Eur. Phys. J. D (2014) 68: 79DOI: 10.1140/epjd/e2014-50035-5

Regular Article

THE EUROPEANPHYSICAL JOURNAL D

Vlasov equation for photons and quasi-particles in a plasma�

Jose Tito Mendoncaa

IPFN, Instituto Superior Tecnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

Received 13 January 2014 / Received in final form 12 February 2014Published online 15 April 2014 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2014

Abstract. We show that, in quite general conditions, a Vlasov equation can be derived for photons ina medium. The same is true for other quasi-particles, such as plasmons, phonons or driftons, associatedwith other wave modes in a plasma. The range of validity of this equation is discussed. We also discussthe Landau resonance, and its relation with photon acceleration. Exact and approximate expressions forphoton and quasi-particle Landau damping are stated. Photon and quasi-particle acceleration and trappingis also discussed. Specific applications to laser-plasma interaction, and to magnetic fusion turbulence, areconsidered as illustrations of the general approach.

1 Introduction

The Vlasov equation was first proposed in 1938 [1] andplays a central role in plasma physics theory. This equa-tion describes the evolution of the distribution function ofcharge particles in a plasma, and neglects short range col-lisions. The particle distributions therefore evolve underthe influence of the plasma electromagnetic field, as deter-mined by the Maxwell’s equations, where the current andcharge distributions are determined by the particle popu-lations. For this reason, the Vlasov equation is intrinsicallynonlinear. The main properties of this equation, and itsdifferent applications, are well known and have been de-scribed in many textbooks (see for instance, [2–4]), andwill not be addressed here. What is less known is that theVlasov equation can also be used to describe the evolu-tion of the electromagnetic wave spectrum in a plasma,as well as the turbulent spectrum of various wave modes,such as plasma waves, ion-acoustic waves and drift waves.All these turbulent spectra can be described as a gasof different kinds of quasi-particles, photons, plasmons,phonons or driftons. Chargeons (for dust-charge fluctua-tions) [5] and alfvenic quasi-particles (for Alfven wave tur-bulence) [6] could also be included in the class of plasmaquasi-particles. It is the aim of this paper to discuss theVlasov equation for such quasi-particles.

Photon kinetic theory took shape in the 90’s of thelast century [7], due to a confluence of work developed in-dependently in plasma physics and nonlinear optics. Thecommon features were associated with the use of intenselaser pulses in these two areas. Due to the large spectralwidth of these intense and ultra-short pulses, the photon

� Contribution to the Topical Issue “Theory and Applica-tions of the Vlasov Equation”, edited by Francesco Pegoraro,Francesco Califano, Giovanni Manfredi and Philip J. Morrison.

a e-mail: titomend@ist.utl.pt

phase was not particularly relevant for the understandingof the observed phenomena. The laser pulses could thenbe seen as a beam of quasi-particles interacting with themedium, and nonlinearly changing its properties. It wassoon realized that this photon kinetic theory was formallysimilar to the turbulence theory of magnetized plasmas,with the aim of understanding anomalous transport [8,9].This converged into a unified view of plasma turbulenceand quasi-particle kinetics, where the Vlasov equation forphotons and quasi-particles plays a central role. From sucha view it emerged the understanding that photons andquasi-particles behave in many ways as charged particlesin a plasma, in the sense that Landau damping and trap-ping effects usually associated with electrons and ions canalso occur for quasi-particles.

Here we mainly focus on the dynamical and kinetic as-pects of quasi-particle physics which are directly relevantto the Vlasov description of laser pulses and plasma turbu-lence. We also stress the limitations of the quasi-particleVlasov equation, and show that this equation can be ob-tained from a more general wave kinetic equation, when wetake the geometric optics limit. It means that the Vlasovequation excludes diffraction and other effects associatedwith the undulatory nature of these quasi-particles. Sucha limitation is however also an advantage, because it al-lows us to understand many relevant aspects of plasmaturbulence and laser plasma interactions. In particular, itallows us to develop a quasi-linear theory of turbulence,based on resonant quasi-particle Landau damping, andto understand instability saturation mechanisms due toquasi-particle trapping.

The content of the paper is the following. In Sec-tion 2 we consider simple dynamical processes of photonsin a non-stationary plasma. This allows us to introducethe basic concepts of photon effective mass and charge,and to discuss photon trapping by electron plasma waves.

Page 2 of 9 Eur. Phys. J. D (2014) 68: 79

By assuming a large number of photon trajectories, wethen obtain a Liouville equation for photons, which is for-mally analogous to the Vlasov equation. An importantpoint emerging from this simple approach is that a forcedue to plasma waves, and plasma density perturbationsin general, acts on the photons. If the force is static, itonly changes the photon momentum and leads to a sim-ple refraction. When however the force is time-dependentit changes the photon frequency, a process that can becalled time-refraction. This temporal extension of the re-fraction concept was first observed in plasmas and calledflash ionization [10–12]. Non-stationary plasmas were dis-cussed in detail by [13]. It has also received some attentionin nonlinear optics [14], as well as in quantum optics [15].

We then consider kinetic photon processes in Section 3,where the Wigner-Moyal procedure is delineated. Such aprocedure allows us to derive a general photon kineticequation, which is exactly equivalent to the wave equa-tion, and which retains all the undulatory aspects of theelectromagnetic radiation. It describes the evolution ofthe photon quasi-distribution, which is a Wigner functiontending to the photon number distribution in the limit ofgeometric optics. In this limit, the photon kinetic equationreduces to a Vlasov equation. It is relevant to notice thatthe force acting on the photons also depends on the photondensity (or electromagnetic wave energy), due to pondero-motive effects that change the amplitude and shape of theplasma density perturbations. This provides intrinsic non-linear properties of the photon Vlasov equation, similar tothose existing for electron or ion Vlasov equations.

An important consequence of the photon kinetic de-scription is that photons can be seen as an additionalparticle species in a plasma, on the same ground as elec-trons and ions. We can then study the dispersion relationof electron plasma waves in the presence of electromag-netic turbulence (as described by the photon number dis-tribution), as discussed in Section 4. In addition to theusual electron susceptibility, we then get an expressionfor a similar photon susceptibility, which also contributesto the dispersion relation. This photon susceptibility con-tains a new Landau resonance, associated with photonswith group velocity equal to the phase velocity of electronplasma waves. Such a resonance is physically relevant tothe case of electron plasma waves with long wavelengthsand supra-thermal phase velocities, as those excited byshort laser pulses [16]. These electrostatic waves will notbe Landau damped by electrons, due to their large phasevelocity, but they can eventually be damped by photonLandau damping, a process first described in [17] using theVlasov equation approach, and then generalized in [18] byincluding photon recoil effects. Another interesting con-sequence of this Vlasov description of the photon gas isthat we can develop a quasi-linear theory, leading to aFokker-Planck equation for the average photon distribu-tion on long time scales. Quasi-linear diffusion of photonsin wavelength or wavenumber space will then take place,built up by an overlap of Landau damping processes as-sociated with a spectrum of electron plasma waves. Thisopens the door to a deeper understanding of the energy

exchanges between photon and plasmon generic spectra ina turbulent plasma. More generally, this can lead to a newparadigm of energy transfer between short and long wave-length scale perturbations in a medium. Sections 2 to 4are partly based on previous reviews [7,19], but include afresh view and new details.

The discussions of these sections concentrated on elec-tromagnetic radiation (seen as a photon gas), in isotropicplasmas. But the wave kinetic approach, based on theWigner-Moyal procedure, can be extended to a genericspectrum of waves or quasi-particles in a plasma. Thisis discussed in Section 5, where a simple but generic ap-proach allows us to understand such a generalization. Theresulting exact wave kinetic equation, reduces again to aVlasov equation in the geometric optics limit. Useful ex-amples of application for magnetized plasmas are relatedwith lower hybrid current drive, microwave reflectrome-try, and to zonal flow excitation by drift wave turbulencein tokamaks. The first of these examples is discussed inSection 6. Finally, in Section 7, we state the conclusionsand make the bridge with other areas of physics, wherea similar theoretical approach can be used, such as ultra-cold matter and Bose Einstein condensates [20], neutrinophysics and quantum vacuum processes [21].

2 Photon rays

We start by using a simple and intuitive approach lead-ing to the Vlasov equation for photons. For simplicity weconsider electromagnetic radiation propagating in a non-magnetized plasma. A more rigorous approach, the gen-eralization to quasi-particles, and the inclusion of magne-tized plasmas will be presented later. In a slowly varyingmedium, the electromagnetic wave propagation can be de-scribed by simple ray equations [22,23]. They are the basisof geometric optics. These ray equations can be written incanonical form as [24]

drdt

= vk ≡ ∂ω

∂k,

dkdt

= Fk ≡ −∂ω

∂r, (1)

where r and k are the photon position and the wavevec-tor, and the Hamiltonian ω ≡ ω(r,k, t) is the photonfrequency, as determined by the dispersion relation ω =(k2c2 + ω2

p)1/2. The quantities vk and Fk are the photon(group) velocity and the force acting of the photon. Ina non-stationary plasma, the electron plasma frequencyωp =

√e2n/ε0m is a function of position and time. Here

we adopt the standard notation where n is the electronplasma density, and e and m are the electron charge andmass. These ray equations are only valid in a slowly vary-ing medium, when the scale of variation of ωp is muchslower than the photon period 1/ω and much larger thanthe photon wavelength, 1/k.

In a static but inhomogeneous plasma, the photon en-ergy (or frequency ω) is a constant of motion, and thephoton momentum (or wavevector k) changes in time dueto the usual refraction process, as described by the aboveray equations. This is the basis of the usual geometric

Eur. Phys. J. D (2014) 68: 79 Page 3 of 9

optics approach. However, if the plasma density changesin time, the photon energy is not conserved, as a resultof energy exchanges with the background medium. Thefrequency then evolves in time as

dω

dt=

∂ω

∂t≡ 1

ω

e2

ε0m

∂n

∂t. (2)

This process is sometimes called time refraction [7], be-cause it is a direct extension of the usual refraction intothe temporal domain. The resulting frequency shift is animportant experimental signature of its occurrence. Moreinteresting for experimental purposes is the case when theelectron density is both space and time dependent, andthe processes of refraction and time-refraction occur si-multaneously. The resulting frequency shift can then besignificantly enhanced, a phenomenon also called photonacceleration [25].

A favorable configuration is provided by a relativisticionization front, as first considered theoretically by refer-ences [26,27]. Experiments were performed in microwavesby reference [11], and in optics by references [12,28]. Ion-ization fronts can be described by a density perturbationof the form n(r, t) = n0H(r − vf t), where H(x) is theHeaviside function, vf is the front velocity. This simplemodel for the ionization front neglects front width andshape, which however leads to the same final frequencyshift, as stated next. It also neglects photon dispersion inthe neutral gas outside the front, which is a good approx-imation for photon frequencies far away from any opticalatomic transition of the background atoms. For photonswith initial frequency ωi, propagating in the direction ofthe front, the frequency shifts are given by [7]

Δω ≡ ωf − ωi =ω2

p0

2ωi

1(1 ± βf )

, (3)

where ωp0 is the plasma frequency inside the ionized gas,and βf = vf/c. This is valid for perpendicular incidenceand the plus and minus sign pertain for counter- and co-propagation. This shows that, for βf � 1 very large fre-quency shifts, proportional to the ionization front densityand to the relativistic factor γ2

f = (1 − β2f )−1/2, can be

attained. Moving plasma mirrors are also relevant to highharmonic generation of intense laser pulses [29,30].

For density perturbations moving without significantshape deformation with constant velocity vf , such as ion-ization fronts and electron plasma waves, the above pho-ton canonical equations show that a constant of the mo-tion exists, as defined by

ω′ =√

k2c2 + ωp(s) − vf · k, (4)

with s = r−vf t. Notice that both the photon momentumand the frequency vary with s. The existence of this invari-ant provides a very useful information, from which we canestablish the trapping condition for photons interactingwith an electron plasma wave

ω′+ < ω′ < ω′

−, ω′± =

ωp0

γf

√1 ± (n/n0), (5)

where n is the wave amplitude. These trapped photonsoscillate in the potential potential well, with a bounce fre-quency ωb defined by [7]

ωb =kpc

γf

√n

2(n0 − n). (6)

This photon bounce frequency is very similar to the elec-tron bounce frequency in the same wave, as given byωbe = ωp0

√n/n0.

Such a strong analogy between the photon and elec-tron behavior, points to the existence of an effective massmph and an effective electric charge qph of the photon.The effective mass can readily be derived from the disper-sion relation, as mph = ωp�/c2. This was first noticed byAnderson [31], who already in 1957 had suggested the exis-tence of a Higgs mechanism to provide mass to elementaryparticles. The effective photon charge was proposed in ref-erences [32,33]. It describes the repulsion of electrons byelectromagnetic radiation due to the ponderomotive force,and is given by qph = −(e�/2mω)(ωp/vk)2. Using these re-sults we can define the characteristic charge to mass ratiofor the photon, as

(qph

mph

)=

12

( e

m

) c2

v2k

ωp

ω. (7)

It is interesting to notice that, although the effective pho-ton charge is much smaller than the electron charge e, thischarge to mass ratio is of the order of the electron ratio(e/m).

The analogies between photons (and other quasi-particles) with electrons and ions will be a central issue inthis paper. They ultimately justify the use of Vlasov equa-tions to describe photons and plasma quasi-particles. Butwe will also be aware of fundamental differences motivatedby the undulatory nature of the photon. Such differenceshowever tend to vanish when we described charged parti-cles as quantum particles, as recently shown for quantumplasmas [34].

Photon trapping was experimentally demonstrated byreference [35], and also indirectly by reference [36]. Sucha mechanism could eventually be used to produce single-cycle optical pulses [37]. Stochastic and Fermi accelera-tion processes can also be studied with this dynamicalapproach [7]. It can easily be extended to magnetized plas-mas, as discussed by reference [38].

If, instead of a single ray, we consider the evolution ofa large number N � 1 of photon ray trajectories, we canuse a discrete photon distribution N(r,k, t), defined as

N(r,k, t) =N∑

j=1

δ(r − rj(t))δ(k − kj(t)), (8)

where rj(t) and kj(t) are solutions of the ray equa-tions (1), corresponding to the different trajectories. TheLiouville theorem is therefore valid in the 6-dimensionalphoton phase space (r,k), which allows us to write the

Page 4 of 9 Eur. Phys. J. D (2014) 68: 79

evolution equation

d

dtN(r,k, t) = 0 ,

d

dt≡ ∂

∂t+ vk · ∇ + Fk · ∂

∂k, (9)

where d/dt is the total time derivative. This equation canbe seen as a Liouville equation. It is formally analogous tothe Vlasov equation. But such analogy should not make usforget that the Liouville equation describes a discrete dis-tribution of N -particles, whereas the Vlasov equation ap-plies to a regular 1-particle distribution function. A moreconvincing derivation of the photon evolution equation,which retains the undulatory properties of the photon,will be described next.

3 Photon kinetics

Let us consider a more convincing approach to photon ki-netic equations, by assuming electromagnetic wave prop-agation in isotropic plasmas, as described by:

[

∇2 − 1c2

∂2

∂t2− ω2

p

c2

]

E = 0, (10)

where E is the wave electric field, and the electron plasmafrequency is assumed to vary in space and time, due to theexistence of plasma density perturbations, n(r, t). It wasshown by references [39,40] that a photon kinetic equa-tion can be derived by introducing the auto-correlationfunction, as:

K(r, s, t, τ) = E(r+s/2, t+τ/2)·E∗(r−s/2, t−τ/2). (11)

Using a simple generalization of the Wigner-Moyal proce-dure [41–44], we define the double Fourier transformation

W (ω,k; r, t)=∫

ds

∫dτ K(r, s, t, τ) exp(−ik · s+iωτ).

(12)This quantity can be called quasi-probability, becauseit tends in the appropriate limit to the photon num-ber density, as discussed below. An exact evolution equa-tion for W (ω,k; r, t) can be derived from the wave equa-tion (10), as:

i

(∂

∂t+

c2kω

· ∇)

W =ω2

p0

n0

∫I(ω,k,q)

dq(2π)3

, (13)

where ω2p0 = (e2n0/ε0m), and n0 is the averaged value of

the electron plasma density. We have used the integral

I(ω,k,q) =∫

n(Ω,q) [W− − W+] exp(iq · r − iΩt)dΩ

2π.

(14)Here, the quantities W± have been defined as W± ≡W (ω±Ω/2,k±q/2). We have also used the Fourier spec-trum of the plasma density fluctuations, such that

n(r, t) =∫

dq(2π)3

∫dΩ

2πn(Ω,q) exp(iq · r− iΩt), (15)

where n(Ω,q) are the Fourier components of the electrondensity fluctuations n(r, t).

This equation can be simplified if plasma density spec-trum reduces to a simple Fourier component, represent-ing an electrostatic wave. In this case, we use n(r, t) =n0 + n cos(q · r− Ωt), where n is the perturbation ampli-tude. The photon kinetic equation (13) becomes

i

(∂

∂t+

c2kω

· ∇)

W = ω2p0

n

n0[W− − W+]

× exp(iq · r − iΩt). (16)

Noting that the photon frequency ω and wavevector k arenot independent, because they locally satisfy a dispersionrelation, we can use a reduced quasi-distribution

W (ω,k, r, t) = Wk(r, t) δ(ω − ωk), (17)

where ωk =√

k2c2 + ω2p(r, t). This is valid for moderate

intensities, but nonlinear corrections can also be intro-duced in the dispersion relation, thus widening the rangeof validity of this reduced function.

Let us now consider the limit of geometric optics. Thislimit is valid when the density fluctuations slowly evolvein space and time, as compared with the photon spaceand time scales 1/k and 1/ωk. The photon kinetic equa-tion (13) is then reduced to the Vlasov equation

(∂

∂t+ vk · ∇ + Fk · ∂

∂k

)Wk = 0, (18)

where Fk is the force introduced in equation (1). Thisis formally identical to the single particle Liouville equa-tion (9), but it now involves a smooth distributionWk(r, t). This quantity is proportional to the photon num-ber density distribution Nk(r, t) and the photon Vlasovequation (18) states that the photon number density is aninvariant

d

dtNk(r, t) = 0. (19)

This can be simply related to the electromagnetic energydensity Uk, by using

Nk(r, t) =Uk

�ωk=

ε04�ω3

k

Wk(r, t). (20)

In this geometric optics limit, the photon quasi-probabilityreduces to a distribution function, and the photon numberdensity Nk is a real positive number, as it should be.

4 Photon Landau damping

An important feature of the Vlasov description of colli-sionless plasmas, is the inclusion of self-consistent elec-tromagnetic fields, which are determined by the distri-bution functions of the different charged particle speciesin the medium. Similar self-consistent fields exist in theVlasov equation for photons, because the force acting on

Eur. Phys. J. D (2014) 68: 79 Page 5 of 9

the photon distribution also consistently depends on suchdistribution. This new self-consistent field, which is not anelectromagnetic field, but a force field acting on photons,is considered here, and its influence on the plasma wavedispersion relation is discussed.

The force acting on the photons depends on the gra-dient of the electron plasma density, as shown by equa-tion (1). We therefore need to determine its value, andthe way it depends on the photon distribution. For thatpurpose, we assume immobile ions, and describe the elec-trons with the Vlasov equation

(∂

∂t+ v · ∇ +

Fe

m· ∂

∂v

)fe = 0, (21)

where fe ≡ fe(r,v, t), v is the electron velocity, and theforce acting on the electrons Fe contains a ponderomotiveforce term, due to the presence of electromagnetic radia-tion. We can define

Fe = −∇φ − e2

2mω20

∇I(r, t), (22)

where ω0 is the average frequency of the electromagneticwave spectrum. The electrostatic potential is determinedby the Poisson’s equation, as usual, as:

∇2φ =e

ε0

(∫fe(r,v, t)dv − n0

), (23)

where n0 is the equilibrium density. In the force Fe wehave also used the integral over the photon distribution

I(r, t) =∫

Wk(r, t)dk

(2π)3. (24)

The second term in equation (22) is the total radiationpressure acting on the electrons, which couples the plasmadensity with the photon field. Let us now assume an elec-trostatic wave perturbation, evolving as exp(ik · r − iωt).To avoid misunderstandings we use (ω′,k′) to character-ize photon modes. The dispersion relation can be derivedusing the standard perturbative analysis, and leads to thefollowing result

1 + χe(ω,k) + χph(ω,k) = 0, (25)

where χe(ω,k) is the usual electron susceptibility, de-fined by:

χe(ω,k) =ω2

p

k2

∫∂f(v)/∂v

(v − ω/k)dv, (26)

where v = v · k/k is the parallel electron velocity, andf(v) is here the equilibrium one-dimensional electron dis-tribution. On the other hand, the photon susceptibilityχph(ω,k) is determined by [18]

χph =e2k

2m2

∫(Gp−k/2 − Gp+k/2)

(p − p0)dp, (27)

where p = k′ · k/k is the parallel photon momentum,and G is the equilibrium one-dimensional photon quasi-probability, defined by:

G(p) =∫

W (p,k′⊥)

(∂u/∂p)0dk′

⊥(2π)2

, (28)

where u is the parallel photon velocity. The derivative inthe denominator is calculated for the resonant situation ofp = p0, such that the parallel photon velocity equals thephase velocity of the electron plasma wave u(p0) = ω/k.The presence of photons brings a small correction to thedispersion relation of the electron plasma waves, as shownbelow, but more interestingly, it can significantly changethe total wave damping coefficient γ. The result is:

γ = γe + γph, (29)

where γe is the well known electron Landau dampingcoefficient

γe =πω2

p

2k2

(∂f

∂u

)

u=ω/k

, (30)

and γph is the photon Landau damping coefficient [18]

γph = −π

4ωp0

e2k

m2[G(p0 − k/2)− G(p0 + k/2)] . (31)

It is obvious from these expressions that, when electronLandau damping is negligible, electron plasma waves canstill be damped by photons. In equilibrium, the photonequilibrium distribution G is a Bose-Einstein distribution,and the quantity under brackets will be positive, leading towave damping. In the opposite case, when G(p0 + k/2) >G(p0−k/2), instability can occur. This is relevant to laserplasma interaction, where a laser pulse can be associatedwith a beam of quasi-particles. We then have photon-beamplasma instability, similar to electron-beam instabilities.

By comparing the expressions for the electron andphoton Landau damping coefficients, it becomes obviousthat the photons are not exactly point-like particles, andthat their undulatory nature implies the existence of re-coil effects, leading to the replacement of the derivative inequation (30) by a population difference in equation (31).These recoil effect is usually associated with quantum par-ticles [34], but we can see from here that it is indeed relatedwith the undulatory properties of photons. Photon recoilis due to momentum loss (or gain) by emission (or absorp-tion) of electron plasma waves by photons at the Landauresonance. However the difference vanishes in the geomet-ric optics limit, when recoil effects become negligible andk � p ∼ k′

0. We can then use the expansion

G0(p0 ± k/2) � G0(p0) ± k

2

(∂G

∂p

)

0

, (32)

and photon Landau damping becomes

γ =π

8ωp0

e2k2

m2

(∂G

∂p

)

0

. (33)

Page 6 of 9 Eur. Phys. J. D (2014) 68: 79

This expression can be directly derived by using theVlasov description for photons [7,17]. Let us now examinethe photon-beam instability. We can use the equilibriumphoton distribution

W (k′) = (2π)3W0δ(k′ − k′0). (34)

The above susceptibilities become [18]

χe = − (ω2p + 3k2v2

the)ω2

, (35)

and

χph =e2k2

2m2

ω2p

ω2

W0k(v′+ − v′−)(ω − kv′+)(ω − kv′−)

. (36)

We have used v′± = v′(k′0 ± k/2). In the quasi-classical

limit of k � k′0, the dispersion relation for the photon

beam plasma system can be approximated by

χph =ω2

p

ω2

e2k4

2m2

W0c2

ω′0

1(ω − kv′0)2

, (37)

with v′0 ≡ v′(k′0). This result could have been directly

derived from the Vlasov equation (18). We see that, in thisapproximation, the presence of a photon beam is similar tothat of an electron beam. In this case, we can also have abeam instability, excited by the photon beam. Assumingthe electron Landau damping is negligible, γe � 0, fora double resonant condition ω = kv′0 � ωp, we get themaximum instability growth rate

γ =√

32

ωp

(e2k4

4m2

W0c2

ω′0ω

2p

)1/3

. (38)

This is proportional to the cubic root of the photon den-sity W0, which is typical of a beam instability. These coldbeam instabilities appear to be much stronger than the ki-netic beam instability associated with an inverse Landaudamping [17]. In the usual language of plane wave prop-agation in plasmas, these photon beam effects would beclassified as Raman stimulated scattering of the transverseelectromagnetic wave.

5 Wave kinetics

The above results can be generalized to the case of ageneric quasi-particle in a plasma. This was shown, forinstance, in references [45,46], but an alternative and sim-pler approach is considered here. We start from the waveequation

[∇× (∇×) +

1c2

∂2

∂t2

]E = −μ0

∂J∂t

, (39)

where E is the wave electric field. The conductive cur-rent J can be defined in a non-homogeneous and non-stationary plasma, by:

J(r, t) =∑

α

∫dt′

∫dr′ ¯σα(r, r′, t, t′) ·E(r′, t′), (40)

where α = i, e refers to the particle species in the plasma(electrons and ions), and ¯σα is the conductivity tensor ofeach species. For a given field mode E ∝ exp(ik · r− iωt),when the scales of inhomogeneity and non-stationary aremuch larger than 1/k and 1/ω, these equations lead to thewell known dispersion relation

D(ω,k) ≡ e∗ω · ¯D(ω,k) · eω = 0, (41)

for a field eigen-mode with unit polarization vector eω,where ¯D(ω,k) is the dispersion tensor as defined by:

¯D(ω,k) =(NN − N2¯1 + ¯κ

), ¯κ = ¯1+

i

ωε0

∑

α

¯σα. (42)

Here we have used the normalized wavevector N =(c/ω)k, and the linear conductivity tensors ¯σα. This isonly valid for linear plasma response and local homogene-ity. In order to generalize this result we can assume linearor nonlinear deviations from such an ideal situation, duefor instance to the existence of space and time dependentdensity perturbations for the different species δnα(r, t),which are added to the equilibrium plasma density n0.Fluctuations in the static magnetic field δB0(r, t) couldalso be included. In this case, the unperturbed result (4)is replaced by:

[D

(ω + i

∂

∂t,k + i∇

)+ δD(r, t)

]E = 0, (43)

where E is the field mode amplitude, and

δD(r, t) =∑

α

(∂D∂nα

)δnα(r, t) +

∂D∂B0

· δB0. (44)

Expanding such an expression around the local linear dis-persion (41), we obtain the evolution equation for the waveamplitude as:

(∂

∂t+ vk · ∇

)E = iV (r, t)E, (45)

where the mode group velocity vk, and the perturbationpotential V (r, t), are defined by:

vk = − (∂D/∂k)(∂D/∂ω)

, V (r, t) =δD(r, t)(∂D/∂ω)

. (46)

This is valid for a local mode approach, where the wavemode is treated locally as in a homogeneous plasma. Wenow introduce the field autocorrelation function, as inequation (11). Going to the Wigner function W (r, t, ω,k),as the double Fourier transform of this auto-correlationfunction, we obtain:

i

(∂

∂t+ vk · ∇

)W =

∫dq

(2π)3

∫dΩ

2πV (Ω,q)

× [W− − W+] exp(iq · r− iΩt),(47)

Eur. Phys. J. D (2014) 68: 79 Page 7 of 9

where W± = W (ω±Ω/2,k± q/2), and V (Ω,q) is the fre-quency and wave vector spectrum of the perturbation po-tential associated with the fluctuations of the backgroundplasma. As before, it is useful to consider the reducedWigner function defined by equation (17), but now re-ferring to a generic quasi-particle quasi-distribution. Theevolution equation of this reduced function will be for-mally identical with the above equation, but where W isreplaced by Wk.

As in the above discussion of photon kinetics, it is nowuseful to take the limit of geometric optics expansion, validwhen the scale length of the background plasma densityis much larger than the wavelength of the wave modesdescribed by the quasi-distribution function Wk, or |k| �|q|. We then have

W±k � Wk ± q

2· ∂Wk

∂k+

12

(q2· ∂

∂k

)2

× Wk ± 13!

(q2· ∂

∂k

)3

Wk. (48)

Replacing this in the Wigner-Moyal equation for quasi-particles, we obtain an approximate kinetic equation forrays with distribution function Wk, such that(

∂

∂t+vk · ∇−∇V (r, t) · ∂

∂k+

13×23

∇3V · ∂3

∂k3

)Wk =0.

(49)This is an improved form of the kinetic equation for raytrajectories, where the diffraction effects associated withthe wave character of these physical objects is taken intoaccount (last term, with a third order derivative). A sim-ilar approach has been recently proposed for a relativis-tic quantum plasma [47]. When diffraction effects are ne-glected, this equation reduces to a Vlasov equation forquasi-particle, as given by:

d

dtWk(r,k, t) = 0, (50)

with the total time derivative defined as:

d

dt≡ ∂

∂t+

drdt

· ∇ +dkdt

· ∂

∂k. (51)

Notice that the quasi-particle ray equations are now de-fined by:

drdt

= vk,dkdt

= −∇V (r, t), (52)

which generalize the ray equations (1) for arbitrary modesin a plasma. As noted before, an additional ray equationshould be added, which is usually ignored in ray tracingcodes. It describes the frequency shift along the ray tra-jectories, due to the temporal variation of the medium,and can be stated as

dω

dt=

∂

∂tV (r, t). (53)

This generalizes equation (2) for an arbitraryquasi-particle.

6 Magnetically confined plasmas

Ray equations are currently used in magnetized plasmasto study, for instance, lower hybrid current drive and mi-crowave reflectometry. In these two examples photon ki-netics and photon Vlasov equation can be applied. Let usillustrate the above discussion with the case of LHCD. Forlower-hybrid waves, we can use the dispersion relation

D(ω,k) = 1 − k2⊥ε⊥(ω) − k2

‖ε‖(ω) = 0, (54)

where k = (k⊥, k‖), with perpendicular and parallel com-ponents with respect to the static magnetic field B0, andthe dielectric functions are:

ε⊥(ω) = 1 +ω2

pe

ω2ce

− ω2pi

ω2, ε‖(ω) = −ω2

pe

ω2� 1. (55)

This corresponds to the so-called slow wave solution. Simi-larly, we could have considered the fast wave solution. Thisallows us to write

δD(r, t) = −k2⊥

ω2pe

ω2ce

1n0

δne(r, t). (56)

The resulting potential perturbation takes the form

V (r, t) =ω3

ω2ce

1(me/mi + k2

‖/k2⊥)

1n0

δne(r, t). (57)

On the other hand, the wave group velocity vk associatedwith these slow LH waves is:

vk = − ω3

k2⊥ω2

pi + k2‖ω

2pe

[k‖ε‖(ω)e‖ + k⊥ε⊥(ω)

]. (58)

For a density perturbation, δne(r, t) = n(q) exp(iq · r −iΩt), associated for instance to a given drift wave modewith frequency Ω = Ω(q), satisfying the drift dispersionrelation with wavevector q, we have the correspondingperturbation in the ray distribution δWk = Wk(q) exp(iq·r− iΩt). Using the kinetic equation (49), we obtain

Wk(q) = − α(ω)(Ω − vk · q)

(q · ∂W0

∂keff

)n(q)n0

, (59)

where W0 ≡ W0(k) is the equilibrium distribution, validin the absence of the drift mode. We have also used thequantity

α(ω) =ω3

ω2ce

1(me/mi + k2

‖/k2⊥)

, (60)

and defined an effective differential operator

∂

∂keff≡

[

1 +1

3 × 23

(q · ∂

∂k

)2]

∂

∂k. (61)

Such an operator includes the lowest order diffraction ef-fects associated with the wave character of the LH rays.

Page 8 of 9 Eur. Phys. J. D (2014) 68: 79

Table 1. Examples of kinetic quasi-particle processes.

Quasi-particle Slow wave Main relevancePhotons plasma waves photon Landau dampingPhotons ionization front photon accelerationPlasmons ion sound waves plasmon beam instability

Alfvenic q-p ion sound waves anomalous ion heatingDriftons zonal flows L-H transition

This allows us to evaluate, for a given ray distributionF0, the importance of diffraction corrections to ray trac-ing. Neglecting the diffraction term, the LH quasi-particlesstart to behave as real particles, and the Vlasov descrip-tion is recovered.

Another fundamental aspect of magnetized plasmas isthe anomalous transport, and the associated L (low con-finement) to H (high confinement)-mode transitions. Thisis crucial for the understanding of particle and energytransport in the plasma of tokamak discharges, and tothe success of magnetic fusion research. Here again theVlasov equation approach can be used, as discussed inreferences [48,49].

7 Conclusions

Here we have discussed the importance of the Vlasov equa-tion for photons and quasi-particles in a plasma, by show-ing its range of valid and its main properties. We haveshown that this equation can be derived from an exact ki-netic equation, if we take the limit of geometric optics.This shows a significant difference with respect to theVlasov equation for particles, which is valid when shortrange particle collisions are neglected. It should howeverbe pointed out that the photon (or quasi-particles) ana-logue to particle collisions would be the nonlinear couplingbetween photon modes, which was ignored in our analysis.Such a neglect can be justified in the case of broadbandturbulent spectrum, where nonlinear coupling usually be-comes negligible due to phase mixing.

The Vlasov description of generic plasma turbulence(photons, plasmons or other quasi-particles) leads to theappearance of a quasi-particle susceptibility, which con-tains a Landau resonance. This resonance leads to photonand quasi-particle Landau damping of long wavelengthmodes in the plasma. It also establishes a privileged chan-nel for energy transfer between long and short scale per-turbations in the plasma. This was shown relevant in dif-ferent physical processes, such as wakefield generation byintense laser pulses, or those related with the excitation ofzonal flows by drift wave turbulence in tokamaks. Exam-ples of such processes in plasmas are included in Table 1,in order to illustrate the importance of this wave kineticapproach to plasma turbulence.

The universality of the wave kinetic approach is notlimited to photons and quasi-particles. Neutrinos in aplasma have also been shown to possess an effective chargesimilar to that of photons [32,33,50]. Neutrino Landaudamping of electron plasma waves is also possible [51],and the Vlasov description of collective neutrino plasmas

has been used to consider neutrino beam instabilities.This could provide a possible mechanism for supernovaexplosions [52].

At very short scales, electrons and ions can also beseen as waves, and their undulatory nature is describedby quantum mechanics. This is relevant to quantum plas-mas, where the electron (and eventually the ion) responseto electromagnetic field perturbation can be described bywave kinetic equations similar to those discussed here [34].A simple approach to relativistic quantum plasmas hasbeen proposed, where spin effects were neglected and theVlasov equation for electrons was derived from a Klein-Gordon equation [47]. Finally, the use of Vlasov equationsfor ultra-cold atoms in Bose Einstein condensates, and theatomic Landau damping of Bogoliubov sound waves hasbeen considered [20]. Given the success of the present ap-proach, its universality and robustness, we believe that itwill lead to a global understanding of plasma turbulence,and of many other collective processes in different media.

References

1. A.A. Vlasov, J. Exp. Theor. Phys. 8, 291 (1938)2. A.I. Akhiezer, I.A. Akhiezer, R.V. Polovin, A.G. Sitenko,

K.N. Stepanov, Plasma Electrodynamics (Pergamon Press,Oxford, 1975)

3. D.R. Nicholson, Introduction to Plasma Physics (JohnWiley & Sons, New York, 1983)

4. R. Balescu, Equilibrium and Nonequilibrium StatisticalMechanics (John Wiley & Sons, New York, 1975)

5. J.T. Mendonca, N.N. Rao, A. Guerreiro, Europhys. Lett.54, 741 (2001)

6. J.T. Mendonca, P.K. Shukla, Phys. Plasmas 14, 122304(2007)

7. J.T. Mendonca, Theory of Photon Acceleration (Instituteof Physics, Bristol, 2001)

8. J.T. Mendonca, R. Bingham, P.K. Shukla, Phys. Rev. E68, 016406 (2003)

9. A.I. Smolyakov, P.H. Diamond, V.I. Shevchenko, Phys.Plasmas 7, 1349 (2000)

10. E. Yablonovitch, Phys. Rev. A 10, 1888 (1974)11. R.L. Savage Jr., C. Joshi, W.B. Mori, Phys. Rev. Lett. 68,

946 (1992)12. J.M. Dias et al., Phys. Rev. Lett. 78, 4773 (1997)13. D.K. Kalluri, Electromagnetics of Complex Media:

Frequency Shifting by a Transient Magnetoplasma Medium(CRC, Boca Raton, 1999)

14. A.B. Shvartsburg, Phys. Usp. 48, 797 (2005)15. V.V. Dodonov, A.B. Klimov, D.E. Nikonov, Phys. Rev. A

47, 4422 (1993)16. S. Shiraishi et al., Phys. Plasmas 20, 063103 (2013)17. R. Bingham, J.T. Mendonca, J.M. Dawson, Phys. Rev.

Lett. 78, 247 (1997)18. J.T. Mendonca, A. Serbeto, Phys. Plasmas 13, 102109

(2006)19. J.T. Mendonca, Phys. Scr. 74, C61 (2006)20. J.T. Mendonca, H. Tercas, Physics of Ultra-Cold Matter

(Springer, New York, 2013)21. A. Di Piazza, C. Muller, K.Z. Hatsagortsyan, C.H. Keitel,

Rev. Mod. Phys. 84, 1177 (2012)22. M. Born, E. Wolf, Principles of Optics (Pergamon Press,

London, 1959)

Eur. Phys. J. D (2014) 68: 79 Page 9 of 9

23. M. Kline, I.W. Kay, Electromagnetic Theory andGeometric Optics (Interscience Publishers, New York,1965)

24. J.T. Mendonca, L.O. Silva, Phys. Rev. E 49, 3520 (1994)25. S. Wilks, J.M. Dawson, W.B. Mori, T. Katsouleas, M.E.

Jones, Phys. Rev. Lett. 62, 2600 (1989)26. V.I. Semenova, Sov. Radiophys. Quant. Electron. 10, 599

(1967)27. M. Lampe, E. Ott, J.H. Walker, Phys. Fluids 21, 42 (1978)28. N.C. Lopes et al., Europhys. Lett. 66, 371 (2004)29. S. Gordienko, A. Pukhov, O. Shorokhov, T. Baeva, Phys.

Rev. Lett. 93, 115002 (2004)30. B. Dromey et al., Phys. Rev. Lett. 99, 085001 (2007)31. P.W. Anderson, Phys. Rev. 130, 439 (1973)32. J.T. Mendonca, L.O. Silva, R. Bingham, N.L. Tsintsadze,

P.K. Shukla, J.M. Dawson, Phys. Lett. A 239, 373 (1998)33. J.T. Mendonca, A. Serbeto, P.K. Shukla, L.O. Silva, Phys.

Lett. B 548, 63 (2002)34. F. Haas, Quantum Plasmas, an Hydrodynamical Approach

(Springer, New York, 2011)35. C.D. Murphy et al., Phys. Plasmas 13, 033108 (2006)36. J. Faure et al., Phys. Rev. Lett. 95, 205003 (2005)37. Z.M. Sheng et al., Phys. Rev. E 62, 7258 (2000)38. S. Eliezer, J.T. Mendonca, R. Bingham, P. Norreys, Phys.

Lett. A 336, 390 (2005)

39. N.L. Tsintsadze, J.T. Mendonca, Phys. Plasmas 5, 3609(1998)

40. J.T. Mendonca, N.L. Tsintsadze, Phys. Rev. E 62, 4276(2000)

41. E. Wigner, Phys. Rev. 40, 749 (1932)42. J.E. Moyal, Proc. Camb. Phil. Soc. 45, 99 (1949)43. W.E. Brittin, W.R. Chappell, Rev. Mod. Phys. 34, 620

(1962)44. M. Hillary, R.F. O’Connel, M.O. Scully, E.P. Wigner,

Phys. Rep. 106, 121 (1984)45. J.T. Mendonca, K. Hyzanidis, Phys. Plasmas 18, 112306

(2011)46. J.T. Mendonca, S.M. Benkadda, Phys. Plasmas 19, 082316

(2012)47. J.T. Mendonca, Phys. Plasmas 18, 062101 (2011)48. P.H. Diamond, S.-I. Itoh, K. Itoh, T.S. Hahm, Plasma

Phys. Control. Fusion 47, R35 (2005)49. A. Yoshizawa, S.-I. Itoh, K. Itoh, Plasma and Fluid

Turbulence (Institute of Physics, Bristol, 2003)50. A. Serbeto, L.A. Rios, J.T. Mendonca, P.K. Shukla, Phys.

Plasmas 11, 1352 (2004)51. L.O. Silva, R. Bingham, J.M. Dawson, J.T. Mendonca,

P.K. Shukla, Phys. Rev. Lett. 83, 2703 (1999)52. R. Bingham et al., Plasma Phys. Control. Fusion 46, B327

(2005)