1 lecture notes jagiellonian university krakow 2007 ...th- · conclusions based on these equations...

October 3, 2007 18:24 WorldScientific/ws-b9-75x6-50 LecKra1˙2

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Lecture Notes JagiellonianUniversity Krakow 2007

Material selected from thebook:

Statistical Thermodynamicsand Stochastic Theory ofNonequilibrium Systems

Werner Ebeling and Igor Sokolov

World Scientific Singapore 2005


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Chapter 1

Introduction(selected topics)

1.1 The task of statistical physics

Statistical Physics is that part of physics which de-rives emergent properties of macroscopic matter fromthe atomic properties and structure and the micro-scopic dynamics.Emergent properties of macroscopic matter mean herethose properties (temperature, pressure, mean flows,dielectric and magnetic constants etc.) which are es-sentially determined by the interaction of many par-ticles (atoms or molecules).Emergent means that these properties are typical formany-body systems and that they do not exist (ingeneral) for microscopic systems.The key point of statistical physics is the introduc-tion of probabilities into physics and connecting themwith the fundamental physical quantity entropy.

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4 Introduction (selected topics)

atomistics → statistical mechanics→ thermodynamics and transport

The task is to construct the bridge between micro-physics, i.e.properties and dynamics of atoms andmolecules with macrophysics, i.e. thermodynamics,hydrodynamics, electrodynamics of media

microphysics → statistical physics→ macrophysics

Mmacrophysical properties = properties which aredetermined by the interaction of very many particles(atoms, molecules), in contrast to properties whichare characteristic for one or a few particles.A macrosystem is a many-body system which is de-termined by the basic equations of classical or quan-tum mechanics. Further basic elements of the theoryshould be the laws of interaction of the particles as,e.g. the Coulomb law, the symmetry principle andthe boundary conditions characterizing the macro-scopic embedding.The problem seems to be practically insolvable, notonly for the impossibility to solve more than 1024 cou-pled usual or partial differential equations but alsodue to the incomplete knowledge about the initial


The task of statistical physics 5

and boundary conditions.New concepts are needed and it has to be shown thatprobabilities and/or density operators and entropiesmay be introduced in a natural way if the dynamicsis unstable.The point of view taken in this textbook is mainlyclassical but the quantum-statistical analysis goes inmany aspects in a quite analogous way.Using probabilities instead of trajectorieswe come to a dynamics of probabilities.In the classical case: Liouville equationin the quantum case: von Neumann equation:

microdynamics + probabilities→ Liouville - von Neumann dynamics

Macroscopic properties may be described as meanvaluesHowever: Liouville - von Neumann equation are for-mally completely equivalent to the original dynamicalequations.have still the property of reversibility of the micro-scopic dynamics.Conclusions based on these equations would not bein accord with the second law of thermodynamicsSo, a second step has to be made.



The basic idea is, that macroscopic processes allowand require a coarse-grained description (Gibbs andthe Ehrenfest’s).1) it makes no sense to describe a macroscopic pro-cess in all microscopic detail, since it completely im-possible to observe all the details and to follow thetrajectories of all particles. 2) a coarse - grained de-scription keeps the relevant macroscopic informationsbut neglects the irrelevant microscopic details.it was only recently understood, that the whole con-cept of coarse - graining is intimately connected withthe instabilities of the microscopic trajectories 3) Fi-nally we arrive at equations for the coarse - grainedprobabilities which are irreversible and yield an ap-propriate basis for the macroscopic physics. Theseequations are called kinetic equations or master equa-tions.Our scheme may now be completed in the followingway:

coarse-graining + dynamic instability→ kinetic / master eqs.

The following chapters are aimed to work out thisprogram, with some special attention to the conceptof Brownian motion . But, before going in the details


On history of fundamentals of statistical thermodynamics 7

we would like to have another more historical orientedlook at the development of the basic ideas and atthe historical facts - admittedly with some bias tothe development in Berlin, largely following earlierwork (Rompe et al., 1987; Ebeling & Hoffmann, 1990,1991).

1.2 On history of fundamentals of statistical thermodynamics

19th century pioneers :Sadi Carnot (1796-1832),Robert Mayer (1814-1878),Hermann Helmholtz (1821-1894),William Thomson (1824-1907)Rudolf Clausius (1822-1888).Evidently Mayer was the first who formulated thelaw of energy conservation. His paper is clearly ex-pressing the equivalence of work and heat .Joule came to similar conclusions which were basedon direct measurements concerning the conversion ofwork into heat.Physicists which worked in the middle of the 19thcentury in Berlin: The genius of Hermann Helmholtzdetermined the direction and the common style of re-search1847 he reported to the ”Berliner Physikalische Gesellschaft”,a new society founded by young physicists, about



his research on the principle of conservation of en-ergy. At 27 years of age Helmholtz was working asa military surgeon to a regiment of Hussars in Pots-dam. He could follow his interest in physics only inhis leisure time, since his family’s financial situationdid not allow him to enjoy full-time study. The ex-perimental research which he carried out from thebeginning of the 1840’s in the laboratory of his ad-viser Professor Magnus was primarily devoted to theconversion of matter and heat in such biological pro-cesses as rotting, fermentation and muscular activ-ity. Helmholtz’s insight led him to infer a new law ofnature from the complexities of his measurements onjuices and extracts of meat and muscles. From exper-iments and brilliant generalization emerged the prin-ciple of conservation of energy or what is now calledthe first law of thermodynamics. Neither J.R. Mayernor J.P. Joule (not to speak of the other pioneers ofthe energy principle) recognized its fundamental anduniversal character as clearly as did Helmholtz, whomust therefore be regarded as one of the discover-ers of the principle, although his talk to the BerlinPhysical Society was given later than the fundamen-tal publications of Mayer and Joule. Both were un-known to Helmholtz at the time. Helmholtz had tofight hard for the recognition of his result - Professor



Poggendorf, the influential editor of the “Annalender Physik und Chemie”, had no wish to publishwhat seemed to him rather speculative and philo-sophical. Magnus also regarded it with disfavor, butat least recommend that it be printed as a separatebrochure, as was very quickly managed with the helpof the influential mathematician C.G. Jacobi. Thenew law of nature quickly demonstrated its fruitful-ness and universal applicability. For instance Kirch-hoff’s second law for electrical circuits is essentiallya particular case of the energy principle. Nowadaysthese laws are among the most frequently appliedlaws in the fields of electrical engineering and elec-tronics. The discovery of the fundamental law of cir-cuits was done early in Kirchhoff’s life in Konigsbergand Berlin.

Rudolf Clausius (1822-1888) also played anessential role in the history of the law of conservationof energy and its further elaboration (Ebeling & Or-phal, 1990). After studying in Berlin, he taught forsome years at the Friedrich-Werdersches Gymnasiumin Berlin and was a member of the seminar of Pro-fessor Magnus at the Berlin University. His reporton Helmholtz’s fundamental work, given to Magnus’colloquium, was the beginning of a deep involvement



with thermodynamical problems. Building on thework of Helmholtz and Carnot he had developed, andpublished 1850 in Poggendorff’s Annalen his formula-tion of the second law of thermodynamics. Clausiuswas fully aware of the impact of his discovery. Thetitle of his paper explicitly mentions ”laws”. Hisformulation of the second law, the first of several,that heat cannot pass spontaneously from a coolerto a hotter body, expresses its essence already. Un-like Carnot, and following Joule, Clausius interpretedthe passage of heat as the transformation of differentkinds of energy, in which the total energy is con-served. To generate work, heat must be transferredfrom a reservoir at a high temperature to one at alower temperature, and Clausius here introduced theconcept of an ideal cycle of a reversible heat engine.In 1851bf William Thomson (Lord Kelvin) formulated inde-pendently of Clausius another version of the secondlaw. Thomson stated that it is impossible to cre-ate work by cooling down a thermal reservoir. Thecentral idea in the papers of Clausius and Thomsonwas an exclusion principle: ”Not all processes whichare possible according to the law of the conserva-tion of energy can be realized in nature. In otherwords, the second law of thermodynamics is a se-



lection principle of nature”. Although it took sometime before Clausius’ and Thomson’s work was fullyacknowledged, it was fundamental not only for thefurther development of physics, but also for sciencein general. 1965 Clausius gave more generalforms of the second law.The form valid today was reported by him at a meet-ing of the “Zuricher Naturforschende Versamm-lung” in 1865. There for the first time, he intro-duced the quotient of the quantity of heat absorbedby a body and the temperature of the body d′Q/Tas the change of entropy . The idea to connect thenew science with the atomistic ideas arose already inthe fifties of the 19th century.August Karl Kronig (1822-1879)extended thermodynamics and started with statisti-cal considerations. In this way Kronig must be con-sidered a pioneer of statistical thermodynamics. In1856, he published a paper in which he describeda gas as system of elastic, chaotically moving balls.Kronig’s model was inspired by Daniel Bernoulli’s pa-per from 1738, where Bernoulli succeeded in derivingthe equation of state of ideal gases from a billiardmodel. Kronig’s early attempt to apply probabilitytheory in connection with the laws of elastic colli-sions to the description of molecular motion, makes



him one of the forerunners of the modern kinetic the-ory of gases. After the appearance of Kronigs paper,Clausius developed the kinetic approachIn a paper “Uber die Art der Bewegung, die wirWarme nennen”, which appeared 1857 in Vol. 100of the Annalen der Physik, Clausius published hisideas about the atomistic foundation of thermody-namics. In fact, his work from 1857 as well as afollowing paper published in 1858 are the first com-prehensive survey of the kinetic theory of gases. Asa result of his work Clausius developed new termslike the mean free path and cross section and intro-duced in 1865 the new fundamental quantity entropy.Further we mention the proof of the virial theoremfor gases, which he discovered in 1870. Parallel toClausius’s work the statistical theory was developedin Great Britain byJames Clerk Maxwell derived in 1860 theprob distribution for velocities of moleculesin Philosophical Magazine (probability distributionfor the velocities of molecules in a gas). In 1866Maxwell gave a new derivation of the velocity dis-tribution based on a study of direct and reversedcollisions and formulated a first version of a trans-port theory. In 1867 Maxwell considered firstthe statistical nature of the second law of



thermodynamics and considered the con-nection between entropy and information.His “Gedankenexperiment” about a demon observ-ing molecules we may consider as the first fundamen-tal contribution to the development of an informa-tion theory. In 1878 Maxwell proposed the new term“statistical mechanics”.

Ludwig Boltzmann (1844-1906)began his studies at the University of Vienna in 1863,he was deeply influenced by Stefan, who was a bril-liant experimentalist and also by Johann Loschmidt(1821-1895) who was an expert in the kinetic theoryof gases. Boltzmanns kinetic theory of gases:In 1866, he found the energy distribution for gases.In 1871 he formulated the ergodic hypothesis ,In 1872 formulation of Boltzmanns famous kineticequation and the H-theorem.

In 1889Max Planck (1858 - 1947)was called to succeed Kirchhoff at the Berlin Chairof Theoretical Physics where he became one of themost famous of theoretical physicists at his time, inparticular a world authority in the field of thermody-namics. He was a pioneer in understanding the fun-



damental role of entropy and its connection with theprobability of microscopic states. Later he improvedHelmholtz’s chemical thermodynamics and his the-ory of double layers, as well as developed theories ofsolutions, including electrolytes, of chemical equilib-rium and of the coexistence of phases. Planck wasespecially interested in the foundations of statisticalthermodynamics. In fact he was the first who wrotedown explicitely the famous formula

S = k log W. (1.1)

The great american pioneerJosiah Willard Gibbs (1839 - 1903):developed the ensemble approach, the entropy func-tional and was the first to understood the role of themaximum entropy method.

The new field is not free of contradictions and math-ematical difficulties: criticized e.g. byHenri Poincare (1854 - 1912) followed byZermelo: stated problems of mathematical foun-dation of Boltzmann’s theory.Probles with the “ergodic hypothesis”. The lattersays that the trajectory of a large system crosses ev-ery point of the energy surface. Zermelo found aserious mathematical objection against Boltzmannstheory which was based on the theorem of Poincare



about the “quasi- periodicity of mechanical systems”published in 1890 in the paper “Sur le probleme detrois corps les equations de la dynamique”. In thisfundamental work Poincare was able to prove un-der certain conditions that a mechanical system willcome back to its initial state in a finite time, the so-called recurrence time. Zermelo showed in 1896 in apaper in the “Annalen der Physik” that BoltzmannsH-theorem and Poincares recurrence theorem werecontradictory. In spite of this serious objection, inthe following decades statistical mechanics was dom-inated completely by ergodic theory. A deep analysisof the problems hidden in ergodic theory was givenbyPaul and Tatjana Ehrenfest in a survey ar-ticle 1911in “Enzyklopadie der Mathematischen Wissenschaften”.Much later it was recognized that the clue for the so-lution of the basic problem of statistical mechanicswas the concept of instability of trajectories devel-oped also by Poincare in 1890 in Paris. Before westudy this new direction of research, we explain firstthe development of some other directions of statisti-cal thermodynamics.

In 1907, Einstein proposed that quantum



effects lead to the vanishing of the specificheat at zero temperature.His theory may be considered as the origin of quan-tum statistics. In 1924 Einstein gave a correctexplanation of gas degeneracy at low temper-atures by means of a new quantum statistics, theso-called Bose-Einstein statistics. In addition to theBose-Einstein condensation his ideas about the in-teraction between radiation and matter should beemphasized. In 1916 his discussion of spontaneousemission of light and induced emission and adsorp-tion forms the theoretical basis of the nonlinear dy-namics and stochastic theory of lasers.

Another important line of the development of ther-modynamics is the foundation of irreversible ther-modynamics.We mention the early work of Thomson, Rayleigh,Duhem, Natanson, Jaumann and Lohr. The finalformulation of the basic relations of irreversible ther-modynamics we owe to the work of Onsager (1931),Eckart (1940), Meixner (1941), Casimir (1945), Pri-gogine (1947) and De Groot (1951). Irreversible ther-modynamics is essentially a nonlinear science, whichneeds for its development the mathematics of nonlin-ear processes, the so-called nonlinear dynamics.



The great pioneers of nonlinear dynamics inthe 19th century were Helmholtz, Rayleigh, Poincareand Lyapunov. John William Rayleigh (1842-1919)is the founder of the theory of nonlinear oscillations.Many applications in optics, acoustics, mechanics andhydrodynamics are connected with his name. Alexan-der M. Lyapunov was a Russian mathematician, whoformulated in 1982 the mathematical conditions forthe stability of motions. Henri Poincare (1854-1912)was a French mathematician, physicist and philoso-pher who studied in the 1890s problems of the me-chanics of planets and arrived at a deep understand-ing of the stability of mechanical motion. His work”Les methodes nouvelles de la mechanique celeste”(Paris 1892/93) is a corner stone of the modern non-linear dynamics. Important applications of the newconcepts were given by the engineers Barkhausen andDuffing in Germany and van der Pol in Holland.Heinrich Barkhausen (1881-1956) studied physics andelectrical engineering at the Technical University inDresden, were he defended in 1907 the dissertation“Das Problem der Schwingungserzeugung” devotedto to problem of selfoscillations. He was the firstwho formulated in a correct way the necessary phys-ical conditions for self-sustained oscillations. Laterhe found worldwide recognition for several technical



applications as e.g. the creation of short electromag-netic waves. Georg Duffing worked at the TechnicalHigh School in Berlin-Charlottenburg. He workedmainly on forced oscillations; a special model, theDuffing oscillator was named after him. In 1918 hepublished the monograph “Erzwungene Schwingun-gen bei veranderlicher Eigenfrequenz und ihre tech-nische Bedeutung”. Reading this book, one canconvince himself that Duffing had a deep knowledgeabout the sensitivity of initial conditions and chaoticoscillations. A new epoch in the nonlinear theorywas opened when A.A. Andronov connected the the-ory of nonlinear oscillations with the early work ofPoincare. In 1929 he published the paper “Les cy-cles limites de Poincare et la theorie des oscil-lations autoentretenues” in the Comptes RendusAcad. Sci. Paris. The main center of the devel-opment of the foundations of the new theory theoryevolved in the 1930s in Russia connected with thework of Mandelstam, Andronov, Witt and Chaikinas well as in the Ukraina were N.M. Krylov, N.N.Bogoliubov and Yu.A. Mitropolsky founded a schoolof nonlinear dynamics.

That there existed a close relation between sta-tistical thermodynamics and nonlinear science was



not clear in the 19th century when these importantbranches of science were born. Quite the opposite,Henri Poincare, the father of nonlinear science, wasthe strongest opponent of Ludwig Boltzmann, thefounder of statistical thermodynamics. In recent timeswe have the pleasure to see that Poincare’s work con-tains the keys for the foundation of Boltzmann’s er-godic hypothesis. The development of this new sci-ence had important implications for statistical ther-modynamics. We have mentioned already the newconcept of instability of trajectories developed byPoincare in Paris in 1890. This concept was in-troduced into statistical thermodynamics by Fermi,Birkhoff, von Neumann, Hopf and Krylov. The firstsignificant progress in ergodic theory was made throughthe investigations of G. Birkhoff and J. von Neumannin two subsequent contributions to the Proceedingsof the national Academy of Science U.S. in 1931/32.The Hungarian Johann von Neumann (1903-1957)came in the 1920s to Berlin attracted by the sphereof action of Planck and Einstein in physics and vonMises in mathematics. Von Neumann, who is oneof the most influential thinkers of the 20th centurymade also important contributions to the statisticaland quantum-theoretical foundations of thermody-namics. Von Neumann belonged to the group of “sur-



prisingly intelligent Hungarians” (D. Gabor, L. Szi-lard, E. Wigner), who studied and worked in Berlinaround this time. The important investigations ofvon Neumann on the connection between microscopicand macroscopic physics were summarized in his fun-damental book “Mathematische Grundlagen derQuantenmechanik” (published in 1932). It is herethat he presented the well known von Neumann equa-tion and other ideas which have since formed the ba-sis of quantum statistical thermodynamics. Von Neu-mann formulated also a general quantum-statisticaltheory of the measurement process, including the in-teraction between observer, measuring apparatus anthe object of observation. This brings us back toMaxwell and in this way to another line of the his-torical development.

The information-theoretical approach tostatistical physicsstart with Maxwell’s speculations about a demon ob-serving the molecules in a gas. Maxwell was in-terested in the flow of information between the ob-server, the measuring apparatus and the gas. Infact this was the first investigation about the rela-tion between observer and object, information andentropy. This line of investigation was continued



by Leo Szilard, prominent assistant and lecturerat the University of Berlin and a personal friend ofvon Neumann. His thesis (1927) ”Uber die En-tropieverminderung in einem thermodynamischenSystem bei Eingriffen intelligenter Wesen” inves-tigated the connection between entropy and informa-tion. This now classic work is probably the first com-prehensive thermodynamical approach to a theory ofinformation processes and, as the work of von Neu-mann, deals with thermodynamical aspects of themeasuring process. The first consequent approachto connect the foundations of statistical physics withinformation theory is due to Jaynes (Jaynes, 1957;1985). Jaynes method was further developed and ap-plied to nonequilibrium situations by Zubarev (Sub-arev, 1976; Zubarev et al., 1996, 1997)) and by Stratonovich.The information-theoretical method is of phenomeno-logical character and connected with the maximumentropy approach.

On history of the concept of Brownianmotion

As observed first by Ingenhousz and Brown, themicroscopic motion of particles is essentially erratic.These observations led to the concept of Brownianmotion which is basic to Statistical Physics. More-



over, the discussion of Brownian motion introducedquite new concepts of microscopic description, perti-nent to stochastic approaches. The description putforward by Einstein in 1905/1906, Smoluchowski in1906 and Langevin in 1908 is so much different fromthe one of Boltzmann: it dispenses from the descrip-tion of the system’s evolution in phase space and re-lies on probabilistic concepts. Marc Kac put is asfollows: ”... while directed towards the same goalhow different the Smoluchowski approach is fromBoltzmann’s. There is no dynamics, no phasespace, no Liouville theorem – in short none ofthe usual underpinnings of Statistical Mechanics.Smoluchowski may not have been aware of it buthe begun writing a new chapter of Statistical physicswhich in our time goes by the name of Stochasticprocesses”The synthesis of the approaches leading to the under-standing of how the properties of stochastic motionsare connected to deterministic dynamics of the sys-tem and its heat bath were understood much later inworks by Mark Kac, Robert Zwanzig and others.

A big part of our book is devoted to Brownian mo-tion. For this reason and also having in mind theanniversary of the fundamentals of stochastic theory



to be noticed in the years 2005-2008, we will preparea separate article on the history of this importantconcept.

Problems:

What is the main difference (with respect to termi-nology and basic equations) between the standardapproach to statistical mechanics (Boltzmann, Gibbs)and the stochastic approach by Smoluchowski, Ein-stein, Fokker, Planck et al.?(See the view of Marc Kac and have a look at someoriginal papers as far as available!)


Chapter 2

Thermodynamic, Deterministic andStochastic Levels of Description

(Selected results)

2.1 Thermodynamic level

First Law:There exists a fundamental extensive thermody-namic variable E. Energy can neither be creatednor be destroyed. In can only be transferred orchanged in form. Energy is conserved in isolatedsystems. The energy production inside the systemis zero.

dE = deE + diE (2.1)

diE = 0 (2.2)

Any process is connected with a transfer deE orwith a transformation of energy. Energy transfermay have different forms as heat, work and chemi-cal energy. The unit of energy is 1J = 1Nm, cor-responding to the work needed to move a body 1meter against a force of 1 Newton. An infinitesimalheat transfer we denote by d′Q and the infinitesimal

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26 Thermodynamic, Deterministic and Stochastic Levels of Description (Selected results)

work transfer by d′A. If there is no other form oftransfer, i.e. the system is closed, we find for theenergy change (balance of energy):

dE = deE = d′Q + d′A (2.3)

In other words, the infinitesimal change of energy of asystem equals the sum of of the infinitesimal transfersof heat and work. This is a mathematical expressionof the principle given above: A change of energy mustbe due to a transfer, since creation or destruction ofenergy is excluded. If the system is open, i.e. theexchange of matter in the amount dNi per sort i isadmitted, we assume

dE = d′Q + d′A +∑

iµidNi (2.4)

Here the so-called chemical potential µi denotes theamount of energy transported by a transfer of a unitof the particles of the chemical sort i . Here µi hasthe dimension of energy per particle or per mole. Theinfinitesimal work has in the simplest case the form

d′A = −pdV (2.5)

In the case that there are also other forms of workwe find more contributions having all a bilinear form

d′A =∑

klkdLk (2.6)


Thermodynamic level 27

where lk is intensive and Lk is extensive and in par-ticular we have l1 = −p and L1 = V . Strictly speak-ing this expression for the infinitesimal work is validonly for reversible forms of work. Later we shall comeback to the irreversible case. In this way the balanceequation for the energy changes (2.6) assumes theform

dE = d′Q +∑

klkdLk +

∑

iµidNi (2.7)

In this equation there remains only one quantity whichis not of the bilinear structure, namely the infinites-imal heat exchange d′Q. The hypothesis, that bilin-earity holds also for the infinitesimal heat, leads usto the next fundamental quantity, the entropy. Weshall assume that d′Q may be written as the productof an intensive quantity and an extensive quantity.The only intensive quantity which is related to theheat is T and the conjugated extensive quantity willbe denoted by S. In this way we introduce entropyals the extensive quantity which is conjugated to thetemperature:

d′Q = TdS (2.8)

This equation may be interpreted also in a different



way by writing following Clausius

dS =d′Q

T(2.9)

The differential of the state variable entropy is givenby the infinitesimal heat d′Q divided by the temper-ature T . In more mathematical therms, the temper-ature T is an integrating factor of the infinitesimalheat.

The variable entropy was introduced in 1865 byClausius. The unit of entropy is 1J/K . One caneasily show that this quantity is not conserved. Letas consider for example two bodies of different tem-peratures T1 and T2 being in contact. Empirically weknow that there will be a heat flow from the hotterbody 1 to the cooler one denoted by 2 . We find

d′Q1 = T1dS1 = d′Q = T2dS2 (2.10)

Due to our assumption T1 > T2 we get dS1 < dS2

, i.e. heat flow down a gradient of temperature pro-duces entropy. The opposite flow against a gradientof temperature is never observed. A generalizationof this observations leads us to the

Second Law of thermodynamics:



Thermodynamic systems possess the extensive statevariable entropy. Entropy can be created but neverbe destroyed. The change of entropy in reversibleprocesses is given by the exchanged heat dividedby the temperature. During irreversible processesentropy is produced in the interior of the system.In isolated systems entropy can never decrease.

Let us come back now to our relation (2.7) whichreads after introducing the entropy by eq. (2.8)

dE = TdS +∑

klkdLk +

∑

iµidNi (2.11)

This equation is called Gibbs fundamental relation.Since the Gibbs relation contains only state vari-ables it may be extended (with some restrictions) alsoto irreversible processes. In the form (2.13) it maybe interpreted as a relation between the differentialsdE, dS, dLk and dNi . Due to the Gibbs relation oneof those quantities is a dependent variable. In otherwords we may write e.g.

E = E(S, Lk, Ni)

or

S = S(E, Lk, Ni).

In order to avoid a misunderstanding, we state ex-plicitely: “The Gibbs fundamental relation was



obtained from a balance, but primarily it is rela-tion between the extensive variables of a system”.Since this point is rather important, let as repeat itagain: Eq.(2.6) expresses a balance between the en-ergy change in the interior dE (the l.h.s of the eq.)and the transfer of energy forms through the border(the r.h.s.). On the other hand eq. (2.13) expressesthe dependence between state variables, i.e. a com-pletely different physical aspect. For irreversible pro-cesses the Gibbs relation (2.13) remains unchanged,at least in cases where the energy can still be ex-pressed by the variables S, Lk, Ni . On the otherhand the balance (2.6) has to be modified for irre-versible processes. This is due to the fact that thereexists a transfer of energy which passes the borderof the system as work and changes inside the sys-tem into heat. Examples are Ohms heat and theheat due to friction. In the following we shall denotethese terms by d′Adis. Taking into account thosecontributions, the balance assumes the form

dE = d′Q +∑

klkdLk + d′Adis +

∑

iµidNi (2.12)

For later applications we formulate now the first andthe second law in a form due to Prigogine (1947). Thebalance of an arbitrary extensive variable X may be



always written in the form

dX = deX + diX (2.13)

where the index ”e” denotes the exchange with thesurrounding and the index ”i” the internal change.Then the balances for the energy and the entropyread

dE = deE + diE, (2.14)

dS = deS + diS (2.15)

Further the first and the second laws respectively as-sume the mathematical forms

dE = deE; diE = 0

deE = d′Q + d′A +∑

iµideNi

deS =d′Q

T

dS ≥ deS; diS ≥ 0. (2.16)

This writing is especially useful for our further con-siderations. We mention that the relation for deS isto be considered as a definition of exchanged heat.The investigations of De Groot, Mazur and Haasehave shown that for open systems other definitionsof heat may be more useful. As to be seen so far, the



”most natural” definition of heat exchange in opensystems is

deS =d∗Q

T+

∑

isideNi (2.17)

The new quantity d∗Q is called the reduced heat;while d′Q is called here the entropic heat. Furthersi is the specific entropy carried by the particles ofkind i . The idea which led to the definition (2.18) is,that the entropy contribution which is due to a simpletransfer of molecules should not be considered as aproper heat. Another advantage of the reduced heatis, that it possesses several useful invariance prop-erties (Haase, 1963; Keller, 1977). The first and thesecond laws of thermodynamics formulated above area summary of several hundred years of physical re-search. They constitute the most general rules ofprohibition in physics.

2.2 Lyapunov Functions: Entropy and Thermodynamic Potentials

As stated already by Planck, the most characteris-tic property of irreversible processes is the existenceof so-called Lyapounov functions. This type of func-tion was defined first by the russian mathematicianLyapunov more than a century ago. A Lyapunovfunction is a non-negative function with the follow-


Lyapunov Functions: Entropy and Thermodynamic Potentials 33

ing properties

L(t) ≥ 0,dL(t)

dt≤ 0. (2.18)

As a consequence of these two relations, Lyapunovfunctions are per definition never increasing in time.Our problem is now to find a Lyapunov function foran arbitrary macroscopic system. Let us assume thatthe system is initially (t = 0) in a nonequilibriumstate, and that we are able to isolate the given sys-tem for t > 0 from the surrounding. From the thedefinition of equilibrium follows that, after isolation,changes will occur. Under conditions of isolation theenergy E will remain fixed, within the natural un-certainities, but the entropy will monotoneously in-crease due to the second law. Irreversible processesconnected with a positive entropy production P ≥ 0will drive the system finally to an equilibrium state lo-cated at the same energy surface. In thermodynamicequilibrium, the entropy assumes the maximal value

Seq(E, X)

which is a function of the energy and certain otherextensive variables. The total production of entropyduring the process of equilibrization of the isolatedsystem may be obtained by integration of the entropy



production over time.

∆S(t) =∫ t

0P (t)dt (2.19)

Due to the condition of isolation, there is no exchangeof entropy during the whole process. Due to the non-negativity of the entropy production

diS

dt= P (t) ≥ 0 (2.20)

the total production of entropy ∆S(t) is a monotonouslynon-decreasing function of time. The concrete valueof ∆S(t) depends on the path γ from the initial tothe final state and on the rate of the transition pro-cesses. However the maximal value of this quantity∆S(∞) should observe some special conditions.

Just for the case that the transition occurs with-out any entropy exchange, this quantity should beidentical with the total entropy difference betweenthe initial state and the equilibrium state at timet → ∞

δS = Seq(E, X) − S(E, t = 0) (2.21)

This is the so-called entropy lowering which is sim-ply the difference between the two entropy values.By changing parameter values infinitely slow alongsome path γ we may find a reversible transition andcalculate the entropy change in a standard way e.g.



by using the Gibbs fundamental relation (2.13). Animportant property of the quantity δS is, that it isindependent on the path γ from the initial state tothe equilibrium state. On the other hand the entropychange on an irreversible path may depend on detailsof the microscopic trajectory. In average over manyrealizations (measurements) should hold

〈∆S(∞)〉 = 〈∫ ∞0

P (t)dt〉 = δS (2.22)

This equality follows from the fact that S is a statefunction, its value should be independent on the pathon which the state has been reached. Assuming for amoment that the equality (2.24) is violated we couldconstruct a cyclic process which contradicts the sec-ond law. The macroscopic quantity

∆S = 〈∆S(∞)〉 (2.23)

may be estimated by averaging the entropy produc-tion for many realizations of the irreversible approachfrom the initial state to equilibrium under conditionsof strict isolation from the outside world. The result

∆S = δS (2.24)

is surprising: It says that we can extract equlibriuminformation δS from nonequilibrium (finite-time) mea-surements of entropy production. We will come backto this point in the next section. Here let us proceed



on the way of deriving Lyapunov functions.By using the relations given above we find for macro-scopic systems the following Lyapunov function

L(t) = ∆S − ∆S(t) (2.25)

which yields for isolated systems (Klimontovich, 1992,1995 )

L(t) = Seq − S(t)

Due todL(t)

dt= −P (t) ≤ 0 (2.26)

and

∆S ≥ ∆S(t) (2.27)

the function L(t) has indeed the necessary properties(2.18) of a Lyapunov function.Let us consider now a system which is in contact witha heat bath of temperature T . Following Helmholtzwe define the characteristic function

F = E − TS (2.28)

which is called the free energy. According to Gibbs’fundamental relation the differential of F is given by

dF = dE − TdS − SdT

=∑

klkdLk +

∑

iµidNi − SdT (2.29)



In this way we see that the proper variables for thefree energy are the temperature and the extensivevariables Lk (note L1 = V ) and Ni with (i = 1, ...s).In other words we have

F = F (T, Lk, Ni)

The total differential (2.31) may also be consideredas a balance relation for the free energy change for aquasistatic transition between two neighboring states.Let us now consider the transition under more gen-eral situations admitting also dissipative elements.Then we find

dF = dE − SdT − TdS =

d′A − d′Q − SdT − TdeS − TdiS.

dF = d′A − SdT − TdiS (2.30)

At conditions where the temperature is fixed andwhere the exchange of work is excluded we get

dF = −TdiS ≤ 0;dF

dt= −TP ≤ 0 (2.31)

As a consequence from eq. (2.31), the free energy isa nonincreasing function for systems contained in aheat bath which excludes exchange of work. At theseconditions the free energy assumes its minimum F



at the thermal equilibrium. Consequently the Lya-pounov function of the system is given by

L(t) = F (t) − Feq (2.32)

which possesses the necessary Lyapunov properties(2.18). Another important situation is a surroundingwith given temperature T and pressure p. The char-acteristic function is then the free enthalpy (Gibbspotential)

G = E + pV − TS (2.33)

with the total differential

dG = dE − pdV − V dp − SdT − TdS (2.34)

and the balance relation

dG = d′A−d′Q−V dp−pdV −SdT −TdeS−TdiS

= V dp − SdT − TdiS. (2.35)

For given temperature and pressure we get

dG = −TdiS ≤ 0

dG

dt= −TP ≤ 0 (2.36)

Consequently the free enthalpy G is a non-increasingfunction for systems imbedded in an isobare and isother-mal reservoir. In thermal equilibrium the minimum



Geq is assumed. Therefore

L(t) = G(t) − Geq (2.37)

is a Lyapunov function possessing the necessary prop-erties (2.27-28). Let us consider now the most generalsituation, where our system is neither isolated nor ina reservoir with fixed conditions during its course toequilibrium. In the general case the Lyapounov func-tion may be defined by

L(t) = Seq(E(t = 0), X)−S(E, t = 0)−∫ t

0P (t′)dt′

(2.38)This function has again the necessary properties of aLyapunov function (2.27-28), i.e. it is non-negativeand non-increasing. However it will tend to zero onlyunder the condition of total isolation during the timeevolution.The definition of the entropy production is in generala non-trivial problem. In the special case howeverthat the only irreversible process is a production ofheat by destruction of mechanical work the definitionof P (t) is quite easy. Since then P (t) is given asthe quotient of heat production and temperature, acalculation of L(t) requires only the knowledge of thetotal mechanical energy which is dissipated.



2.3 Energy, entropy, and work

Energy, entropy and work are the central categoriesof thermodynamics and statistical physics. The fun-damental character of these phenomenological quan-tities requires our full attention. The entropy conceptcloses the gap between the phenomenological theoryand the statistical physics. In spite of the central po-sition of energy, entropy and work in physics, thereexist many different definitions and interpretations(Zurek, 1990). Below we will be concerned with sev-eral these interpretations. We introduce mechanicalenergy, heat and work. Furtheron we talk about theClausius entropy, the Boltzmann entropy, the Gibbsentropy, the Shannon entropy and Kolmogorov en-tropy. Extending the categories of energy and en-tropy to other sciences the confusion may even in-crease. In order to avoid any misinterpretation onehas to be very careful when talking about these cate-gories. However there should be no dought, that en-ergy and entropy are central quantities. But due totheir fundamentality a specific difficulty of philosoph-ical character arises: It is extremely difficult or evenimpossible to avoid tautologies in their definition.In conclusion we may say that energy and entropyshould be elements of an axiomatics of science. As wememntioned above, the difficulty in defining funda-


Energy, entropy, and work 41

mental quantities was already discussed by Poincarewith respect to energy. In his lectures on thermody-namics (1893) Poincare’ says:

“In every special instance it is clear what energyis and we can give at least a provisional definitionof it; it is impossible however, to give a generaldefinition of it. If one wants to express the (first)law in full generality,... , one sees it dissolve be-fore one’s eyes, so to speak leaving only the words:There is something, that remains constant (in iso-lated systems).”

We may translate this sentence to the definitionof entropy in the following way: “In every specialinstance it is clear what entropy is and we cangive at least a provisional definition of it; it isimpossible however, to give a general definition ofit. If one wants to express the second law in fullgenerality, ... , one sees it dissolve before one’seyes, so to speak leaving only the words: Thereis something, that is non-decreasing in isolatedsystems”.

In this way our definition of entropy is finally: En-tropy is that fundamental and universal quantity char-acterizing a real dynamical system, that is non-decreasingin isolated systems. Energy and entropy are not in-dependent, but are connected in a rather deep way.



Our point of view is based on a valoric interpreta-tion (Ebeling, 1993). This very clear interpretationwill be taken as the basis for a reinterpretation ofthe various entropy concepts developed by Clausius,Boltzmann, Gibbs, Shannon and Kolmogorov. Asthe key points we consider the value of energy withrespect to work. The discussion about this relationstarted already in the last century and is continuingtill now. The valoric interpretation was given firstby Clausius and was worked out by Helmholtz andOstwald. But then, due to a strong opposition fromthe side of Kirchhoff, Hertz, Planck and others, itwas nearly forgotten except by a few authors (Schopf,1984; Ebeling and Volkenstein, 1990). As a matterof fact however, the valoric interpretation of the en-tropy, was for Clausius itself the key point for theintroduction of this new concept in 1864-65. Whatmany physicists do not know is that the entropy con-cept taught in universities as the Clausius concept ismuch nearer to the reinterpretation given by Kirch-hoff than to the original Clausius’ one (Schopf, 1984).Here we try to develop the original interpretation interms of a value concept in connection with somemore recent developements. We concentrate on pro-cesses in isolated systems, i.e. with given energy. Letus start with a comparison of the entropy concepts of



Clausius, Boltzmann and Gibbs. In classical thermo-dynamics the entropy difference between two statesis defined by Clausius in terms of the exchanged heat

dS =d′Q

T; δS = S2 − S1 =

∫

2

1

d′Q

T. (2.39)

Here the transition 1 → 2 should be be carried outon a reversible path and d′Q is the heat exchangealong this path. In order to define the entropy of anonequilibrium state we may construct a reversible”Ersatzprozess” connecting the nonequilibrium statewith an equilibrium state of known entropy. Let usassume in the following that the target state 2 is anequilibrium state. By standard definition an equilib-rium state is a special state of a system with the prop-erties that the variables are uniquely defined, con-stant in time and remain the same after isolation fromthe surrounding (compare section 2.1). The state 1is by assumption a nonequilibrium state, i.e. a statewhich will not remain constant after isolation. Dueto internal irreversible processes, the process start-ing from state 1 will eventually reach the equilib-rium state 2 which is located (macroscopically) onthe same energy level. This is due to the condition ofisolation which is central in our picture. Now we mayapply eq. (2.39) finding in this way the nonequlib-



rium entropy.

S1(y; E, X, t = 0) = Seq(E, X) − δS(y; E, X)(2.40)

The quantity δS is the so-called entropy lowering incomparison to the equilibrium state with the sameenergy. It was introduced by Klimontovich as a mea-sure of organization contained in a nonequilibriumsystem (Klimontovich, 1982, 1989, 1990; Ebeling andKlimontovich, 1984; Ebeling, Engel and Herzel, 1990).Several examples were given as e.g. the entropy low-ering of oscillator systems, of turbulent flows in atube and of nonequlibrium phonons in a crystal gen-erated by a piezoelectric device.We shall assume in the following that the entropylowering depends on a set of order parameters y =y1, y2, ..., yn as well as on the energy E and on otherextensive macroscopic quantities X . The equilib-rium state is characterized by y1 = y2 = ... = yn = 0.There are some intrinsic difficulties connected withthe construction of an “Ersatzprozess”; therefore Muschik(1990) has developed the related concept of an ac-companying process. By definition this is a projec-tion of the real path on a trajectory in an equilibriumsubspace. Since the entropy is as a state function in-dependent on the path, the concepts “Ersatzprozess”or accompanying process give at least a principal pos-



sibility of calculating the nonequilibrium entropy. Inpractice these concepts work well for nonequilibriumstates which are characterized by local equilibrium,which is valid e.g. for many hydrodynamic flows andchemical reactions (Glansdorff and Prigogine, 1971).In more general situations the exact definition of thethermodynamic entropy remains an open question,which is the subject of intensive discussions (Ebelingand Muschik, 1992). Let us consider now another ap-proach which is based on the concept of entropy pro-duction. Assuming again that the initial state 1 is anonequilibrium state, we know from the definition ofequilibrium that, after isolation, changes will occur.Under conditions of isolation the energy E will re-main fixed, within the natural uncertainties, but theentropy will monotoneously increase due to the sec-ond law. Irreversible processes connected with a pos-itive entropy production P > 0 will drive the systemfinally to an equilibrium state located at the sameenergy surface. In thermodynamic equilibrium, theentropy assumes the maximal value Seq(E, X) whichis a function of the energy and certain other exten-sive variables. According to eq. (2.37) the entropychange is a Lyapunov functions and may be obtained



by integration of the entropy production over time:

δS = Seq(E, X) − S(E, t = 0) =∫

2

1P (t)dt (2.41)

In the special case that production of heat by de-struction of mechanical work is the only irreversibleprocess application of eq. (2.41) is quite easy. Sincethen P (t) is given as the quotient of heat produc-tion and temperature a calculation of requires theknowledge of the total mechanical energy which isdissipated. Eq. (2.41) is another way to obtain theentropy lowering and in this way the entropy of anynonequilibrium state. As above we may consider thisdifference as a measure of order contained in the bodyin comparison with maximal disorder in equilibrium.Eq. (2.41) suggests also the interpretation as a mea-sure of distance from equilibrium. So far the ther-modynamic meaning of entropy was discussed, butentropy is like the face of Janus, it allows other inter-pretations. The most important of them with respectto statistical physics is the interpretation of entropyas measure of uncertainty or disorder. In the pio-neering work of Boltzmann, Planck and Gibbs it wasshown that in statistical mechanics the entropy of amacrostate is defined as the logarithm of the thermo-dynamic probability W

S = kB log W (2.42)



which is defined as the total number of equally proba-ble microstates corresponding to the given macrostate.Further kB is the Boltzmann constant. In the sim-plest case of classical systems, the number of stateswith equal probability corresponds to the volume ofthe available phase space Ω(A) divided by the small-est accessable phase volume h3 (h - Planck’s con-stant). Therefore the entropy is given by

SBP = kB log Ω∗(A), (2.43)

Ω∗(A) = Ω(A)/h3 (2.44)

Here A is the set of all macroscopic conditions. Inisolated systems in thermal equilibrium, the availablepart of the phase space is the volume of the energyshell enclosing the energy surface

H(q, p) = E. (2.45)

If the system is isolated but not in equilibrium onlycertain part of the energy shell will be available. Inthe course of relaxation to equilibrium the probabil-ity is spreading over the whole energy shell fillingit finally with constant density. Equilibrium meansequal probability, and as we shall see, least informa-tion about the state on the shell. In the nonequi-librium states the energy shell shows regions withincreased probability (attractor regions). We may



define an effective volume of the occupied part of theenergy shell by

S(E, t) = kB log Ω∗eff(E, t), (2.46)

Ω∗eff(E, t) = exp (S(E, t)/kB) (2.47)

In this way, the relaxation on the energy shell may beinterpreted as a monotoneous increase of the effectiveoccupied phase volume. This is connected with adevaluation of the energy.

Let us discuss now in more detail the relation be-tween free energy and work. The energetic basis ofall human activities is work, a term which is alsodifficult to define. The first law of thermodynamicsexpresses the conservation of the energy of systems.Energy may assume various forms. Such forms of en-ergy as heat or work appear in processes of energytransfer between systems. They may be of differentvalue with respect to their ability to perform work.The (work) value of a specific form of energy is mea-sured by the entropy of the system. As shown firstby Helmholtz, the free energy

F = E − TS (2.48)

represents the amount of energy in a body at fixedvolume and temperature which is available for work.Before going to explain this in more detail we go



back for a moment to a system with fixed energy andwith fixed other external extensive parameters. Thenthe capacity to do work (the work value) takes itminimum zero in thermodynamic equilibrium, wherethe entropy assumes the maximal value Seq(E, X).Based on this property Helmholtz and Ostwald de-veloped a special entropy concept based on the term“value”. In the framework of this concept we con-sider the difference

δS = Seq(E, X) − S(E, X) (2.49)

as a measure of the “value” of the energy containedin the system. In dimensionless units we may definea “lowering of entropy” by

Le = [S(E, X) − S(E, X)]/NkB (2.50)

where N is the particle number. We consider Le asa quantity which measures the distance from equilib-rium or as shown above the (work) value of the en-ergy contained in a system. Further the lowering ofentropy Le should also be connected with the nonoc-cupied part of the phase space. As shown above, anynonequilibrium distribution is concentrated on cer-tain part of the energy surface only. Therefore therelaxation to equilibrium is connected with a spread-ing of the distribution and a decrease of our knowl-edge on the microstate.



In terms of the phase space volume of statisticalmechanics this measure has the following meaning.It gives the relative part of the phase space in theenergy shell which is occupied by the system. Thesecond law of thermodynamics tells us that entropycan be produced in irreversible processes but neverbe destroyed. Since entropy is a measure of value ofthe energy this leads to the formulation that the dis-tance from equilibrium and the work value of energyin isolated systems cannot increase spontaneously. Inother words Le and w are Lyapunov functions ex-pressing a tendency of devaluation of energy.In order to increase the value of energy in a sys-tem one has to export entropy. In this way we haveshown, that the meaning of the thermodynamic con-cept of entropy may be well expressed in terms ofdistance from equilibrium, of value of energy or ofrelative phase space occupation instead of the usualconcept of entropy as a measure of disorder.Now let us come back to the free energy, the term in-troduced into thermodynamics by Helmholtz. As wewill shown in detail, the concept of Helmholtz maybe interpreted in the way that the total energy con-sists of a free part which is available for work and abound part which is not available. A related conceptis the exergy, which is of much interest for technical



applications.Due to the relation

E = F + TS = Ef + Eb (2.51)

the energy in a body consists of two parts.

E = Ef + Eb; Ef = F ; Eb = TS (2.52)

Correspondingly, the first part Ef = F may be inter-preted as that part of the energy, the “free energy”,that is available for work. The product of entropywith the temperature Eb = TS may be interpretedas the bound part of the energy. On the other handdue to

H = G + TS = Hf + Hb (2.53)

it gives also the bound part of the enthalpy (G - beingthe free enthalpy). From the second law follows asshown in the previous section that under isothermalconditions the free energy is a non-increasing functionof time

dF

dt≤ 0 (2.54)

and that under isobaric-isothermal condition the freeenthalpy is non-increasing

dG

dt≤ 0 (2.55)



The tendency of F and G to decrease is in fact de-termined by the general tendency expressed by thesecond law, to devaluate the energy (or the enthalpy)with respect to their ability to do work.Let us study now the work W performed on a systemduring a finite transition from an initial nonequilib-rium state to a final equilibrium state. Then as wehave shown

W = δF = Fne − Feq (2.56)

is the work corresponding to a process when the pa-rameters are changed infinitely slowly along the pathγ from the starting nonequilibrium point to the fi-nal equilibrium state. This relation is not true, ifthe parameters are switched along the path γ at afinite rate. At that conditions the process is irre-versible and the work W will depend on the micro-scopic initial conditions of the system and the reser-voir, and will, on average exceed the free energy dif-ference (Jarzynski, 1996)

〈W 〉 ≥ δF = Fne − Feq (2.57)

The averaging is to be carried out over an ensembleof transitions (measurements). The difference

[〈W 〉 − W ] ≥ 0 (2.58)

is just the dissipated work Wdis associated with the


Deterministic Level 53

increase of entropy during the irreversible transition.In recent work Jarzynski (1996) discussed the aboverelations between free energy and work from a newperspective. The new relation derived by Jarzynski(1996) instead of the inequality (2.57) is an equality

〈exp[−βW ]〉 = exp[−βW ] (2.59)

where β = 1/kBT . This nonequilibrium identity,proven by Jarzynski (1997) using different methods,is indeed surprising: It says that we can extract equi-librium information

δF = W = −kBT [ln〈exp(−βW )〉] (2.60)

from an ensemble of nonequilibrium (finite-time) mea-surements. In this respect eq.(2.60) is an equivalentof eq.(2.22).

2.4 Deterministic Level

Description by variables:

x(t) = [x1(t), x2(t), ..., xn(t)] (2.61)

xi(t) = Fi(x1, ..., xn(t)), i = 1, 2, ..., n (2.62)



A detailed pictues you will find in the book, also inmany available textbooks.

2.5 Stochastic Level of Description

Due to stochastic influences the future state of a dy-namical system is in general not uniquely defined.In other words the dynamic map defined by (2.62)is non-unique. A given initial point x(0) may be thesource of several different trajectories. The choice be-tween the different possible trajectories is a randomevent.Description developed by Paul Langevin (1911):Add stochastic forces with zero mean value

xi = Fi(x) +√

2Dξi(t) (2.63)

where xi(t) is a delta-correlated Gaussian randomvariable.

< ξ(t) >= 0; < ξi(t)ξj(t′) >= δijδ(t − t′)

(2.64)By averaging we find

<x > =< Fi(x) >' Fi(< x >) (2.65)

This way, in average, the deterministic dynamics isreproduced at least approximately.After all the term trajectory looses its precise mean-ing and should be supplemented in terms of proba-


Stochastic Level of Description 55

bility theory. We describe the state of the systemat time t by a probability density P (x, t; u). Perdefinition P (x, t; u)dx is the probability of findingthe trajectory at time t in the interval (x, x + dx).Instead of the deterministic equation for the statewe get now a differential equation for the probabilitydensity P (x, t; u).

Define G as the probability flow vector. Based onthe equation of continuity we get

∂tP (x, t; u) = −divG(x, t; u) (2.66)

In the special case that there are no stochastic forcesthe flow is proportional to the deterministic field i.e.

Gi(x, t, ; u) = Fi(x, t; u)P (x, t; u)

Including now the influence of the stochastic forceswe assume here ad hoc an additional diffusive con-tribution to the probability flow which is directeddownwards the gradient of the probability

Gi(x, t; u) = Fi(x, t; u)P (x, t; u) − D∂

∂xiP (x, t; u)(2.67)

This is the simplest “Ansatz” which is consistent witheq.(2.73) for the mean values. The connection ofthe ”diffusion coefficient” D with the properties ofthe stochastic force will be discussed later. Introduc-ing eq.(2.5) into eq.(2.66) we get a partial differential



equation.

∂

∂tP (x, t; u) =

∑

i

∂

∂xi

D∂

∂xiP (x, t; u) − Fi(x, t; u)P (x, t; u)

(2.68)We will use the following notation:(i) If x1, ...xn → x1, ..., xf are usual mechanical coor-dinates, we call the equation Smoluchowski equationto honour the contribution of Marian Smoluchowski(18.. - 1917).(ii) If x1, ..., xn → x1, ..., xf , v1, ...vf represent coor-dinates and velocities (momenta) we denote the equa-tion as Fokker-Planck equation, since Fokker andPlanck wrote down the first version. Alternatively wemay call the equation Klein-Kramers equations af-ter the scientists which formulated the standard formused nowadays.(iii) In the general case that the meaning of the x1, ..., xn

is not specified at all, we speak about the it Chapman-Kolmogorov equation.In the literature all these equations are often calledthe Fokker-Planck equation but this is historicallynot fully correct.

In this way we have found a closed equation forthe probabilities. The found stochastic equation isconsistent with the deterministic equation (2.62) and



will, at least approximately, take into account stochas-tic influences.

On the basis of a given probability distributionP (x, t; u) we may define now mean values of anyfunction f(x) by

〈f(x)〉 =∫

dxf(x)P (x, t; u).

Further we may define the standard statistical ex-pressions as e.g. the dispersion and in particular themean uncertainty (entropy) which is defined as

H = −〈log P (x, t; u)〉 (2.69)

The Fokker-Planck equation has a unique station-ary solution

P0(x; u)

which is the target of evolution

P (x, t; u) → P0(x; u)

There exists a non-negative functonal

K(P ; P0) = 〈log P (x, t; u)P0(x; u)〉such that

dK(P ; P0)

dt≤ 0

This is a very general stochastic inequality which hasmany special cases.



Let us first study a system withFor systems with overdamped potential dynamics

in configuration space and friction

ρ = mγ0

Fi(x, t; u) = −∂V (x, t; u)

ρ∂xi

we get the Smoluchowski equations and the solutionreads

P0(x; u) = const exp

−V (x, t; u)

Dρ

With the Einstein relation

D =kT

ρ

this gives the Boltzmann distribution.

P0(x; u) =1

Qexp

−V (x; u)

kBT

This gives

kBTK =< v(x) > −Ux(T ; u)

where Ux is the configurational part of the internalenergy defined by

Ux = Fx+TSx; Fx = kBT ln Q; Sx = −∫

dxP (x) lnP (x)

In the case of an hamiltonian dynamics we findthe Fokker-Planck-Klein-Kramers equation and the



the stationary solution reads

P0(x, v; u) =1

Zexp

−H(x, v; u)

kbT

Then we find the non-negative functonal

K(P ; P0) = 〈log P (x, v, t; u)P0(x, v; u)〉 = β(F (t)−F0)

such that

dF (t))

dt≤ 0; F (t) → F0

The approach based on the Fokker-Planck equa-tion is the simplest but not the only one. There ex-ists a different approach due to Markov, Chapmanand Kolmogorov, which is based on transition prob-abilities and the idea of a so-called Markov chain...............................

Let us still underline that the concept of the Markovprocess is rather a property of the model we apply forthe description than a property of the physical sys-tem under consideration (Van Kampen, 1981). If acertain physically given process cannot be describedin a given state space by a Markov relation, often itmay be possible, by introducing additional compo-nents, to embed it into a Markov model. This way anon-Markovian model can be converted, by enlarge-ment of the number of variables, into another modelwith Markovian properties. We note already here,



that the basic equation of statistical physics, the Li-ouville equation which will be introduced in the nextChapter, is of markovian character.

By some manipulation of the Chapman-Kolmogorovequation we get the following equation for the prob-ability density.

∂P (x, t)

∂t=

∫

dx′W (x|x′)P (x′, t) − W (x′|x)P (x, t)

(2.70)This equation is called Pauli-equation or master equa-tion since it plays a fundamental role in the theoryof stochastic processes. The integration is performedover all possible states x′ which are attainable fromthe state x by a single jump. It is a linear equationwith respect to P and determines uniquely the evo-lution of the probability density. The r.h.s.consistsof two parts, the first stands for the gain of proba-bility due to transitions x′ → x whereas the seconddescribes the loss due to reversed events. Eq.(2.70)needs still further explanation by the determinationof the transition probabilities per unit time corre-spondingly to the special physical situation. It will bethe subject of chapters 7. and 8. The transition prob-ability is in many cases a quickly decreasing functionof the jump ∆x = x− x′ . By using a Taylor expan-



sion with respect to ∆x and moments of the tran-sition probability one can transform eq.(2.70) to aninfinite Taylor series. This is the so-called Kramers-Moyal expansion.

∂P (x, t)

∂t=

∞∑

1

(−1)m

m!

∑ ∂mM(x)P (x, t)

∂xi1...∂xim

(2.71)

with

Mi1...im(x) =∫

d∆xd∆xi1d∆ximW (x + ∆x|x)(2.72)

being the moments of the transition probabilities perunit time. According to Pawula there are just twopossibilities considering homogeneous Markov pro-cesses(1) All coefficients of the Kramers-Moyal expansionare different from zero.(2) Only two coefficients in the expansion are differ-ent from zero.In the first case we have to deal with the full masterequation. In the latter one the Markovian process iscalled difusive, which is of special interest to us, andleads to the following second-order partial differentialequation:

∂

∂tP (x, t) =

∂

∂xi[Mi(x)P ] +

∑

j

∂2

∂xi∂xj[Mij(x)P ]



(2.73)

This is a generalization of the Fokker-Planck equa-tion given above. For its solution we need of courseinitial conditions P (x, t = 0) and boundary condi-tions which take into account the underlying physics.Writing eq.(??) again in the form of a continuityequation (2.97) we find for the vector of the prob-ability flow the components

Gi(x, t) = Mi(x)P (x, t) +∑

j

∂

∂xi[Mij(x)P (x, t)]

(2.74)

The strong mathematical theory of the given stochas-tic equations was developed by Chapman, Kolmogorovand Feller; therefore one speaks often about the Chap-man - Kolmogorov - Feller equation. In physicshowever, this equation was used much earlier by Ein-stein, Smoluchowski, Fokker and Planck for the de-scription of diffusion processes and Brownian mo-tion respectively (Chandrasekhar, 1943). Due to thisoriginal physical relation the coefficients Mi(x) andMij(x) are often called drift coefficients and diffu-



sion coefficients respectively. Let us still mention thatan alternative mathematical foundation of the theoryof stochastic processes may be based on the theoryof stochastic differential equations (Gichman et al.,1971).Another large class of Markovian processes containssystems with a discrete state space. This concernsthe atomic processes or extensive thermodynamic vari-ables, like e.g. particle numbers in chemical reactingsystems. The Pauli equation which was developedoriginally for the transitions between atomic levelshas the form

∂

∂tP (N , t) =

∑

[W (N |N ′)P (N ′, t) − W (N ′|N )P (N , t)]

(2.75)where N is the vector of possible discrete events(population numbers, occupation numbers). As ex-ample we refer to the large class of birth and deathprocesses where N are natural numbers which changeduring one transition by ∆N = ±1. Let us still sum-marize the new tools in comparison with the deter-ministic models. Obviously the stochastic approachcontains more information about the considered sys-tems due to the inclusion of fluctuations into the de-scription. Besides moments of the macroscopic vari-ables it enables us to determine correlation functions,



spectra which will give knowledge between the func-tional dependence of the fluctuational behaviour atdifferent times.Some physical phenomena can be explained only bytaking into account fluctuations. The stochastic ap-proach on a mesoscopic level delivers often more el-egant solutions than the microscopic statistical ap-proach. Inclusion of fluctuations of the macroscopicvariables does not necessarily enlarge the numberof relevant variables but changes only their charac-ter by transforming them into stochastic variables.The main difference compared with the deterministicmodels is the permeability of separatrices. This state-ment concerns especially non-chaotic dynamics, forinstance if dealing with one or two order parameter.With certain probability stochastic realizations reach(or cross) unstable points, saddle points, and sepa-ratrices what is impossible in the deterministic de-scription. Stochastic effects make possible to escaperegions of attraction around stable manifolds. Phys-ical situations which make use of that circumstamceare e.g. nucleation or chemical reactions where ener-getically unfavourable states has to be overwhelmed.



Problem:

Study a 1D Rayleigh particle with the dynamics

x = v (2.76)

v = (a − bv2)v +√

2Dξ(t) (2.77)

(i) Find solutions without noise D = 0 and(ii) study the Fokker-Planck equation in the case withnoise D > 0.Investigate first the case a < 0, b = 0 (so-calledOrnstein-Uhlenbeck process) and then the case a >0, b > 0 (free active Brownian particles) Find sta-tionary solutions of the Fokker-Planck equation.

1 lecture notes jagiellonian university krakow 2007 ...th- · conclusions based on these equations...

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