1

(ζ − y)2 + ν2t1

)dy (A.41)

+ ı1

k2λ2Dd

2√π

∫ ∞

0ζ ye−y2

(arctan

(ζ + y

νt1

)− arctan

(ζ − y

νt1

))dy

− 1√π

∫ ∞

0νt1 ye−y2

ln(

(ζ + y)2 + ν2t1

(ζ − y)2 + ν2t1

)dy . (A.42)

In the fully ionized case, νn,d = 0 and due to the absence of a strong dissipationmechanism ωi → 0, thus the dimensionless effective damping rate tends to zero,νt1 → 0. In the real part of the response, the second term will vanish, while for thelast term (with P. denoting the Cauchy principal value of the integral around itssingular point ζ = y):

limν1→0

ℜχd,eqk,ω = 1

k2λ2Dd

1 − 1√

πP.

∫ ∞

0ζye−y2

ln[

ζ + y

ζ − y

]2

dy

We split into two integrals using the properties of the logarithmic function and weemploy integration by parts due to ye−y2 = − 1

2d

dy (e−y2).

limνt1→0

Reχd,eqk,ω

=1

k2λ2Dd

1 −

2ζ√

πP.

∫ ∞

0ye−y2

ln[∣∣∣ ζ + y

ζ − y

∣∣∣]dy

=

1k2λ2

Dd

1 −

2ζ√

π

∫ ∞

0ye−y2

ln (ζ + y)dy +2ζ√

πP.

∫ ∞

0ye−y2

ln |ζ − y|dy

=

1k2λ2

Dd

1 +ζ

√π

∫ ∞

0

(e−y2

)′ln (ζ + y)dy

−ζ

√π

P.

∫ ∞

0

(e−y2

)′ln |ζ − y|dy

=1

k2λ2Dd

1 −ζ

√π

∫ ∞

0

e−y2

ζ + ydy −

ζ√

πP.

∫ ∞

0

e−y2

ζ − ydy

+ζ

√π

[e−y2

ln (ζ + y)]∞

0−

ζ√

π

[e−y2

ln |ζ − y|]∞

0 .


The limits to infinity are zero with the use of L’Hospital’s rule, while the limits tozero cancel each other, we set y → −y in the first integral.

limνt1→0

Reχd,eqk,ω = 1

k2λ2Dd

1 − ζ√

π

∫ ∞

0

e−y2

ζ + ydy − ζ√

πP.

∫ ∞

0

e−y2

ζ − ydy

= 1k2λ2

Dd

1 − ζ√

π

∫ 0

−∞

e−y2

ζ − ydy − ζ√

πP.

∫ ∞

0

e−y2

ζ − ydy

= 1k2λ2

Dd

1 − ζ√

π

[∫ 0

−∞

e−y2

ζ − ydy + P.

∫ ∞

0

e−y2

ζ − ydy

]

= 1k2λ2

Dd

1 − ζ√

πP.

∫ +∞

−∞

e−y2

ζ − ydy

,

which is the real part of the plasma dispersion function as expected.As far as the imaginary part is concerned, in the limit νt1 → 0 the second partvanishes. The first part might seem to vanish since (arctan (∞) − arctan (∞)) =π2 − π

2 = 0, yet for z = y, which lies in the parameter space, the argument of theinverse tangent is of indeterminate form. Use of integration by parts, will reveal anascent delta function sequence of the Lorentz line form, δa(x) = 1

πa

x2+a2 , with the

property δ(x) :∫ +∞

−∞ f(x)δ(x)dx = lima→0

∫ +∞

−∞f(x)δa(x)dx = f(0) or less strictly

δ(x) = lima→0

δa(x). We have

limνt1→0

ℑχd,eqk,ω = 1

k2λ2Dd

2ζ√π

limνt1→0

∫ ∞

0ye−y2

arctan

[ζ + y

νt1

]− arctan

[ζ − y

νt1

]dy

= − 1k2λ2

Dd

ζ√π

limνt1→0

∫ ∞

0

(e−y2

)′ arctan

[ζ + y

νt1

]− arctan

[ζ − y

νt1

]dy

= 1k2λ2

Dd

ζ√π

limνt1→0

∫ ∞

0e−y2 d

dy

arctan

[ζ + y

νt1

]− arctan

[ζ − y

νt1

]dy

= 1k2λ2

Dd

ζ√π

limνt1→0

∫ ∞

0e−y2

1

1 +(

ζ+yνt1

)2 + 11 +

(ζ−yνt1

)2

dy

= 1k2λ2

Dd

√πζ

∫ ∞

0e−y2

lim

νt1→0

1π

ν2ti

ν2t1 + (ζ + y)2 + 1

π

ν2ti

ν2t1 + (ζ − y)2

dy

= 1k2λ2

Dd

√πζ

∫ ∞

0e−y2

δ(ζ + y) + δ(ζ − y) dy

= 1k2λ2

Dd

√πζ

∫ ∞

0e−y2

δ(ζ + y) dy +∫ ∞

0e−y2

δ(ζ − y) dy

.

The first integrand is zero, since the negative root does not belong to the integrationinterval, hence overall

limνt1→0

ℑχd,eqk,ω = 1

k2λ2Dd

√πζe−ζ2

. (A.43)



which is the imaginary part of the plasma dispersion function, after the applicationof the Plemelj formula.

The integral response χd,chk,ω

The responses is related to the charging process and is defined by

χd,chk,ω =

∫ı

ω − k · v + ı(νch + νn,d)Φd

p

d3p

(2π)3 . (A.44)

We separate in real and imaginary parts and define νtot2 = ωi + νch + νn,d as thetotal effective damping of the response,

χd,chk,ω =

∫ı

ωr − k · v + ı(ωi + νch + νn,d)Φd

p

d3p

(2π)3

=∫

ı[ωr − k · v − ıνtot2](ωr − k · v)2 + ν2

tot2Φd

p

d3p

(2π)3

= +∫

νtot2

(ωr − k · v)2 + ν2tot2

Φdp

d3p

(2π)3 + ı

∫ωr − k · v

(ωr − k · v)2 + ν2tot2

Φdp

d3p

(2π)3 .

For the real part we use the integral I1

ℜχd,chk,ω

=∫

νtot2

(ωr − k · v)2 + ν2tot2

Φdp

d3p

(2π)3

=∫ ∞

0

∫ π

0

sin θ


dθ

2πνtot2v2Φd(v)dv

=∫ ∞

0

2πνtot2v2

kvνtot2

(arctan

(ωr + kv

νtot2

)− arctan

(ωr − kv

νtot2

))Φd(v)dv

=2π

k

∫ ∞

0

(arctan

(ωr + kv

νtot2

)− arctan

(ωr − kv

νtot2

))vΦd(v)dv

=2π nd

k

(md

2π Td

)3/2

×∫ ∞

0

(arctan

(ωr + kv

νtot2

)− arctan

(ωr − kv

νtot2

))v exp

(−

mdv2

2Td

)dv

=nd

k

√2π

√md

Td

×∫ ∞

0ye−y2

arctan

ωr + k

√2Tdmd

y

νtot2

− arctan

ωr − k

√2Tdmd

y

νtot2

dy

=

√2π

nd

kvT d

∫ ∞

0ye−y2

(arctan

(ζ + y

νt2

)− arctan

(ζ − y

νt2

))dy .


For the imaginary part of the response we use the integral I5

ℑχd,chk,ω =

∫ωr − k · v

(ωr − k · v)2 + ν2tot2

Φdp

d3p

(2π)3

=∫ ∞

0

(∫ π

0



dθ

)2π v2Φd(v)dv

=∫ ∞

0

2π v2

2kvln(



)Φd(v)dv

= ndπ

k

(md

2π Td

)3/2 ∫ ∞

0ln(



)v exp

(−mdv2

2Td

)dv

= nd

k

√md

2π Td

∫ ∞

0ye−y2

ln

(ωr + k√

2Td

mdy)2 + ν2

tot2

(ωr − k√

2Td

mdy)2 + ν2

tot2

dy

= nd√2πkvT d

∫ ∞

0ye−y2

ln

(ωr + k√

2Td

mdy)2 + ν2

tot2

(ωr − k√

2Td

mdy)2 + ν2

tot2

dy

= nd√2πkvT d

∫ ∞

0ye−y2

ln(

(ζ + y)2 + ν2t2

(ζ − y)2 + ν2t2

)dy .

Overall, we have

χd,chk,ω =

√2π

nd

kvT d

∫ ∞

0ye−y2

(arctan

(ζ + y

νt2

)− arctan

(ζ − y

νt2

))dy

+ ı nd√2πkvT d

∫ ∞

0ye−y2

ln(

(ζ + y)2 + ν2t2

(ζ − y)2 + ν2t2

)dy . (A.45)

In the low frequency regime we have ωr < ωpd, while for the charging frequencywe always have ωpd ≪ νch < ωpi. Hence, the total effective damping term isdominating the denominator of the response (with the exception of very strongdamping, where ωi will acquire large negative values and reduce νtot2 significantly).Using νt2 ≫ y, ζ, it is obvious that the imaginary part vanishes, while for the realpart we can use the Taylor expansion of the inverse tangent and keep the first order



term only, arctan x =∞∑

n=0

(−1)n

2n + 1x2n+1 ≃ x for x ≪ 1.

χd,chk,ω =

√2π

nd

kvT d

∫ ∞

0ye−y2

arctan

(ζ + y

νt2

)− arctan

(ζ − y

νt2

)dy

≃√

2π

nd

kvT d

∫ ∞

0ye−y2

ζ + y

νt2− ζ − y

νt2

dy

≃√

2π

nd

kvT d

∫ ∞

02ye−y2

(y

νt2

)dy

≃√

2π

nd

kvT d

2νt2

∫ ∞

0y2e−y2

dy

≃√

2π

nd

kvT d

2√

2kvT d

νtot2

∫ ∞

0y2e−y2

dy

≃√

2π

2√

2nd

νtot2

√π

4≃ nd

νch + νn,d + ωi,

which in absence of neutrals gives the familiar χd,chk,ω ≃ nd

νch. Alternatively, in the

initial response, we can ignore the real part of the denominator, which results in

χd,chk,ω ≃

∫ıΦd(v)ıνtot2

d3v ≃ 1νtot2

∫Φd(v) d3v ≃ nd

νch + νn,d + ωi. (A.46)

The integral response deqk,ω

A response similar to the one we just encountered has the form,

deqk,ω =

∫Φd(eq)

p (q)ω − k · v + ıνn,d

dqd3p

(2π)3 . (A.47)

This response is generated by the BGK collision term in the dust Klimontovichequation, due to the narrowness of the dust equilibrium distribution around theequilibrium charge we have q = qeq and the integration is over the momentum spaceonly. The differences from the charging response are that: the only dissipative termis νn,d and hence the total effective damping term will be νtot1 = ωi + νn,d, thedistribution is normalized to unity and hence the previous results will be dividedby nd, the imaginary factor is missing leading to the previous results multiplication


by −ı.

deqk,ω = 1√

2πkvT d

∫ ∞

0ye−y2

ln[

(ζ + y)2 + ν2t1

(ζ − y)2 + ν2t1

]dy

− ı

√2π

1kvT d

∫ ∞

0ye−y2

arctan

[ζ + y

νt1

]− arctan

[ζ − y

νt1

]dy . (A.48)

Finally, despite the similarities, the approximate expressions are not valid for thisresponse, since the charging frequency is absent. In the fully ionized case theresponse does not exist, since it is connected to collisions with neutrals (alwaysmultiplied by νn,d).

Appendix B

Fluid Description of Dust AcousticWaves

As aforementioned, the partially ionized complex plasma system consists of fourdistinct "species": the ions, the electrons, the dust grains and the neutral gas. In thehydrodynamic description of waves, the perturbed fluid equations are used for theelectrons, the ions and the dust species, whereas the neutral gas fluid quantities areusually considered unperturbed from equilibrium. The fluid equations are derivedby the moments of the generalized Boltzmann equations for the ensemble averagedpart of each distribution function, the equations used are the continuity equationand the momentum equation, thus we have truncation before the second momentand a relation for the pressure tensor has to be assumed. Most commonly, neglectinganisotropy, the pressure is assumed scalar obeying a relation of the form pj

nγjj

=

const , with γj = 1 for isothermal species and γj = 5/3 for adiabatic processes.The hydrodynamic equations for the plasma species have the form, with α =

e, i,

∂nα

∂t+ ∇ (nαvα) = Qsα − Qlα ,

∂vα

∂t+ (vα · ∇)vα = − qα

mα∇Φ − 1

mαnα∇ pα − vα

nαQlα

−∑

j=i,e,d,n=α

να,j(vα − vj) − να,dvα .

where

1. Qsα describes the production of electrons/ions due to ionization processes(electron impact ionization, photo-ionization) or due to emission processesfrom the dust surface (photoelectric, secondary, thermionic, field emission)

147

148 APPENDIX B. FLUID DESCRIPTION OF DUST ACOUSTIC WAVES

2. Qlα describes the loss of electrons/ions due to recombination processes in thevolume, due to collection on the dust surface after inelastic collisions, due toloss in the boundary surface of the complex plasma configuration

3. vα

nαQlα describes momentum loss due to particle loss (only for external to

the plasma system processes, since inelastic collision losses are accounted ina separate term), while there is no similar term for momentum gain due toparticle production because the newly created particles can be assumed atrest (since the dust or neutral thermal velocities are much smaller than thespecies thermal velocities)

4. the first collisional term describes momentum transfer in elastic Coulombcollisions between the species

5. the second collisional term describes momentum transfer in inelastic collisionswith dust particles.

Similarly the hydrodynamic equations for the dust particles have the form

∂nd

∂t+ ∇(ndvd) = 0 ,

vd

∂t+ (vd · ∇)vd = −Zde

md∇Φ − 1

mdnd∇Pd −

∑j=i,e,n

νd,j(vd − vj) +∑

κ=e,i

νd,κvκ ,

where one should note that there is no source or sink of dust particles and that Zd

is the characteristic charge number with its sign. The above system of equations isself-consistently closed by the Poisson equation and the charging equation,

∇2Φ = −4πe(−ne + ni + Zdnd) ,

∂Zd

∂t+ (vd · ∇)Zd =

∑j

Ifj ,

where Ifj are the particle fluxes emitted or absorbed by the grain, with the lastequation being the equation for the characteristic charge with its sign, that isequivalent to the charging equation. The latter equation is of hydrodynamic naturecontaining the convective derivative of the charge, it can be derived by multiplyingthe dust kinetic equation with the charge q and integrating over the momentumspace only.

The linearization of the system is achieved by decomposing each quantity toits equilibrium and perturbed value, and by neglecting high order perturbationterms. We assume un-magnetized homogeneous plasma with non-drifting isother-mal species and set

pj = pj0 + pj1 , Φ = Φ1 , nj = nj0 + nj1 ,

vj = vj1 , E = E1 , Zd = Zd0 + Zd1 ,

(B.1)

149

where the ground state values are determined by the quasi-neutrality condition, theparticle flux balance on the grain surface and the absorption / generation balancefor the plasma particles. The perturbed Poisson equation will acquire the form

∇2Φ1 = −4πe(ni1 − ne1 + Zd1nd0 + Zd0nd1) . (B.2)

The presence of sources/sinks as well as of collisional terms are depending on thephysical scenario of the complex plasma configuration. Certain types of collisionscan be neglected for short wavelengths, where the characteristic mean free pathsare longer than the characteristic scale of the problem. For sufficiently short wave-lengths, ion-electron Coulomb collisions, species-dust elastic and inelastic collisionsas well as electron-neutral collisions can be ignored. On the other hand, ion-neutraland mostly dust-neutral collisions have large corresponding frequencies and shortmean free paths and cannot be neglected in most cases.

Since dust-species inelastic collisions are ignored, there is no need for sourceterms to preserve the plasma. As an approximation the dust charge can be con-sidered non-fluctuating and the charging equation is removed from the system.Moreover, assuming that particle emission from the grain surface can be neglected,the dust charge is negative Zd → −Zd. Therefore, the system of hydrodynamicequation will become, for s = i, e, d ,

∂ns

∂t+ ∇ · (nsvs) = 0 ,

∂ve

∂t+ (ve · ∇)ve = e

me∇Φ − 1

mene∇pe ,

∂vi

∂t+ (vi · ∇)vi = − e

mi∇Φ − 1

mini∇pi − νi,n(vi − vn) ,

∂vd

∂t+ (vd · ∇)vd = Zde

md∇Φ − 1

mdnd∇pd − νd,n(vd − vn) . (B.3)

The strategy is to use the three hydrodynamic equations, linearize and assume wave-like perturbations (exp (−ıωt + ık · r)), express the perturbed density as a functionof the electrostatic potential perturbation by eliminating vs1 , ps1 and substitute inthe Poisson equation, searching for a non-trivial solution. In the case of short wave-lengths the total permittivity can be expressed as the sum of the contribution ofeach species, mixed terms that appear in the kinetic theory treatment are absent.Therefore, the dispersion relation will have the form 1+χe+χi+χd = 0 , comparingwith the perturbed Poisson equation that is

Φ1

1 + 4πe

k2Φ1ne1 − 4πe

k2Φ1ni1 + 4πZde

k2Φ1nd1

= 0 ,

we end up with

χe = 4πe

k2Φ1ne1 , χi = − 4πe

k2Φ1ni1 , χd = 4πZde

k2Φ1nd1 . (B.4)


In the case of the dust acoustic wave, the propagation regime is kvT d ≪ ω ≪kvT i, kvT e and further simplifications can be made. We proceed with a scale analysisbetween the inertial and the pressure gradient force term,∣∣∣∣∣msns

∂vs

∂t

∇ps

∣∣∣∣∣ ∼ msnsωvs

knsTs∼ msv2

s

Ts∼

v2ph

v2T s

, (B.5)

for electrons/ions v2ph ≪ v2

T s and hence the two inertial terms in the momentumequation can be neglected, for dust particles v2

ph ≫ v2T d and hence the pressure

gradient force can be neglected.For the electron species, the momentum equation will be 0 = ene∇Φ − ∇pe,

for isothermal electrons, a substitution from the ideal gas law will result in theBoltzmann equilibrium relation, ne = ne0 exp ( eΦ

Te). We decompose the density

in both sides and use the smallness of the density perturbation to expand theexponential to a Taylor series and keep the first two terms, the result is ne1 =ene0Te

Φ1 , and the electron permittivity will be

χe = 1k2λ2

De

. (B.6)

For the ion species, the continuity equation will give ni0k · vi1 = ωni1, while themomentum equation after taking the inner product with the wavenumber will giveni1 = eni0k2Φ1

mi(ıνi,nω−v2T i

k2) , and the ion permittivity will be

χi =ω2

pi

−ıνinω + k2v2T i

. (B.7)

For the dust particles, continuity will give nd0k · vd1 = ωnd1 and the momentumequation nd1 = − eZd

mdω(ω+ıνd,n) nd0 leading to the dust permittivity

χd = −ω2

pd

ω(ω + ıνd,n). (B.8)

Hence, the form of the dispersion relation will be

1 + 1k2λ2

De

+ω2

pi

k2v2T i − ıνinω

=ω2

pd

ω(ω + ıνd,n). (B.9)

Depending on the strength of the dissipation on neutrals we have the solutions;

1. For νi,n = 0, νd,n = 0, we have the collisionless result with ω = ωr. The

151

dispersion relation becomes

1 + 1k2λ2

De

+ 1k2λ2

Di

=ω2

pd

ω2r

1 + 1k2λ2

D

=ω2

pd

ω2r

ω2r =

ω2pdλ2

D

1 + k2λ2D

k2 .

In the short wavelength limit k2λ2D ≫ 1: we get that ωr ≃ ωpd, stating

that the dust plasma frequency is the highest frequency the dust acousticeigenmodes can reach. In the long wavelength limit k2λ2

D ≪ 1: we get thatωr ≃ ωpdλD k =

√P ZdTi

md(1+P +τ) k, stating that the phase velocity of the dustacoustic waves is approximately constant.

2. For νi,n = 0, νd,n = 0, the dispersion relation in the long wavelength limitbecomes

1 + 1k2λ2

De

+ 1k2λ2

Di

−ω2

pd

(ω + ıνd,n)ω= 0

1 + 1k2λ2

D

−ω2

pd

(ω + ıνd,n)ω= 0

ω2 + ıνd,nω −ω2

pdλ2D

1 + k2λ2D

k2 = 0

ω2 + ıνd,nω − ω2pdλ2

D k2 = 0 .

In the case ν2d,n < 4ω2

pdk2λ2D; the solution to the dispersion relation has both

imaginary and real parts with ωr =√

ω2pdk2λ2

D − ν2d,n

4 , ωi = − νd,n

2 . In thecase ν2

d,n > 4ω2pdk2λ2

D; we have high dissipation and the solution to the dis-

persion relation will be pure imaginary with ωi = − νd,n

2 −√

ν2d,n

4 − ω2pdk2λ2

D.There is a critical wavelength, above which the solutions turn from dampedoscillations to aperiodic damping, by setting the discriminant equal to zerowe get kcrit = νd,n

2ωpdλD= νd,n

2

√md(1+P +τ)

P ZdTi.

3. For νi,n = 0, νd,n = 0, by Taylor expanding in the small ωνi,n

v2T i

k2 quantityand keeping the first order term only, the ion susceptibility will becomeχi ∼ ω2

pi

k2v2T i

(1−ıωνi,n

k2v2T i

)∼ 1

k2λ2Di

(1−ıωνi,n

k2v2T i

)∼ 1

k2λ2Di

(1 + ıωνi,n

k2v2T i

) . Therefore, the


dispersion relation will now be

1 + 1k2λ2

D

+ ıωνi,n

k4λ2Div

2T i

=ω2

pd

(ω + ıνd,n)ω, (B.10)

and if we set A(k) = 1 + 1k2λ2

D

(non-dimensional) and B(k) = 1k4λ2

Div2

T i

(di-mensions of s2) ,

[ıνi,nB(k)] ω3 + [A(k) − νi,nνd,nB(k)] ω2 + [ıνd,nA(k)] ω − ω2pd = 0 ,

the above equation is a third degree polynomial equation for ω = ωr + ωi

which is solvable analytically, through the Cardano formula for monic poly-nomials. It is also worth noticing that with respect to k2 the above equationis quadratic.

Bibliography

Aleksandrov, A. F., L. S. Bogdankevich and A. A. Rukhadze, (1984), Principlesof Plasma Electrodynamics, Springer, New York.

Allen, J. E., (1992), Probe theory - the orbital motion approach, Phys. Scr. 45,497.

Allen, J. E., B. M. Annaratone and U. de Angelis, (2000), On the orbital motionlimited theory for a small body at floating potential in a Maxwellian plasma, J.Plasma Phys. 63, 299.

Allison, J., and B. Chambers, (1966), Reflex discharge in argon using brush cath-odes, Electronic Letters 2, 443.

Al’pert, Y. L., A. V. Gurevich and L. P. Pitaevskii, (1965), Space Physics withArtificial Satellites, Consultants Bureau, New York.

Balescu, R., (1960), Irreversible processes in ionized gases, Phys. Fluids 3, 52.

Balescu, R., (1975), Equilibrium and Non-equilibrium Statistical Mechanics, JohnWiley & Sons, New York.

Barkan, A., R. L. Merlino and N. D’Angelo, (1995), Laboratory observation ofthe dust acoustic wave mode, Phys. Plasmas 2, 3563.

Barkan, A., N. D’Angelo and R. L. Merlino, (1996), Experiments on ion-acousticwaves in dusty plasmas, Planet. Space Sci. 44, 239.

Benedikt, J., (2010), Plasma-chemical reactions: low pressure acetylene plasmas,J. Phys. D: Appl. Phys. 43, 043001.

Berndt, J., S. Hong, E. Kovacevic, I. Stefanovic and J. Winter, (2003), Dustparticle formation in low pressure Ar/CH4 and Ar/C2H4 discharges used for thinfilm deposition, Vacuum 71, 377.

Bhatnagar, P. L., E. P. Gross and M. Krook, (1954), A model for collision pro-cesses in gases. I. Small amplitude processes in charged and neutral one-componentsystems, Phys. Rev. 94, 511.

153

154 BIBLIOGRAPHY

Bingham, J. P., (1967), The construction and general properties of a brush cathodedischarge, Technical Report 719, University of Maryland, Department of Physicsand Astronomy.

Bogoliubov, N. N., (1946), Problems of a Dynamical Theory in Statistical Physics,State Technical Press, Moscow.

Bonitz, M., W. Ebeling and Yu. M. Romanovsky, (2003), Contributions of YuriL. Klimontovich to the kinetic theory of nonideal plasmas, Contrib. Plasma Phys.43, 247.

Born, M., and H. S. Green, (1949), A general kinetic theory of liquids, CambridgeUniversity Press, Cambridge.

Bouchoule, A., A. Plain, L. Boufendi, J. Ph. Blondeau and C. Laure, (1991),Particle generation and behavior in a silane argon low pressure discharge undercontinuous or pulsed radio frequency excitation, J. Appl. Phys. 70, 1991.

Bouchoule, A., (1999) Dusty Plasmas: Physics, Chemistry and Technological Im-pact in Plasma Processing, Wiley, New York.

Boufendi, L., and A. Bouchoule, (2002), Industrial developments of scientific in-sights in dusty plasmas. Plasma Sources Sci. Technol. 11, A211.

Caron, P. R., (1971), Plasma generation using a large V-groove cathode discharge,Appl. Sci. Res. 23, 409.

Castaldo, C., U. de Angelis and V. N. Tsytovich, (2006), Screening and attractionof dust particles in plasmas, Phys. Rev. Lett. 96, 075004.

Chang, J.-S., and J. G. Laframboise, (1976), Probe theory for arbitrary shape ina large Debye length stationary plasma, Phys. Fluids 19, 25.

Chu, J. H., and I. L, (1994), Direct observation of Coulomb crystals and liquidsin strongly coupled rf dusty plasmas. Phys. Rev. Lett. 72, 4009.

D’Angelo, N., (1997), Ionization instability in dusty plasmas, Phys. Plasmas 4,3422.

de Angelis, U., A. Forlani, V. N. Tsytovich and R. Bingham, (1992), Scattering ofelectromagnetic waves by a distribution of charged dust particles in space plasmas,J. Geophys. Res. 97, 6261.

de Angelis, U., (1992), The physics of dusty plasmas, Phys. Scr. 45, 465.

de Angelis, U., (1998), Correlations of dust particles in plasmas, Phys. Scr. T75,75.

BIBLIOGRAPHY 155

de Angelis, U., R. Bingham, A. Forlani and V. N. Tsytovich, (2002), Scatteringand transformation of waves in dusty plasmas, Phys. Scr. T98, 163.

de Angelis, U., A. V. Ivlev, G. E. Morfill and V. N. Tsytovich, (2005), Stochasticheating of dust particles with fluctuating charges, Phys. Plasmas 12, 052301.

de Angelis, U., G. Capobianco, C. Marmolino and C. Castaldo, (2006), Fluctua-tions in dusty plasmas, Plasma Phys. Control. Fusion 48, B91.

de Angelis, U., G. Regnoli and S. Ratynskaia, (2010), Long range attraction ofnegatively charged dust particles in weakly ionized dense dust clouds, Phys. Plas-mas 17, 043702.

Do, H. T., G. Thieme, M. Frölich, H. Kersten and R. Hippler, (2005), Ion moleculeand dust particle formation in Ar/CH4, Ar/C2H2 and Ar/C3H6 radio frequencyplasmas, Contrib. Plasma Phys. 45, 378.

Draine, B. T., (2003), Interstellar dust grains, Annu. Rev. Astron. Astrophys. 41,241.

Else, D., R. Kompaneets and S. V. Vladimirov, (2009), Instability of the ionization-absorption balance in a complex plasma at ion time scales, Phys. Rev. E80,016403.

Feher, G., and M. Weissman, (1973), Fluctuation spectroscopy; determination ofchemical reaction kinetics from the frequency spectrum of fluctuations, Proc. Nat.Acad. Sci. U.S.A 70, 870.

Fortov, V. E., G. Morfill, O. Petrov, M. Thoma, A. Usachev, H. Höfner, A. Zob-nin, M. Kretschmer, S. Ratynskaia, M. Fink, K. Tarantik, Yu. Gerasimov and V.Esenkov, (2005), The project ’Plasmakristall-4’ (PK-4), a new stage in investi-gations of dusty plasmas under microgravity conditions: first results and futureplans, Plasma Phys. Control. Fusion 47, B537.

Fortov, V. E., A. V. Ivlev, S. A. Khrapak, A. G. Khrapak, and G. E. Morfill,(2005), Complex (dusty) plasmas: Current status, open issues, perspectives. Phys.Reports 421, 1.

Fortov, V. E., and G. E. Morfill (eds), (2009), Complex and Dusty Plasmas: FromLaboratory to Space (Series in Plasma Physics), CRC Press, London.

Goertz, C. K., (1989), Dusty plasmas in the solar system, Rev. Geophys. 27, 271.

Gross, E. P., and M. Krook, (1956), Model for collision processes in gases: Smallamplitude oscillations of charged two-component systems, Phys. Rev. 102, 593.

Hutchinson, I. H., and L. Patacchini, (2007), Computation of the effect of neutralcollisions on ion current to a floating sphere in a stationary plasma, Phys. Plasmas14, 013505.

156 BIBLIOGRAPHY

Ichimaru, S., (1964), Hydrodynamic fluctuations and correlations in a plasma, J.Phys. Soc. Japan 19, 1207.

Ichimaru, S., (1973), Basic principles of plasma physics, W. A. Benjamin Inc.,New York.

Ichimaru, S., (1982), Strongly coupled plasmas: high-density classical plasmas anddegenerate electron liquids, Rev. Mod. Phys. 54, 1017.

Ichimaru, S., (1992), Statistical Plasma Physics, Volume I: Basic Principles, Ad-dison Wesley Publishing Company, Tokyo.

Ignatov, A. M., S. A. Trigger, W. Ebeling and P. P. J. M. Schram, (2002), Kineticsof dusty particulates with growing mass, Phys. Lett. A 293, 141.

Ignatov, A. M., S. A. Trigger, S. A. Maiorov and W. Ebeling, (2002), Rotationalkinetics of absorbing dust grains in neutral gas, Phys. Rev. E 65, 046413.

Ivlev, A. V., D. Samsonov, J. Goree and G. Morfill, (1999), Acoustic modes in acollisional dusty plasma, Phys. Plasmas 6, 741.

Ivlev, A. V., S. K. Zhdanov, B. A. Klumov, V. N. Tsytovich, U. de Angelis andG. E. Morfill, (2004), Kinetics of ensembles with variable charges, Phys. Rev. E70, 066401.

Ivlev, A. V., S. A. Khrapak, S. K. Zhdanov and G. E. Morfill, (2004), Force on acharged test particle in a collisional flowing plasma, Phys. Rev. Lett. 92, 205007.

Ivlev, A. V., S. K. Zhdanov, S. A. Khrapak and G. E. Morfill, (2005), Kineticapproach for the ion drag force in a collisional plasma, Phys. Rev. E 71, 016405.

Ivlev, A. V., A. Lazarian, V. N. Tsytovich, U. de Angelis, T. Hoang and G. E. Mor-fill, (2010), Acceleration of small astrophysical grains due to charge fluctuations,ApJ 723, 612.

Jana, M. R., A. Sen and P. K. Kaw, (1993), Collective effects due to charge-fluctuation dynamics in a dusty plasma, Phys. Rev. E 48, 3930.

Green, M. S., (1954), Markov random processes and the statistical mechanics oftime dependent phenomena. II. Irreversible processes in fluids, J. Chem. Phys. 22,398.

Kennedy, R. V., and J. E. Allen, (2003), The floating potential of spherical probesand dust grains. II: Orbital motion theory, J. Plasma Physics 69, 485.

Khrapak, S. A., A. V. Ivlev, G. E. Morfill and H. M. Thomas, (2002), Ion dragforce in complex plasmas, Phys. Rev. E 66, 046414.

BIBLIOGRAPHY 157

Khrapak, S. A., and G. E. Morfill, (2004), Dusty plasmas in a constant electricfield: Role of the electron drag force, Phys. Rev. E 69, 066411.

Khrapak, S. A., S. V. Ratynskaia, A. V. Zobnin, A. D. Usachev, V. V. Yaroshenko,M. H. Thoma, M. Kretschmer, H. Höfner, G. E. Morfill, O. F. Petrov and V. E.Fortov, (2005), Particle charge in the bulk of gas discharges, Phys. Rev. E 72,016406.

Khrapak, S. A., G. E. Morfill, A. G. Khrapak and L. G. D’yachkov, (2006), Charg-ing properties of a dust grain in collisional plasmas, Phys. Plasmas 13, 052114.

Khrapak, S. A., and G. E. Morfill, (2008), An interpolation formula for the ionflux to a small particle in collisional plasmas, Phys. Plasmas 15, 114503.

Khrapak, S. A., and G. E. Morfill, (2009), Basic processes in complex (dusty)plasmas: charging, interactions and ion drag force, Contrib. Plasma Phys. 49,148.

Khrapak, S. A., and G. E. Morfill, (2010), Ionization instability of ion-acousticwaves, Phys. Plasmas 17, 062111.

Klimontovich, Yu. L., (1958), On the method of second quantization in phasespace, Sov. Phys. JETP 6, 753.

Klimontovich, Yu. L., (1959), Relativistic kinetic equations, ZhETF 37, 375.

Klimontovich, Yu. L., (1967), The Statistical Theory of Non-equilibrium Processesin a Plasma, Pergamon, London.

Klimontovich, Yu. L., and W. Ebeling, (1972), Quantum kinetic equations, ZhETF63, 905.

Klimontovich, Yu. L., (1982), Kinetic Theory of Non-ideal Gases and Non-idealPlasmas, Pergamon Press, New York.

Klimontovich, Yu. L., (1983), Kinetic Theory of Electromagnetic Processes,Springer, Heidelberg.

Klimontovich, Yu. L., (1986), Statistical Physics, Harwood Academic Publishers,New York.

Klimontovich, Yu. L., D. Kremp and W. D. Kraeft, (1987), Kinetic theory forchemically reacting gases and partially ionized plasmas, Adv. Chem. Phys. 68,175.

Klimontovich, Yu. L., H. Wilhelmsson, I. P. Yakimenko and A. G. Zagorodny,(1989), Statistical theory of plasma-molecular systems, Phys. Reports 175, 263.

Klimontovich, Yu. L., (1997), Physics of collisionless plasma, Phys. Usp. 40, 21.

158 BIBLIOGRAPHY

Klimontovich, Yu. L., (1998), Two alternative approaches in the kinetic theory offully ionized plasmas, Contrib. Plasma Phys. 38, 387.

Kompaneets, R., and V. N. Tsytovich, (2005), Collective electrostatic interactionof particles in a complex plasma with ion flow, Contrib. Plasma Phys. 45, 130.

Krasheninnikov, S.I., A. Y. Pigarov, R. D. Smirnov and T. K. Soboleva, (2010),Theoretical Aspects of Dust in Fusion Devices, Contrib. Plasma Phys. 50, 410.

Krasheninnikov, S.I., R. D. Smirnov, A. Y. Pigarov, T. K. Soboleva and D. A.Mendis, (2010), Dust in fusion devices: The state of theory and modelling, J.Plasma Phys. 76, 377.

Krügel, E., (2003), The physics of Interstellar Dust (Series in Astronomy andAstrophysics), Institute of Physics Publishing, Bristol.

Kubo, R., (1957), Statistical mechanical theory of irreversible processes. I. Generaltheory and simple applications to magnetic and conduction problems, J. Phys. Soc.Japan 12, 570.

Kubo, R., The fluctuation-dissipation theorem, (1966), Rep. Prog. Phys. 29, 255.

Lampe, M., V. Gavrishchaka, G. Ganguli and G. Joyce, (2001), Phys. Rev. Lett.86, Effect of trapped ions on shielding of a charged spherical object in a plasma,5278.

Lampe, M., R. Goswami, Z. Sternovsky, S. Robertson, V. Gavrishchaka, G. Gan-guli and G. Joyce, (2003), Trapped ion effect on shielding, current flow, and charg-ing of a small object in a plasma, Phys. Plasmas 10, 1500.

Landau, L. D., (1937), Kinetic equation in case of Coulomb interaction, Zh. Eksp.Teor. Fiz. 7, 203.

Landau, L. D., and E. M. Lifshitz, (1980), Course of theoretical physics, Vol. V:Statistical Physics part I, Pergamon Press, Oxford.

Landau, L. D., and E. M. Lifshitz, (1984), Course of theoretical physics, Vol. VIII:Electrodynamics of continuous media, Pergamon Press, Oxford.

Landau, L. D., and E. M. Lifshitz, (1980), Course of theoretical physics, Vol. IX:Statistical Physics part II, Pergamon Press, Oxford.

Landau, L. D., and E. M. Lifshitz, (1981), Course of theoretical physics, Vol. X:Physical Kinetics, Pergamon Press, Oxford.

Lenard, A., (1960), On Bogoliubov’s kinetic equation for a spatially homogeneousplasma, Ann. Phys. 10, 390.

BIBLIOGRAPHY 159

Liboff, R. L., (1990), Kinetic theory: classical, quantum and relativistic descrip-tions, Springer, Berlin.

Liebermann, M. A., and A. J. Lichtenberg, (1994), Principles of Plasma Dis-charges and Materials Processing, Wiley, New York.

Maksimovic, M., S. Hoang, N. Meyer-Vernet, M. Moncuquet, J.-L. Bougeret, J. L.Philips and P. Canu, (1995), Solar wind electron parameters from quasi-thermalnoise spectroscopy and comparison with other measurements on Ulysses, J. Geo-phys. Res. 100, 19881.

Marmolino, C., U. de Angelis, A. V. Ivlev and G. E. Morfill, (2008), Stochasticacceleration of dust particles in tokamak edge plasmas, AIP Conf. Proc 1061,105.

Marmolino, C., U. de Angelis, A. V. Ivlev and G. E. Morfill, (2009), On the roleof stochastic heating in experiments with complex plasmas, Phys. Plasmas 16,033701.

Marmolino, C., (2011), Stochastic heating of dust particles in complex plasmasas an energetic instability of harmonic oscillator with random frequency, Phys.Plasmas 18, 103701.

Matsoukas, T., and M. Russell, (1995), Particle charging in low pressure plasmas,J. Appl. Phys. 77, 4285.

Matsoukas, T., M. Russell and M. Smith, (1996), Stochastic charge fluctuationsin dusty plasmas, J. Vac. Sci. Technol. A 14, 624.

Melandsø, F., T. Aslaksen and O. Havnes, (1993), A new damping effect for thedust-acoustic wave, Planet. Space Sci. 41, 321.

Merlino, R. L., (2009), Dust-acoustic waves: visible sound waves, AIP Conf. Proc.1188, 141.

Merlino, R. L., (2009), Dust-acoustic waves driven by an ion-dust streaming in-stability in laboratory discharge dusty plasma experiments, Phys. Plasmas 16,124501.

Melrose, D. B., and R. C. McPhedran, (1991), Electromagnetic processes in dis-persive media, Cambridge University Press, Cambridge.

Meyer-Vernet, N., (1979), On natural noises detected by antennas in plasmas, J.Geophys. Res. 84, 5373.

Meyer-Vernet, N., (1983), Quasi-thermal noise corrections due to particle impactsor emission, J. Geophys. Res 88, 8081.

160 BIBLIOGRAPHY

Meyer-Vernet, N., and C. Perche, (1989), Tool kit for antennae and thermal noisenear the plasma frequency, J. Geophys. Res. 94, 2405.

Meyer-Vernet, N., S. Hoang, K. Issautier, M. Maksimovic, R. Manning, M. Mon-cuquet and R. Stone, (1998), Measuring plasma parameters with thermal noisespectroscopy. In: Geophysical Monograph Series 103, Measurement Techniques inSpace Plasmas: Fields, American Geophysical Union, Washington DC, 205.

Meyer-Vernet, N., M. Maksimovic, A. Czechowski, I. Mann, I. Zouganelis, K.Goetz, M. L. Kaiser, O. C. St. Cyr, J.-L. Bougeret and S. D. Bale, (2009), Dustdetection by the wave instrument on STEREO: nano-particles picked up by thesolar wind? Solar Phys. 256, 463.

Mihaila, B., S. A. Crooker, K. B. Blagoev, D. G. Rickel, P. B. Littlewood and D.L. Smith, (2006), Spin noise spectroscopy to probe quantum states of ultracoldfermionic atom gases, Phys. Rev. A 74, 063608.

Moncuquet, M., H. Matsumoto, J.-L. Bougeret, L. G. Blomberg, K. Issautier,Y. Kasaba, H. Kojima, M. Maksimovic, N. Meyer-Vernet and P. Zarka, (2006),The radio waves and thermal electrostatic noise spectroscopy (SORBET) exper-iment on BEPICOLOMBO/MMO/PWI: Scientific objectives and performance,Adv. Space Res. 38, 680.

Montgomery, D. C., and D. A. Tidman, (1964), Plasma Kinetic Theory, McGrawHill, New York.

Morfill, G. E., V. N. Tsytovich and H. Thomas, (2003), Complex Plasmas: II.Elementary processes in complex plasmas, Plasma Phys. Rep. 29, 1.

Musal, H. M., (1966), An inverse brush cathode for the negative glow plasmasource, J. Appl. Phys. 37, 1935.

Nefedov, A. P., G. E. Morfill, V. E. Fortov, H. M. Thomas, H. Rothermel, T.Hagl, A. V. Ivlev, M. Zuzic, B. A. Klumov, A. M. Lipaev, V. I. Molotkov, O.F. Petrov, Y. P. Gidzenko, S. K. Krikalev, W. Shepherd, A. I. Ivanov, M. Roth,H. Binnenbruck, J. A. Goree and Y. P. Semenov , (2003), PKE-Nefedov: Plasmacrystal experiments on the International Space Station, New J. Phys 5, 33.

Nicholson, D. R., (1983), Introduction to plasma theory, Wiley, New York.

Pekarek, L., (1968), Ionization waves (striations) in a discharge plasma, Sov. Phys.Usp. 11, 188.

Persson, K.-B., (1965), Brush cathode plasma - a well behaved plasma, J. Appl.Phys. 36, 3086.

Pieper, J. B., and J. Goree, (1996), Dispersion of plasma dust acoustic waves inthe strong coupling regime, Phys. Rev. Lett. 77, 3137.

BIBLIOGRAPHY 161

Rao, N. N., P. K. Shukla and M. Y. Yu, (1990), Dust acoustic waves in dustyplasmas, Planet. Space Sci. 38, 543.

Ratynskaia, S., S. Khrapak, A. Zobnin, M. H. Thoma, M. Kretschmer, A. Usachev,V. Yaroshenko, R. A. Quinn, G. E. Morfill, O. Petrov and V. Fortov, (2004), Ex-perimental determination of dust-particle charge in a discharge plasma at elevatedpressures, Phys. Rev. Lett. 93, 085001.

Ratynskaia, S., U. de Angelis, S. Khrapak, B. Klumov and G. E. Morfill, (2006),Electrostatic interaction between dust particles in weakly ionized complex plasmas,Phys. Plasmas 13, 104508.

Ratynskaia, S., C. Castaldo, E. Giovannozzi, D. Rudakov , G. Morfill, M. Horanyi,J. Yu and G. Maddaluno, (2008), In situ dust detection in fusion devices, PlasmaPhys. Control. Fusion 50, 124046.

Ratynskaia, S., C. Castaldo, K. Rypdal, G. Morfill, U. de Angelis, V. Pericoli-Ridolfini, A. Rufoloni and E. Giovannozzi, (2008), Hypervelocity dust impacts inFTU scrape-off layer, Nucl. Fusion 48, 015006.

Ratynskaia, S., M. De Angeli, U. de Angelis, C. Marmolino, G. Capobianco, M.Lontano, E. Lazzaro, G. E. Morfill and G. Gervasini, (2007), Observation of theEffects of Dust Particles on Plasma Fluctuation Spectra, Phys. Rev. Lett. 99,075002.

Ratynskaia, S., M. De Angeli, E. Lazzaro, C. Marmolino, U. de Angelis, C.Castaldo, A. Cremona, L. Laguardia, G. Gervasini and G. Grosso, (2010), Plasmafluctuation spectra as a diagnostic tool for submicron dust, Phys. Plasmas 17,043703.

Ricci, P., G. Lapenta, U. de Angelis and V. N. Tsytovich, (2001), Plasma kineticsin dusty plasmas, Phys. Plasmas 8, 769.

Rosenberg, M., (1993), Ion- and dust-acoustic instabilities in dusty plasmas,Planet. Space Sci. 41, 229.

Rosenberg, M., (2002), A note on ion-dust streaming instability in a collisionaldusty plasma, J. Plasma Phys. 67, 235.

Schram, P. P. J. M., (1991), Kinetic theories of gases and plasmas, Kluwer Aca-demic Publishers, Dordrecht.

Schram, P. P. J. M., A. G. Sitenko, S. A. Trigger and A. G. Zagorodny, (2000),Statistical theory of dusty plasmas: Microscopic equations and Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, Phys. Rev. E 63, 016403.

Schram, P. P. J. M., S. A. Trigger and A. G. Zagorodny, (2003), New microscopicand macroscopic variables in dusty plasmas, New J. Phys. 5, 27.

162 BIBLIOGRAPHY

Shukla, P. K., (1992), Low-frequency modes in dusty plasmas, Phys. Scr. 45, 504.

Shukla, P. K., and V. P. Silin, (1992), Dust-ion acoustic wave, Phys. Scr. 45, 508.

Shukla, P. K., G. T. Birk and G. E. Morfill, (1997), Dust-acoustic waves in partiallyionized dusty plasmas, Phys. Scr. 56, 299.

Shukla, P. K., and A. A. Mamun, (2002), Introduction to dusty plasmas, Series inPlasma Physics, IOP, Bristol.

Sitenko, A. G., A. G. Zagorodny, Yu. I. Chutov, P. Schram and V. N. Tsytovich,(1996), Statistical properties and relaxation of dusty plasmas, Plasma Phys. Con-trol. Fusion 38, A105.

Spinicchia, N., G. Angella, M. De Angeli, G. Gervasini and E. Signorelli, (2006),Growth of thin hydrocarbon films and hydrogen production in a cusp plasmadevice, Surf. Coat. Technol. 200, 6434.

Sternovsky, Z., S. Robertson and M. Lampe, (2003), Ion collection by cylindricalprobes in weakly collisional plasmas: Theory and experiment, J Appl. Phys. 94,1374.

Su, C. H., and S. H. Lam, (1963), Continuum theory of spherical electrostaticprobes, Phys. Fluids 6, 1479.

Sunyaev, R. A., and Y. B. Zeldovich, (1970), Small-scale fluctuations of relicradiation, Astrophys. & Space Sci. 7, 3.

Thoma, M. H., M. A. Fink, H. Höfner, M. Kretschmer, S. A. Khrapak, S. V.Ratynskaia, V. V. Yaroshenko, G. E. Morfill, O. F. Petrov, A. D. Usachev, A. V.Zobnin and V. E. Fortov, (2007), PK-4: Complex plasmas in space - The nextgeneration, IEEE Trans. Plasma Sci. 35, 255.

Thomas, H., G. E. Morfill, V. Demmel, J. Goree, B. Feuerbacher and D.Möhlmann, (1994), Plasma crystal: Coulomb crystallization in a dusty plasma,Phys. Rev. Lett. 73, 652.

Thomas, H., and G. E. Morfill, (1996), Melting dynamics of a plasma crystalNature (London) 379, 806.

Thomas, H., G. E. Morfill and V. N. Tsytovich, (2003), Complex Plasmas: III.Experiments on strong coupling and long-range correlations, Plasma Phys. Rep.29, 895.

Thompson, C., A. Barkan, N. D’Angelo and R. L. Merlino, (1997), Dust acousticwaves in a direct current glow discharge, Phys. Plasmas 4, 2331.

Tsytovich, V. N., (1989), Description of collective processes and fluctuations inclassical and quantum plasmas, Sov. Phys. Usp 32, 911.

BIBLIOGRAPHY 163

Tsytovich, V. N., U. de Angelis and R. Bingham, (1989), Transition scattering ofwaves on charged dust particles in a plasma, J. Plasma Phys. 42, 429.

Tsytovich, V. N., (1995), Lectures on Nonlinear Plasma Kinetics, Springer, Berlin.

Tsytovich, V. N., (1997), Dust plasma crystals, drops and clouds, Sov. Phys. Usp.40, 53.

Tsytovich, V. N., (1998), One dimensional self-organized structures in dusty plas-mas, Aust. J. Phys. 51, 763.

Tsytovich, V. N., and U. de Angelis, (1999), Kinetic theory of dusty plasmas. I.General approach, Phys. Plasmas 6, 1093.

Tsytovich, V. N., and U. de Angelis, (2000), Kinetic theory of dusty plasmas. II.Dust - plasma particle collision integrals, Phys. Plasmas 7, 544.

Tsytovich, V. N., and U. de Angelis, (2001), Kinetic theory of dusty plasmas. III.Dust - dust collision integrals, Phys. Plasmas 8, 1141.

Tsytovich, V. N., U. de Angelis and R. Bingham, (2001), Low frequency responsesand wave dispersion in dusty plasmas, Phys. Rev. Lett 87, 185003.

Tsytovich, V. N., and U. de Angelis, (2002), Kinetic theory of dusty plasmas. IV.Distribution and fluctuations of dust charges, Phys. Plasmas 9, 2497.

Tsytovich, V. N., U. de Angelis and R. Bingham, (2002), Low frequency responses,waves and instabilities in dusty plasmas, Phys. Plasmas 9, 1079.

Tsytovich, V. N., G. E. Morfill and H. Thomas, (2002), Complex Plasmas: I.Complex plasmas as unusual state of matter, Plasma Phys. Rep. 28, 623.

Tsytovich, V. N., and K. Watanabe, (2003), Universal instability of dust-ion soundwaves and dust acoustic waves, Contrib. Plasma Phys. 43, 51.

Tsytovich, V. N., and U. de Angelis, (2004), Kinetic theory of dusty plasmas. V.The hydrodynamic equations, Phys. Plasmas 11, 496.

Tsytovich, V. N., and G. E. Morfill, (2004), Non-linear collective phenomena industy plasmas, Plasma Phys. Control. Fusion 46, B527.

Tsytovich, V. N., G. E. Morfill and H. Thomas, (2004), Complex Plasmas: IV.Theoretical approaches to complex plasmas and their application, Plasma Phys.Rep. 30, 816.

Tsytovich, V. N., U. de Angelis, A. V. Ivlev and G. E. Morfill, (2005), Kinetictheory of partially ionized complex (dusty) plasmas, Phys. Plasmas 12, 082103.

Tsytovich, V. N., (2006), New paradigm for plasma crystal formations, J. Phys.A: Math. Gen. 39, 4501.

164 BIBLIOGRAPHY

Tsytovich, V. N., (2007), The development of physical ideas concerning the in-teraction of plasma flows and electrostatic fields in dusty plasmas, Phys. Usp 50,409.

Tsytovich, V. N., G. E. Morfill, S. V. Vladimirov and H. M. Thomas, (2008),Elementary physics of complex plasmas, Lect. Notes Phys. 731, Springer, Berlin,Heidelberg.

Uddholm, P., (1983), Chemical contributions to the fluctuation spectrum in aplasma, Phys. Scr. 28, 625.

Varma, R. K., P. K. Shukla and V. Krishan, (1993), Electrostatic oscillations inthe presence of grain-charge perturbations in dusty plasmas, Phys. Rev. E 47,3612.

Williams, R. H., and W. R. Chappell, (1971), Microscopic theory of density fluc-tuations and diffusion in weakly ionized plasmas, Phys. Fluids 14, 591.

Whittet, D. C. B., (2002), Dust in the Galactic Environment (Series in Astronomyand Astrophysics), Institute of Physics Publishing, Bristol.

Zagorodny, A. G., P. P. J. M. Schram and S. A. Trigger, (2000), Stationary velocityand charge distributions of grains in dusty plasmas, Phys. Rev. Lett. 84, 3594.

Zakrzewski, Z., and T. Kopiczynski, (1974), Effect of collisions on positive ioncollection by a cylindrical Langmuir probe, Plasma Phys. 16, 1195.

Zobnin, A. V., A. P. Nefedov, V. A. Sinelshchikov and V. E. Fortov, (2000), Onthe charge of dust particles in a low-pressure gas discharge plasma, JETP 91, 483.

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