Statistical theory of plasma-molecular systems

STATISTICAL THEORY OFPLASMA-MOLECULAR SYSTEMS

Yu.L. KLIMONTOVICH

MoscowState University, Moscow,USSR

H. WILHELMSSON

Institutefor ElectromagneticField Theory andPlasma Physics,EURATOMINFR Association,ChalmersUniversity of Technology,S-41296 Göteborg,Sweden,

Laboratoire de Physiquedes Milieux lonisés, Ecole Polytechnique,F-91128Palaiseau,Franceand Laboratoire de PhysiqueThéoriqueet Mathématique,Université Paris VII, Tour centrale,

3 étage,2 PlaceJussieu,Paris cedex05, France

I.P. YAKIMENKO and A.G. ZAGORODNY

Institutefor TheoreticalPhysics,Academyof Sciencesof the Ukrainian SSR,Kiev, USSR

INORTH-HOLLAND - AMSTERDAM

PHYSICSREPORTS(Review Sectionof PhysicsLetters)175, Nos. 5 & 6 (1989) 263—401. North-Holland, Amsterdam

STATISTICAL THEORY OF PLASMA-MOLECULAR SYSTEMS

Yu.L. KLIMONTOVICHMoscowState University, Moscow,USSR

H. WILHELMSSONInstitutefor ElectromagneticField Theoryand Plasma Physics,EURATOM/NFRAssociation,Chalmers Universityof Technology.

S-41296 GOteborg,Sweden,Laboratoire de Physiquedes Milieux lonisés, Ecole Polytechnique,F-91128Palaiseau, France

and Laboratoire de PhysiqueThéoriqueetMathématique,Université Paris VII, Tour centrale.3 etage, 2 Place Jussieu,Paris cedex 05, France

IF. YAKIMENKO and A.G. ZAGORODNYInstitutefor TheoreticalPhysics,Academyof Sciencesofthe Ukrainian SSR.Kiev, USSR

ReceivedJuly 1988

Contents:

1. Introduction 265 4. Large-scalefluctuationsin plasma—molecularsystems 3512. The basic Set of equations 269 4.1. Equationsfor the smoothedmicroscopic phaseden-

2.1. Maxwell—Lorentz equations for plasma—molecular sities 351systems 269 4.2. Transitionprobabilities in collisional systems 355

2.2. Transitionprobabilities of particlesin a systemwith- 4.3. Dielectric responseand thecorrelation of fluctuationout electromagneticinteraction 276 sourcesin systemswith collisions 356

2.3. Dielectric responsefunctions for plasma—molecular 5. Influence of large-scalefluctuationson theelectromagneticmatter 278 processesin plasma—molecularmatter 359

2.4. Correlationfunctions of fluctuation sources 279 5.1. Infinite plasma—molecularmatter 3593. Microscopic electromagnetic fluctuations in plasma— 5.2. Half-space 378

molecularsystems 283 5.3, Layer 3873.1. Infinite medium 283 6. Concluding remarks 3983.2. Half-space 309 References 3983.3. Plasma—molecularlayer 338

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Yu.L. Klimontovicheta!., Statisticaltheory ofplasma—molecularsystems 265

Abstract:The basic principles of kinetic theoryare formulated for combinedplasma—molecularsystemsconsisting of both free and bound charged

particles.The descriptionof thesubsystemof boundparticlesis basedon theclassicalmodel of atomicoscillators(this makesit impossibleto takeinto accountionization and recombinationprocesses,but thegeneralformalism is still quite useful). In theframeworkof sucha model, collectiveelectromagneticprocessesin infinite andboundedplasma—molecularmatterarestudied.The influenceof boundarieson thecollision integrals,onthekinetic coefficientsand on thespacedistributionsof particlesis investigatedin detail.

The theoryof electromagneticfluctuationsin boundedplasma—molecularsystemsis developedas well. This theory is usedto obtain thecorrelationfunctions of electrondensity fluctuationsand spontaneousemissionspectra.A numericalanalysisof spontaneousemissionspectraispresentedfor systemswith plane-parallelboundaries.The generalizationof the fluctuationtheoryof bremsstrahlungradiationin aplasmato thecaseof plasma—molecularmatteris also discussed.

1. Introduction

Many processesoccurringin naturalandin laboratoryconditionsare determinedby effectsproducedby both a plasmaandamolecularsubsystem.Suchcombinedsystemsmaybe calledplasma—molecularsystems.The simplestexampleof such a systemis a partially ionizedhydrogenplasmaconsistingof fourcomponents:electrons,ions, atoms,andelectromagneticfield.

It is naturalthata rigorousstatisticaltheoryevenof this “simplest” plasma—molecularsystemmustbe developedon the basis of a quantumdescription.In manycases,however,it is usefulto usea moreapproximateapproachbasedon the classicalmodel of atomicoscillators.The developmentof such asimple (yet in many casesquite effective) classical statistical theory of a boundedplasma—molecularsystemis one of the aimsof the presentwork. The importanceand necessityof taking into accountboundaryconditionswill be shownfor thesesystemswhen both dynamicandfluctuation phenomenaaredescribed.

Consideringplasma—molecularsystemsit is usefulto pointout the two limiting cases,correspondingto a fully ionizedelectron—ionplasmaandan atomicsystemwith ionization degreeequalto zero. Thestatisticaltheory of nonequilibriumprocessesis well developedfor theselimiting cases.The develop-ment of a statistical theory for a descriptionof the processesin fully ionized plasmasand gasesissimplified essentiallydue to the existenceof small parameters.For example, in the caseof a fullyionizedplasmaa spherewith the particleinteractionradius(DebyeradiusrD) containsmanyparticles.This means that the interaction of chargedparticles in a fully ionized plasma is collective. Thisphenomenonmay be characterizedby the so-calledplasmaparameterg = 1 Inr~,which is inverselyproportionalto the averagenumberof particlesin a spherewith the particleinteractionradius,i.e., tothe numberof effectively interactingparticles.It is importantthat such a plasmais typical ratherthanexceptionalin nature andlaboratories.

In contrast,in the caseof a gasbinary particleinteractionsplay the main role overa wide rangeofpressures(Boltzmannapproximation)becausethe propervolume of anatom or a moleculeis muchlessthanthevolume permolecule1/n. This maybecharacterizedby thedensityparametere = nr~(r0 is theeffectivesizeof an atomor a molecule).Let us notethat for atmosphericpressure~ i0~.Thus a gasis ranfiedover a wide rangeof pressures.

In the statisticaltheory of rarefied gasesthe following methodsof descriptionare now current.Historically the Boltzmannapproachwas the first. But it was basedon brilliant physicalintuition ratherthanon rigorousmathematics.

A moregeneralkinetic theory of rarefiedgaseswas developedalmostsimultaneouslyby Bogoliubov[1], Born and Green [2], and Kirkwood [3], using an approximate(with respect to the densityparameter)solutionof a hierarchyof equationsfor a sequenceof distributionfunctions[Bogoliubov—Born—Green—Kirkwood—Yvon(BBGKY) equations](see also refs. [4—8]).That madeit possible to

266 Yu. L. Klimontovich et al,, Statisticaltheory of plasma—molecularSystenis

obtain the Boltzmannequation(in a mathematicallymorerigorousmanner)andto outline the way toobtain moregeneralkinetic equations.The Chou—Uhlenbeckequation[7, 8], taking into accountbothtwo- andthree-particlecollisions, is an exampleof such an equation.The advancesof thetheorybasedon the approximatesolution of the BBGKY equationswere quite impressive. They representanimportantstepin the developmentof the statisticaltheory of nonequilibriumprocesses,but new majorproblemsappearedat the sametime.

Firstly, it was found that perturbationtheory in termsof the densityparametercould not beusedtocalculate higher approximationsbecausein the next approximation(when a four-particlecollision isdescribed)the collision integral in the kineticequationsbecomesdivergent[9—12].Oneof the ways toremovethis problem is basedon partially summingthe most divergent diagrams [131.A simplerphysicalanalysiswasperformedin refs. [8, 14, 151 dealingwith the kinetictheory of nonidealgasesandnonidealplasmas.Two basicpointsof the BBGKY theory werenot usedin such an approach.Namely,the conditionthat the initial correlationsbe completelydisregardedwas replacedby the conditionthatthey be partially considered.The importanceof defining physicallyinfinitesimal spaceandtime intervals1ph’ Tph~a relevantvolume VPh andthe averagenumberof particlesin the physically infinitesimal volumeNPh = flVPh, which dependupon the chosenapproximation,was also indicated.

Thus,the possibility of usingthe continuumapproximationin the kinetic theory of gasesandplasmaswas considered.Such a considerationis also necessaryfor the developmentof a kinetic theory offluctuations.

Let us notethat in M.A. Leontovich’swork (seeref. [16] and also refs. [17,181) the first basis of akinetic fluctuationtheory waslaid. This paperwas written in 1935, i.e. long beforethe BBGKY theoryappeared.Without usingthe dynamicaltheory he employedthe Smoluchowskiequationto derive anequationfor the many-particle distribution function of a rarefied gas (Boltzmann gas). A similarequation(masterequation)was consideredlater by many authors (seerefs. [18—201).

In a multiplicative approximation,when the many-particle distribution function is written as aproduct of single-particledistributions (kinetic fluctuations are neglected),the Boltzmann equationfollows from the Leontovichequation.But on the basis of the Leontovichequationit is alsopossibletodevelop a more generaltheory which will take into accountkinetic fluctuations.For this purposeit isnecessaryto usea hierarchyof equationsfor a sequenceof distributionfunctionswhich follows from theLeontovich equation.However, unlike the BBGKY equationsthis hierarchywill be dissipative.

However, the kinetic theory of fluctuationswas developedin a different approach.The Boltzmannequationwith relevantLangevinsourceswas chosenas a startingpoint for the theory [211.Just thesame basic idea was used by Uhlenbeckand Fox [22]. Then the kinetic theory of nonequilibriumfluctuationsin a Boltzmanngaswas also formulated [8,23—291.

In the kinetic theory of a fully ionized plasma it is convenient to use the equationsfor themicroscopic phase densities of electronsand ions Na(R,P, t) (a = e,i), and for the microscopicelectromagneticfields EM(R, t) andBM(R, t) as a basicsetof equations.Suchan approachbeganto bedevelopedin 1954. A detaileddescriptionof it may be found in refs. [8, 18, 24,30, 31].

The kinetic equationsfor a rarefiedplasmamaybe derivedin the small fluctuation approximationdue to the smallnessof the plasmaparameterg~ 1, becausethe numberof particles in a spherewiththe particle interaction radiusis large and collective effects are fundamental.Such an approximationmay be called the polarizationapproximation.It leadsto a kinetic equationwith the Balescu—Lenardcollision integrals[32—351.The collision integral for chargedparticlesproposedby Landauin 1937 [361leadsto a rougherbut very effective approachto the kinetic theory of plasmas.

The theory of nonequilibriumprocesseshasrecently been developedin different directions. The

Yu.L. Klimontovichet a!., Statistical theoryofplasma—molecularsystems 267

Balescu—Lenardand Landauequationsarevalid only for ideal plasmas.This meansthat the contribu-tion of particle interactionsto the thermodynamicalfunctionsis not takeninto account(zeroth-orderapproximationin the plasmaparameter).In the polarizationapproximation(the first-orderapproxima-tion in the plasmaparameter)it is possibleto obtainthe kinetic equationsalso for a nonidealplasma[37,38]. The statistical theoryof a strongly nonidealplasmahasnot yet beendevelopedin detail.

Just as the kinetic theory of gases,the kinetic theory of fluctuations was developedalso fornonequilibriumplasmas[8, 18,23, 24,39, 40]. Considerableprogresshasbeenachievedin the theory ofweakly turbulent plasmas,too [41—47].One of the important problemshere is the calculation ofanomalouskinetic coefficients, which aredefined by large-scalecollectivefluctuationsratherthan bymicroscopic(single-particle)ones [42,43, 48,49].

Until recently much attentionhasbeenpaidto the calculationof nonequilibriumfluctuationsin aninfinite medium. A systematicstudy of nonequilibriumfluctuationsin boundedplasmas(such as asemi-infinite plasma,a plasma layer, a plasmacolumn and a plasmasphere)are presentedin refs.[50—56].

The statistical theory of plasma—molecularsystems(especiallyfor boundedsystems)has not beendevelopedin equally great detail. At the same time, as was mentionedabove, boundedplasma—molecularsystemsarefrequently met in natureandlaboratories.Their investigationis very importantfor solving manyproblemsin such fields as nonlinearoptics, quantumelectronics,the physics of theatmosphereandof space,astrophysics,thermonuclearfusion, MHD transformationof energy,plasmachemistry,etc.

In recentyearsmany problemsconcerningthe radiationof combinedsystemsconsisting of a freeelectronlaseranda molecularlaser[57—59],theelectromagneticinteractionof moleculeswith surfaces[60—62],wave phaseconjugationin matter [63,64] etc. have been investigated.As a result, newpossibilitiesof generatingand controlling the propertiesand using the electromagneticradiation inplasma—molecularsystemshavebeendemonstrated.A detailedstudy of thesepossibilitiesrequiresafurtherdevelopmentof the statisticaltheory of plasma—molecularsystemsin a direction describingtheplasmaand molecularsubsystemson an equalfooting.

The kinetictheory of a partially ionizedplasmaproposedin refs. [65,66] and thendevelopedin refs.[8, 24] maybe usedas a basisfor the abovegeneralization.In theseworks aset of kineticequationsforthe distributionfunctionsof electrons,ions andatomswas obtainedin the polarizationapproximation.The general structure of the collision integralswas investigatedand the theory of molecular linebroadeningwas developedas well. In refs. [67,68], the kinetic theory was generalizedto the caseofnonidealchemically reactingsystems.

It shouldbe pointed out that in the frameworkof the classicalmodelof the molecularsubsystemtobe used in the presentpapernonideality cannotbe describedsystematically.Nonideality would bedisplayed only by quantitiesthat do not dependon chemical transformations(suchas ionization andrecombination),for example,in the effective Lorentz field.

Naturally, the polarization approximation enablesus to take into account a weak collectiveinteraction. It is of greatinterestto describesimultaneouslystrong interactionsat small distancesandweak, yet collective interactionsat large distances.In order to solve this problem, it is important torepresentthe Boltzmanncollision integralin termsof fluctuationspectra(seesections6 and7, chapter24 in ref. [18],and ref. [69]). Similar problemsexist in nonequilibriumthermodynamics[70] andin thetheoryof the equilibrium statefor systemsof interactingparticles[71—73].Many of the otherproblemsin the kinetictheory of a nonequilibriumlow-temperatureplasmaareconsideredin ref. [74],andalsointhe series“Problemsof PlasmaTheory” (English translation“Reviewsof PlasmaPhysics”) [75].

268 Yu.L. Klimontovich etal., Statistical theoryof plasma—molecularsystems

The different problemsmentionedabovehaverecently beenconsideredalso for the more generalcaseof a spatially inhomogeneousplasma—molecularsystem.

A systematicrepresentationof the statisticaltheory of plasma—molecularsystemsincluding the basicresults of the above investigationsis the aim of the presentpaper. Since the theory of boundedplasma—molecularsystemsis very complicated,we useherethe classicalmodel of atomicoscillatorstodescribethe molecularsubsystems.It makes the results obtainedin this paper less general (chemicaltransformations,ionization and recombinationcannotbe describedin the framework of the classicalmodel). In spiteof this the results are still quite useful.

The paperis organizedin the following manner.In section2 wegive a generalizationof the transitionprobability approach[50,51,53] to the caseof

plasma—molecularmatter. This approachis basedon the generalideas of Langevin’s descriptionoffluctuation phenomena[31,76—82] and on the methodof inverting the fluctuation—dissipationrelation[31,83—85]. The transitionprobability of aparticlein a systemwith no electromagneticinteractions(theGreenfunction, dielectricpropagator)plays the centralrole in this approach.This quantity mayeasilybe calculatedfor unboundedand boundedmedia.In the latter case,the appropriatecalculationsmaybe doneusingthe alternativetrajectorysummationmethod[511.The physicalcontentof the transitionprobability approachis equivalentto the dressedtestparticleprinciple developedby NozièresandPines[86], Thompsonand Hubbard[87], Rostokerand Rosenbluth[88,89], and Ichimaru[48,90, 91]. Thisapproachmakesit possible to obtain general relations for the dielectric responsefunctions and thecorrelation functions of microcurrentsin the caseof plasma—molecularsystems(bounded systemsincluded) [92,93].

The third section of the paperis devotedto a calculationof the microscopicfluctuation spectrainplasma—molecularsystems. On the basis of the general relationsobtained in section 2, the elec-tromagneticprocessesin plasma—molecularsystemswith planeparallelboundarieshavebeeninvesti-gatedandthe spectraof the electromagneticfield and electricchargefluctuationsin suchsystemshavebeen calculated [92—95].The collision integrals and kinetic coefficients have been found and theinfluenceof the boundarieson the electromagneticandkinetic processesin the near-boundaryregion isindicated. It has been found that the equilibrium distributions for boundedsystemsare spatiallyinhomogeneous.The influenceof the parametersof the plasmaand of the externalmedium on thespatial and orientationaldistribution of particles nearthe boundarieshasbeen investigated[96—991.

In section 4 the general questionsof the large-scalefluctuation theory as applied to boundedplasma—molecularsystemsareconsidered.This treatmentis basedon the kinetic theory of fluctuationsin plasmasand gases,which was developedin refs. [8, 18, 24]. The equationsfor microscopicphasedensitiesand electromagneticfields smoothedwith respectto physically infinitesimal spaceand timeintervalsare the basic set of equationsin this theory. It is important that, unlike the equationsfornonaveragedmicroscopicquantities,theseequationsare dissipativeand the dissipationof smoothedmicroscopicphasedensityperturbationsis definedby thecollision integralscalculatedin the frameworkof microscopicfluctuation theory.

Because the analytic calculationsusing kinetic equationswith Boltzmann, Landau or Balescu—Lenard collision integralsare a very sophisticatedproblem, in the presentpaperwe usethe modelcollision integral introduced by Bhatnagar,Grossand Krook [1001.This model approachhasbeenfound to beuseful in investigationsof the electromagneticfluctuationsin plasmas[49,101—105].Therealso exist othermodel collision integrals(see,for example,refs. [106,1071). Using the BGK collisionintegral, wecalculatethe fluctuationpropagatorsandderivegeneralrelationsfor the dielectricresponseand correlationfunctionsof randomcurrents[92,93, 108].

Yu.L. Klimontovich et a!., Statisticaltheory of plasma—molecularsystems 269

In the fifth section the general approachobtainedin section 4 is used to describe large-scalefluctuationsin boundedplasma—molecularsystemswith plane parallel boundaries.The spectraandspaceand time distributionsof suchfluctuationsare investigatedin detail for a semi-infiniteplasma—molecular system. It is shown that the presenceof boundariesleads to additional channelsofcorrelationstaking into account wave and particle reflection from the boundaries.The fluctuationtheory of bremsstrahlungin a plasma—molecularsystemis developed.The spectraof spontaneousemission of electromagneticradiation from a semi-infinite systemand a plasma—molecularlayer arecalculated[94,95].

2. The basic set of equations

2.1. Maxwell—Lorentzequationsforplasma—molecularsystems

Let usconsidera systemconsistingof free electronsandions, atomsandmolecules.The atomsandmoleculesin their turnconsistof negativelyandpositivelychargedparticles.In particular,an atom maybe consideredas a boundstateof a singly chargedion andan electron.

Whenone considerscertainphenomenaatomscould be describedwithout taking into accounttheirinternal structure.However, in many casesthis model is very rough. A study of the basis of thestatisticaltheory of combinedplasma—molecularsystemswith both boundand free statesof chargedparticlestakeninto accountis the aim of ourpaper.The subsystemsof free electronsandions would bedescribedon the basisof a classicalapproach.The subsystemof boundchargedparticles(calledbelowthe molecularsubsystem)is treatedin theframeworkof the classicaltwo-centremodel. In sucha modeldescriptionthe atom (the molecule)is regardedasa pair of classicalchargedparticlesboundwith somemodel(for instance,harmonic)potential.

A microscopicstateof the systemunderconsiderationwithin the framework of the classicalatommodel (in particular, atomic oscillator model) may be given by the following microscopicphasedensities[8, 18,24]:

Na(Xa,t)=~~[Xa~Xia(t)], a=e,i;

NUb

Nab(Xa,Xb, t) = ~ 5[Xa — XtU(t)]5 [Xb — X16(t)]

whereXa = (Ra,I’s), ~a(t) andX~6(t)arethe phasetrajectoriesof particlesof the relevantspecies,Na

is the numberof free electronsandions (a = e,i), Nab is the numberof boundpairs.The quantitieswithindexa correspondto free chargedparticles(a = e,i) andthequantitieswith index ab to electronsandions boundin pairs. Further, the indicesa, b markingthe phasevariableswill be omitted in all casesthat cannotbe misunderstood.

To describeboundpairs it is convenientto introducenew phasevariables,x (r, p), X (R,P),where

T=Ra —Rb, R=(maRa + mbRb)I(ma+ mb),

270 Yu.L. Klimontovicheta!., Statistical theory ofplasma—molecularsystems

p = (mamb/M)(V~— Vb)~my, P = maVa+ mhVb MV,

M=ma+mb, m=mamhlM,

ma andmb arethe massesof the boundparticles.The new variablesdescribethe motion of a pair as awhole (R, P) andthe motion of a particle with reducedmassin the centreof masssystemof the pair(r, p). In termsof thesevariables

RUb = R ± (m5aIM)r, ‘~a,h= (m~~IM)P±

and the microscopicphasedensityof pairs maybe written in the following manner:

~ ~as(r,p,R,P)~(x,X).

The equationsof motion for the microscopicphasedensitiesare as follows:

(~+V~+Fa(R,V,t)~)Na(X~t)=0, (2.1)

~

(2.2)

wherea~= mbIM, ab = —m~IM,Fa(R, V, t) is the Lorentz force,

Fa(R, V, t) = ea[E(R, t) + (V/c) X B(R, t)]

e~is the chargeof the relevantparticles. In eq. (2.2) for the microscopicphasedensityof pairs weintroducedthe additionalterm — ab(1~)~T~This term takesinto accountthat in the classicalmodel ofan atom the particlesin a pair interactwith a potential 4)ab(T), which we introducein the modelatominsteadof the Coulombpotential in a real atom.

The fields E(R, t) andB(R,t) in the Lorentz force are the microscopicelectric andmagneticfieldssatisfyingthe Maxwell—Lorentzequations,

curlE(R,t)= —~ -~-B(R,t), curlB(R,t)= ~ -~—E(R,t)+~ ~ J.,(R,t)c 8t C’ 9t c o’=~a.h.ab

divE(R,t)=4i~ ~ p~(R,t), divB(R,t)=0, (2.3)g=a,b,ab

whereJ,.J(R, t) andp~(R,t) arethe microscopiccurrentsandchargedensitiesexpressedin termsof thefunctionsNa(X, t) andN~5(~1f,t).*) In the caseof free chargedparticles (a= e, i)

* Below, insteadof i,~ , j,,, we will also use themore explicit notation~

Yu.L. Kilmontovichet a!., Statistical theoryof plasma—molecularsystems 271

1a(R,t) = ea fdPVNa(X,t), a= e,i, pa(R, t) = eaJ dPN~(X,t), (2.4)

and in the caseof boundpairs

Jab(’~’ t) = ~1a(R,t) + Jb(’~’ t) = J dXa dXb [e~V~5(R— Ra) + ebVbô(R— Rb)]NUb(XU,Xb, t),

Pab(’~’t)=Pa(R, t)+pb(R, t)=JdX~dXb [ea~a)+ eb~(R Rb)]Nab(Xa,Xb, t). (2.5)

Equations(2.5) may be written in the following form:

‘~b(’~,t) = e1 Jdx dPVNUb(x,R— a1r, P — a1(MIm)p, t) ~ e1 dx dPVNab(~,t),

Pab(R, t) = t~b e1f dx dPNUb(x, R — a1r, P — a~(M/m)p,t) ~etJ dx dPN~b(~,t), (2.6)

where ~ (x, X1), X~ (R — a1r, P — a1(M/m)p).Here and in what follows we usethe notation E~for summationover the particlesbound in pairs.

The index 1 marksthe quantities(charge,Lorentz force, etc.) relatedto boundparticles.The latter equationmay be expandedinto a seriesin the atomic parameter~ -~ r~IA(r0 is the

effective size of an atom, A is the characteristicspatialscaleof an electromagneticfield in the system,for example,the wavelength).Formally the expansioncorrespondsto that in the operator(r. t9taR),

JUb(R, t) = e1 dx dP(V + a1v)~~)~‘~‘ (r ~)~Nab(~’, t), (2.7)

pUb(R, t) = e1 Jdx dP~ (—1) a1 (r -~) N~6(~t’,t). (2.8)

Introducingthe microscopicpolarization andmagnetizationvectors [24,109]

P~b(R,t) = et f dx dP~ a~(_1)’~~(~~ t),

Mab(R,t)=)_~etJdXdPE cr~(—1) (~~)

x - ((rxV)+at n+1 (TXV))Nab(~,t), (2.9)

the currentsand charge densitiesin the molecularsubsystemmay be presentedin the followingstandardform:

P~b(1~,t) = —div Pab(R, t) , Jab(R, t) = Pab(R, t) + c curlMab(R, t)

Thus, eqs. (2.7), (2.8) includeall termsof a multipoleexpansionof the molecularcurrentand charge.

272 Yu.L. Klimontovich etal., Statistical theory of plasma—molecularsystems

Let us note that in accordancewith eqs. (2.3)—(2.5),the microscopicself-consistentfields E(R, t)

andB(R, t) areproducedby all the particlesof the systemincluding the particlesboundin pairs.At thesame time, if we take into accountthat the interaction in a pair of quantum-mechanicalparticles isdescribedby the model potential 4~iUb(r),it is necessaryto subtractthe Coulombforce acting betweenthe boundparticlesfrom the total electromagneticforce. The relevantequationfor Nab(~’,t) shouldbewritten as

~ ~

at ai~ or Or Op

+ ~ [F1(R+ a1r, V + a1v, t) — F~(r)](~+ a1 ~)]NQb(~, t), (2.10)

whereF~(r)= —F~(r)= (eaeblr3)r is the Coulombforceactingon an electron(ion) in apair dueto the

presenceof an ion (electron).Taking into accountthe explicit form of theseforces,eq. (2.10) maybewritten in the following form:

[a a a+ V —~ + y — ~ [~at~fr) — 4~ab(T)] ~

+~F/(R+alr,V+atv,t)(~+a(~—)]NQb(~,t). (2.11)

wherewe introducethe notation 4Qb(r) = eaeb/r.The equationsfor macroscopicquantities may be obtained from the correspondingmicroscopic

equationsby taking the statisticalaverage.The result is

curl(E(R, t)) = — ~ -~- (B(R, t)) , curl(B(R, t)~= ~ -~- (E(R, t)~+ ~ ~J0(R,t))

c Ot c Ot c ~abab

div~E(R,t))=4ir ~ ~p~(R,t)~, divKB(R,t)~=0, (2.12)o’—a,b,ab

(~- +V + ~Fa(R,V,t)) ~)f~(X,t)=—-~- K~F~(R,V,t)~ ~N~(X,t)), a=e,i, (2.13)

[a a a a a

+ V ~ + v ~- — — [~Ub(r) — ~abfr)1

+ E (F,(R + a1r, V + atv, t))(~ + a1 ~)]f~b(~~ t)

— -~— ~ ~F1(R+ a1r, V + a1y, t)(~ + a1 ~Nab(~~ t)), (2.14)

where~a = NaIV~~ = NabIV~

fa(X, t) (Na(X, t)~, f~b(~’t) ~Nab(~, t)~,ha ~ab

Yu.L. Klimontovichet al., Statistical theoryofplasma—molecularsystems 273

~Na(X, t) = Na(X, t) — (Na(X,t)) , tjNab(2t’, t) = N~b(~’,t) — (NQb(~,t))

SFa(R,V,t)=Fa(R,V,t)~ (Fa(R,V,t)) ,

V is the volume of the system.Theaveragevaluesof thecurrentsandchargesarestill describedby eqs.(2.6), in which the quantitiesNa(X, t) and Nab(~’,t) are replacedby ~‘~afa(A’, t) and nabfab(fl~,t),

respectively.The equationsfor microscopicphasedensity fluctuationsfollow from eqs. (2.1), (2.11), (2.13) and

(2.14). They maybe written as

La ~Na(X, t) + 3Fa(R, V, t) [nafa(X, t) + SNa(X,t)]

=(~Fa(R,V, t) ~N~(X, t)), a =e,i, (2.15)

“ab ~Nab(~, t) + ~ ~iF,(R+ a1r, V + a1v,t)(~ + a1 ~)[‘~abfab(~, t) + ~3Nab(~,t)]

= SF~(R+ a1r, V + a1v, t)(_J~+ a1 ~_)~Nab(~, t)), (2.16)

where

a aLaas~+V.~j+(Fa(R,V,t))~, a=e,i,

‘~ab as + V + v — [c1~(r)— ~jr)] ~- + ~ (F~(R+ a

1r, V + a1v, t))(_~+ a1

3Fa(R,V, t) = e~[~E(R,t) + (V/c) x ~B(R,t)],

~E(R,t)=E(R,t)—(E(R,t)~ , ?iB(R,t)=B(R,t)—(B(R,t)~

If thereis the possibility to representthe quantities

Ia=_IK6Fa(R,V,t) j~3Na(X~t)), a=e,i,a (2.17)

‘ab = — ~ ~ I~’~3FI(R+ a1r, V + a1v, t)(~ + a1 ~_)6Nab(~, t))

as functionalsof the single-particledistribution functionsfa(X, t) andfab(~,t), the equationsfor thedistribution functions

Lafa(X,t)=Ia , a=e,i, (2.18)

Labfab(’~” t) = ‘ab (2.19)

maybe consideredas kinetic equationsandthe quantities‘a and ‘ab as collision integrals.

274 Yu.L. Klimontovichet a!., Statistical theory ofplasma—molecularsystems

Since the quantities KFU~and (F1~ are the averagedself-consistentfields, it is obvious that thecollision integralsdescribethe particleinteractionsat shortdistanceswhenthe Coulombforce betweenindividual particles(pairs of particles)aremuchlargerthanthe averagedself-consistentfields. In otherwords,the collision integralstake into accountthat part of the interaction that cannotbe describedby aself-consistentfield.

Forour furtherdiscussionit is importantto indicatethe following point. The collision integralsin thekineticequations(closedequationsfor the distributionfunctionsfe’ fi’ fei) aredefinedby thesmall-scale(microscopic) fluctuation correlations. Let us consider this idea in more detail by employing theexampleof a neutralgas (ionization degreeequalto zero).

For statesslightly out of equilibrium the characteristiccorrelationradiusis of the order of the atomsize, lear —~ r0. The relaxationlength in the kinetic equationis of the orderof the meanfree pathof anatom, ire! 1/nr~.The physically infinitesimal time interval TPh. the length

1p5’ and the numberofparticles in the physically infinitesimal volume NPh are defined by the relations

Tph=~TreI~ lph=~lre!~ N~5=l/~, E=nr~.

This meansthat for the quantitieslcor,1p5 and IrC! the following inequalitiesare satisfIed:

!‘cor<’ph<ire! . (2.20)

An appropriateinequalitymayalso be formulatedfor time scales,

Tcor < Tpii ~ Trel . (2.21)

In the caseof a fully ionized plasma,

Tc~r~l/Wp~ ICOrTD, i-Ph’-_.l/WP, i’—~r~, N~h—.l/g.

Here, g is the plasmaparameter,w~,is the Langmuirfrequency.For a rarefiedplasmag ~ 1, andeqs.(2.20), (2.21) should be replacedby

‘car ~ 1ph ~ ire! Tcor ~ Tp~~ Tre!

Examplesof the parametersTPh~1ph and NPh for plasma—molecularsystemsmay be found in ref. [24](~1,2, chapter3).

Notice that the averaging over the physically infinitesimal time interval introducesa relevantGibbs ensemble.Sometimesit is moreconvenientto perform statisticalaveragingover the physicallyinfinitesimal volume VPh insteadof time averaging.This enablesoneto describethe fast processesin thekinetic approach.

To derive the kinetic equation,it is necessaryto usethe following doubleinequality:

Tear ~ TPh < ~ ~ Tre!

The inequalityzi -‘ ‘~ ~ definesthe term “collisionlessapproximation”,becausewithin the frameworkof such an approachwe takeinto considerationonly the fluctuationswith correlationtime of the orderof ~‘, which is much less than the time betweenthe collisions.

Yu.L. Klimontovicheta!., Statisticaltheory ofplasma—molecularsystems 275

The following point is also very important. The quantity 4 ~ 1 ITph may be consideredas theparameterwhich divides the fluctuations into large-scalefluctuations (Tcor < 1/4) and small-scale(microscopic) ones (Tcor> 114).

The calculationin the collisionlessapproximationis the zeroth-orderapproximationin the quantity4r~.However, the limit 4—*0 should be takenonly in the final expressions.

In accordancewith thisrule theright-handsideof eqs. (2.15), (2.16), which arecharacterizedby therelaxationtime Trei~maybe omitted, but in orderto distinguishmicroscopicfluctuationsit is necessaryto introducedissipativeterms —4 ~Na and —4 ~Nab~respectively.The limit 4—*0, as was mentionedabove, shouldbe takenin the final results.

If one supposesalso that fluctuationsaresmall, the nonlineartermsin the equationsfor ~Na(X, t)and ~1NUb(~,t) may be neglected.This leads us to an approximation which is equivalent to thepolarizationapproximationin the theoryof chargedparticlesystems[8, 18,24]. Statisticalaveragingineqs. (2.17) in theframeworkof suchan approximationis equivalentto averagingover timeintervalsTph

greaterthan the microscopiccorrelationtime Tcor but muchless thanthe relaxationtime Trei•

In the approximationunderconsiderationthe evolutionequationsfor thefluctuationsare as follows:

k ~Na(X, t) = ~~Fa(R, V, t) nafa(X,t), a = e, i, (2.22)

~ab 6Nab(~,t) = —E ~FI(R+ a1r, V + a1v, t)(~ + a1 ~—)nUbfab(~,t), (2.23)

curl~E(R,t)= _! ~ ~iB(R,t),

(2.24)

curl~B(R,t)=— -~—aE(R,t)+——~~

at C

wherewe use the following notationfor the evolutionoperators:

- a a a a~ab — ~ +4) + V + V — ~ [4a~fr) — ~abfr)]

+ E (F1(R + a1r, V + a1v, t))(~ + a1

The fluctuation microcurrentsare definedby the relations

&ta(R,t)aJNa(X,t), a=e,i,

~JUb(R, t) = e1f dx dPV ~ t).

Equations(2.22) and (2.23) differ from the equationsfor the perturbationsof distributionfunctions(which describe the interaction of particles through a self-consistentfield) by the terms with thedissipativefactor 4.


The set of equations(2.22)—(2.24)describethe electromagneticfluctuationsin plasma—molecularsystemsin the collisionlessapproximation.Unlike this approach,the theory of large-scale(kinetic)fluctuationsshould be basedon appropriatefluctuationalkinetic equations[8, 18,24].

We notethat eqs. (2.22)—(2.24) areequivalentto a closedset of equationsfor the secondmoments~iNa(X, t) ~Nb(X’, t’)~,(~Na(X,t) ~NUb(~l~’,t’)~ and (~NUh(~1~,t) aNUh(O~’,t’)~.Such a systemmaybederivedin thefollowing way. It is necessaryto multiply eqs. (2.22), (2.23)by arelevantquantity andtoperform statisticalaveraging.The absenceof nonlineartermsin theseequationsmakesit possibletorestrict ourselvesto the calculationof the secondmoments.In contrast,if nonlinearfluctuation termsarepresentin the evolutionequationfor the fluctuations,the approachmentionedabovewill lead to aninfinite hierarchyof equationsconnectingthe multi-time, centralmomentsof order of n and n + 1(n=2,3,. .

2.2. Transitionprobabilities ofparticles in a systemwithout electromagneticinteraction

The formal solutionsof eqs. (2.22) and (2.23) maybe representedin the following form:

~Na(X, t) = ~N~°1(X, t) — ~a J dt’ f dX’ Wa(X,X’, r) ~Fa(R’~ V’, t’) a~(x’,t’) , a = e, i.

(2.25)

~Nab(~, t) = ~ t) — hay f dt’ f d~’~ ~‘, T)

x E ~F1(R’+ a1r’, V’ + a1v’, t’)(~ + a1 ~ t’), (2.26)

where ~N~°~(X,t) and ~ t) are the solutions of the homogeneousequations(2.22), (2.23)describing the fluctuations in a system without electromagneticinteractions, Wa(X,X’, r) andWab(~,~“, T) are the Greenfunctionsof the relevantequations,which describethe transitionof aparticle (or a boundpair) from the phasepoint X’ (or ~“) to the point X (or ~) in a time T = t — t’.

Accordingto the above definition, the transitionprobabilitiessatisfy the equations

~a14’aCX’,X’, T) = 0 (2.27)

with initial conditions

Wa(X,X’, r) = 6(X— X’), (2.28)

and

‘~ab~’ab(~,f”, r) = 0 (2.29)

with initial conditions

Way(~,~“, T) = 3(~~~“). (2.30)


In the caseof boundedsystemstheseequationsshouldbe supplementedwith appropriateboundaryconditionswhich describethe interactionof particles(in free or boundstates)with the boundaries.Inthe presentpaper we restrict ourselvesto the investigationof systemswith specularly reflectingboundaries.In such a caseit is natural to introducethe modelboundaryconditionsin the followingform:

Wa(X,X’, r) asWa(R, P, X’, T)~s= Wa(R, pt X~,r)I~, (2.31)

Wab(~,~“, r) = Wab(T, p, R, P, ~“, T)I~= Wab(Tt, p~, R,pt ~ T)~s, (2.32)

whereS is the surfaceof the system,

Pt=’ — \ t( — \ t.......( ——\ ~, ~ p —~p1, pa), r —kT~, re),

Pr,, p,~andr~arethe normal components(with respectto the boundary)of the relevantvectors.Thismodel correspondsto thesimultaneouschangeof sign ofthenormalmomentumcomponentsP~andp~,as well as of the normal componentof the relative coordinater~,when the particle collides with aboundary.

In accordancewith theexplicit form of eqs. (2.25), (2.26), fluctuationcurrentsare representedbythe two termscorrespondingto the sourceand inducedcomponents,

6J~(R, t) rrr.~jJ~,.°~(R,t)+&1~(R, t), ~pif(R,t)=~p~.°~(R,t)+~p~(R, t), (2.33)

where~ t) and~ t) arethecurrentandchargefluctuationsrelatedto the first termsin eqs.(2.25) and (2.26). These quantitiescorrespondto fluctuationsin a systemwithout electromagneticinteractions,

~J~°~(R,t) = eaJdP V 3Na(X,t),

~p(O)(Rt)=eJdP~N(Xt) a=e,i,

(2.34)

&t~°~(R,t) = e1 Jdx dPV SNab(~,t),

Sp~°~(R,t) = e1f dx dP ~1Nab(~, t).

The quantities~3J~(R, t) and ~p~”(R, t) may be consideredas fluctuationsof currentsand chargesinducedby the self-consistentfields ~E(R,t) andbB(R,t),

~Jind(R t) = ~eana J dt’ f dX’ J dPVWa(X,X’, r) ~Fa(R’,V’, t’) fa(X’, t’), a = e, i,

~nd(R t) = ~eanaJ dt’ J dX’ JdP’ Wa(X,X’, T) ~Fa(R’,V’, t’) f~(X’,t’), (2.35)

278 Yu.L. Klimontovich et a!., Statistical theory ofplasma—molecularsystems

~J1~d(R t) = —E ein~~fdt’ Jd~’f dPdxVW~~(~1,~., r)

X ~F,(R’, V’, t’)(~ + a1 ~)fah(~.~ t’),

ap~(R t) = —E elnab f dt’ f d~’f dPdx Wah(~,~., r)

X ~F1(R’,V’, t’)(~ + a1. ~ t’). (2.36)

Let us rememberthat herewe usethe notation

= (x, X1), X1 (R — a1r, P — a1(MIm)p).

Substituting the fluctuation currents and chargesinto the Maxwell—Lorentz equationsfor thefluctuation fields andintroducingthe microscopicelectric induction

~D(R,t)~E(R,t)+4ir f dt’ ~ ~J~(R,t’), (2.37)—~ ua.h,ah

eqs. (2.23) may be rewritten as follows:

curl~E(R,t)= —~ ~- ~B(R,t),

curl~B(R,t)=~—~—~D(R,t)+~-~~ ~J~,°~(R,t), (2.38)c Ot C’ u—ahab

div~D(R,t)=4~~ ~p~(R,t), divaB(R,t)=0.ua,h,ah

Accordingto definition(2.37),we includein the electric inductionthetotal responseof thesystemtoa microscopicelectromagneticfield. In sucha casethe magneticfield strengthcoincidesidentically withthe magneticinduction, ~B(R,t) as~H(R, t).

The chosendefinition of ~D(R,t) is adoptedfrom plasmatheory. It is not the only one, becausethere is an arbitrarinessin the definition of the additional quantity— the magnetic field strengthaH(R, t). Sincein the presentpaperwe considerplasma—molecularsystems,which includea subsystemof free chargedparticles,the introduceddefinition is used in what follows. This is the mostconvenientrepresentationto describethe electromagneticpropertiesof a plasma.Just for the samereasonsweshall useappropriaterelationsfor averaged(macroscopic)quantities,too.

2.3. Dielectric responsefunctionsfor plasma—molecularmatter

In the quasi-stationary(stationary)casewhen the distribution functionsare weakly dependentontime, it is possibleto perform a Fourier transformationandcalculate the Fourier componentsof the

Yu.L. Klimontovichetal., Statisticaltheory ofplasma—molecularsystems 279

inducedcurrents,

~Jrind(R) = JdR’ o~(R,R’, w) ~E1~(R’) ~- J dR’ ~ R’, w) ~E1~(R’),

where

~~(R,R’, w) — ~ JdX’JdPViWa*U(X,X’)

< [ôm(a’+” ~ —iV; ~-~T] ~ fa(X’, t), a=e,i,

~ R’, w) = —~

4lTeIelnUbJdx dP dx’ dP’ V~Wflb~(~,~)

x (~ + a~~~-~)fab(~’, t)[~Jk(w + W’ ~.)— iV ~]. (2.39)

Waw(X,X’) and Wabw(~‘,a”) are theFourier componentsof the Greenfunctions,

Waw(X,X’) = f dr e”~Wa(X,X’, r),

(2.40)

Wabw(~’,~‘) = Jdr e’~TW~(~,~“, r).

The quantitieso~(R,R’, w) and ~ R’, w) may be regardedas the tensorsof conductivity anddielectric susceptibility,respectively.The latter gives the relation betweenthepartial contribution ofthe subsystemof speciesa~to the electric induction and the Fourier componentof the microscopicelectric field,

~D~~(R)= ~E1~(R)+ ~ JdR’ ~ R’, w) 8E1~,~,(R’). (2.41)

a =a,b,abV

Becausethe equationsfor fluctuationsof microscopicphasedensities,which arethe basicequationsforthe calculationof o~(R, R’, w) and ~ R’, w), coincidewith the linearizedkinetic equationin thecollisionless approximation, the above tensors may be consideredas macroscopiclinear responsefunctions (if we describethe electromagneticprocessesduring a time interval lessthanthe relax~ationtime). Naturally, in the oppositecasethe quantitieso~0~(R,R’, w) and~ R’, w) will be differentfrom the relevantmacroscopicresponsefunctions.

2.4. Correlationfunctionsoffluctuation sources

In order to develop a closed statistical theory of the systemsunder consideration, it is necessarytocalculate the correlationfunctionsof sourcesof electromagneticfield fluctuations.This mayeasilybe

280 Yu.L. Klimontovicheta!., Statistical theoryof plasma—molecularsystems

performedif we take into accountthat in the given approximation(small-scalefluctuations)therandommotion of particles in a systemwithout electromagneticinteractionsis the source of microscopicfluctuation fields. Thus, to calculatethe correlation function of field sourceswe haveto know thequantities(~N~°~(X,t) ~N~°~(X’, t’)~ (a = e,i) and ~ t) ~ t’)~.It is easyto performtheappropriatecalculationsin terms of transitionprobabilities.The final result is

~ t) ~N~°~(X’, t’)~= na[ W’a(X,X’, T)fa(X’, t’) + 14’a(X”, X, ~T)fa(X, t)]

~ t) 8N~°~~(~”,t’)~= nab[ Wab(c~’,~“, ~ t’) + 147ab(~’,~‘, ~T)fab(~C, t)] . (2.42)

Using this relation, one obtains

J~°~(R,t) ~J~°~(R’,t’)~ = e~n~JdPf dP’ V1V~

Wa(X,X’, T)fa(X’, t’) + W~(X’,X, ~T)fa(X, t)] , a = e,i,

(~J~°~(R,t) aJ~°~(R’,t’)~~= n~~e1e1f dx dPf dx’ dP’ VIV;

x ~ ~,, T)fab(~~,t’) + ~ ~‘ T)fay(~’i, t)] . (2.43)

The correlationfunctions of electric chargedensity fluctuationsmay be calculatedtoo. The result ofsuch calculationsis written in the form

(~p~°~(R,t) ~p~°~(R’,t’)~= e~n~f dPf dP’

x [Wa(X, X’, T)fa(X’, t’) + Wa(X’, X, ~T)fa(X, t)1 , a = e, i,

p10~(R,t) ~p~°~(R’,tP)yb = e

1e,n~~f dx dPf dx’ dP’

x [Wab(~, ., T)fUh(~., t’) + Way(~,~ T)f~~(~,t)1. (2.44)

In the stationarycase(slow time dependencealso included)the spectraldensitiesof the correlationfunctionsunderconsiderationmaybe representedin the following form:

KaJ~°~(R)~J~°~(R’)) as J dT e~T(~J~°~(R,t) ~J~°~(R’,t’)) u

= I,~(R, R’, w) + I~*(Rf, R, w) - (2.45)Here,

I~(R, R’, w) = e~n~J dPf dP’ ~V~Waw(X, X’)fa(X’, t), a = e, i,

(2.46)Iab(R R’, w) = elelnUbJdx dPJ dx’ dP’ VVWabJ~I, ~‘)fab(~’ t).

Yu.L. Klimontovichet al., Statistical theoryof plasma—molecularsystems 281

The relevantresult for the correlationfunctionsof chargedensityfluctuationsis as follows:

~ asJ dre’°~(~p~°~(R,t)6p(~)(Rl,t’)Y’

= I°(R, R’, cv) + If*(RP, R, cv), (2.47)

where

I°(R, R’, cv) = e~n~f dPJdP’ WGW(X, X’)fa(X’, t),

(2.48)Iab(R R’, cv) = elelnabf dx dPJdx’ dP’ Wabw(~’l, ~)fab(~’ t).

Notice that the resultsobtaineddo not take into accountthe retardationof distributionfunctions,becausewe havesupposedthat the correlationfunctionsare constantquantitiesduringa time of theorder of the correlationtimes.

This meansthat the particle interaction is takeninto account,when one calculatesthe dissipativecharacteristics.Nondissipativecharacteristics,for example,thethermodynamicfunctions,aredescribedin the frameworkof the ideal gasapproximation.A morecompletedescriptionof particleinteractionsin thekinetic theory of plasmasandgasesis presentedin refs. [14,15] andin ref. [24]. Someresultsfora nonidealpartially ionized plasmamay be found in ref. [67].

The effectsof nonidealityaredefinedboth by spatial nonlocality(the distribution functionchangingover somecharacteristicdistance)and time retardation.Nonlocality is takeninto accountby theaboverelations.If it is necessaryto describethe time retardationof distribution functions,this maybe doneby making the following replacements:

W~~(X,X’)fa(X’, t)—* (W~~~(2~’,X’) + ~jj Waw(X,X’) ~)fa(2~”, t),

Wabw(~,~‘)f~b(~’~ t)-* (WUb~(~,~,) + ~i~-14~(~,~) ~)fab(r~., t).

A similar replacementshouldalso be madein eqs.(3.4), (3.7) for thesusceptibilities,whereinsteadof f0(X’, t) and fab(~” t) their momentumderivativesare applied.

In the caseof thermodynamicequilibrium with a temperatureT the distribution functionsfa (X)(a = e, i) and fab(~) are describedby equilibrium single-particledistributions.For free electronsandions theseare the Maxwellian distributions,

fa(X)f~°~(i°) 1 3/2 exp(—P2/2maT), a=e,i.

(2irm~T)

In the caseof a bound pair the equilibrium distribution is as follows:

282 Yu.L. Klimontovichet a!.. Statistical theory ofplasma—molecularsystems

fay(~°)=fah(X, P) =f~(P)f~(p)f~(r)

f~(P)= (2~T)32exp(—P2/2MT), fah(P) = (2~rnT)32exp(—p2/2mT).

— exp(—~,~(r)/T)f~~(r)-Idrexp(-&~(r)/T)

Using thesedistributions it is easyto see that eqs. (2.46) may be rewritten as

I~(R,R’, w) = ~— T~~(R,R’, cv).

The spectraldensitiesof the correlationfunctions areexpressedin termsof the antihermitianpartsofthe susceptibilitytensors,

(~J~°~(R)~J~°~(R’)~= i ~ [~(R’, R, cv) — ~ R’, w)1, (2.49)

in accordancewith the fluctuation—dissipationtheorem[41,76,78].It should be noted that the correlation functions ~N~°~(X, t) ~N~°~(X’, t’)) and

~ t) ~ t’)) expressedin transitionprobabilities satisfy the equations

~a(X, t) ~N~(X’, t’)~ = 0, a = e,~

~ah (~N~(~’,t) ~ t’)~ = 0

with the initial conditions

~~i ~ t) ~N~°1(X’, t)) = n~f~(X,t) S(.X— X’) , a = e, I

~ t) ~ t)) = ~ t)8(~ —

This correspondsto the resultsobtainedon the basis of the hierarchy of equationsfor the centralmoments[8, 24].

Thus,we haveexpressedthe dielectricresponsetensorsand the correlationfunctionsof fluctuationsourcesin termsof rathersimplequantities— the transitionprobabilitiesof particlesin a systemwithoutelectromagneticinteractions.This meansthat the correlationfunction of plasma—molecularsystemsinthe polarizationapproximationcan be developedif the transitionprobabilitiesareknown. It shouldbepointed out that all the aboverelationswere obtainedwithout anyassumptionaboutthe shapeof theboundarysurface.Therefore,all theserelationsmaybe used for studyingthe fluctuationspectraif thetransitionprobabilitiesare calculatedwith particle reflection from the boundariestakeninto account.This is the main advantageof the transitionprobability approach.Below we give exampleswhichdemonstratethe possibility to solve someproblemsof the theory of electromagneticfluctuationsin aplasma—molecularsystemon the basis of the abovemethod.

Yu.L. Klimontovichet a!., Statistical theoryof plasma—molecularsystems 283

3. Microscopic electromagneticfluctuations in plasma—molecularsystems

3.1. Infinite medium

3.1.1. Electromagneticprocessesin plasma—molecularmatterThe efficiency of the approachdevelopedabovemay be illustratedby examplesof the solution of

electrodynamicand fluctuational problemsin the caseof an infinite medium.One can easily show that in this casethe solutionsof eqs. (2.27) and (2.28) for the transition

probabilitiesare of the form

Wa(X,X’, T) = 6(R — R’ — V’r)6(P — P’) e’~, (3.1)

Wab(~,~‘, T) = 6(R — R’ — Vr)ô(P — P’)6(r — r’ cosw~r— (p1mw0)sinw0r)

xô(p_pFcoswor+rlmwosjnwor)e_~T. (3.2)

In the final relationincluding integrandswith smoothfunctions(distributionfunctions) 4—-*0. Here wereproducethe result for Wab(~,~‘, r) correspondingto the oscillator potential

4ab(r) = mw~r2I2(classicalmodel of an atomicoscillator).

Using eqs. (2.39), (3.1) and (3.2) one easilyfinds the Fourier componentsof the linear responsefunctions,

4ire2n I V. of (X t)IaP~~

1(k,w)= ~ a j dP z.l [~k(wkV)+kk~1, a=e,i,

~ cv) = —i ~4~eteinabJdx dPf dr exp[i(w — k~V + i4)r]

x exp{ik. [a1,r— a1r coscv0r — a1(plmw0)sin w0r]}

x (V.+ a1v~cosw0r — a1r1w0sin WoT)[~kI(w— k V—a,,k~v) + kkVJ + a,kkvJ]

(a

X~—~-+aI——)fUb(~’,t)

= ~4~e1e~n~f dx dP ~ (—i)~J~(a1kr)Jm(aik• v/cu0) exp(ik. Ta1)

x [8~1(w— k . V—a1k v) + kkV) + a,kkvJ]

V+vmw0/(k.v)+r~nw0/(k.r)( a a ‘~

X cv — k.V—(n+m)w0+i4 ~ + a1~Op )fab(~~0. (3.3)

For the spectraldensitiesof the correlationfunctionsof fluctuationsources,one has

(~j~O)~J~O))~ = I~(k,cv) + I(7*(k cv), o = e, i, ei, (3.4)

where

284 Yu.L. Klimontovich eta!., Statistical theoryof plasma—molecularsystems

I f(Xt)I,~(k,cv) = ie~n~J dP U V + i4 a = e, t~

,ab(k w) = elelnab f dx dPJ dT(l~+ a1v~coscv0r — a1cv0r~sin w0r)

X (T’ + a1v1) exp[i(cv — k . V + i4)]

x exp{ik. [a1.r— a1r coscv0r — ct1(p/mw0)sin w0T]}f~~(~,t)

= i E etetnabJdx dP ~ (—i)~J~(a1k~r) Jm(aik~v/cu0)

(V, + v~mw0l(k~v) + r, nw0/(k~r))(V1 + a,v~)f~b(~,t)x exp(ia1k r) . . (3.5)

cv — k . V— (n + m)w0+ i4

In order to take into accountthe retardationof distributionfunctions, the following replacementsshould be made:

fa(X,t) ~1+~ 1 O~ fa(X,t

)

cv—k~V+i4~~r2 i(cv—k.V+i4) at) cv—k~V+i4

fab(~,t) ~ 1 8w—k.V—(n+m)w0+i4 ‘r 2 i(w—kV—(n+m)w0+i4) Ot

fab(~’ t)X w—kV—(n+m)w0+i4 -

A similar replacementshould be madein the relationsfor the susceptibilitytensors.The general relationsfor the correlation functionsof fluctuation sourcescan be simplified, if one

takesinto accountthat the distribution functionsfa(X, t) andfab(~” t) dependon the phasevariablesthrough integralsof the motion of the particles (namely, through the energyof the particles), i.e.,fa(X, t) =fa(X0, t0), fab(~’ t)=fUb(Y.l’o, t0), wherethe relationsbetweenX andX0, rand ~ aregivenby the equationsof motion. In otherwords, the distributionfunctionsshould be constantquantitiesalongthe phasetrajectoriesover time intervalslessthanthe time betweencollisions. In the collisionlesslimit this requirementis satisfied,becausein such alimit the equationsfor the distributionfunctionsandthe equationfor the transitionprobabilitieshavethe samecharacteristics(namely, the particle phasetrajectories)[seeeqs. (3.1), (3.2)]. Taking into accountthis fact one obtains

j~°)~j~O) )~ = 2~e~n~JdPV1V1fa(X,t)~(cv— k~V), a = e,i,

j~°)~j~°) )~ =21T ~ elelnUbf dx dP~ (—i)~J~(a

1k.r)Jm(aik v/cu0)exp(ia1k.r)

x (v + + r~~~)(V1 + a1.v1)f~~(~,t)6(w — kV— (n + m)c00). (3.6)

In our further calculations we shall use the longitudinal parts of the susceptibility tensors and thecorrelationfunctionsof fluctuation sources,which describethe quasi-staticand potentialfluctuations.

Yu.L. Klimontovicheta!., Statistical theory ofplasma—molecularsystems 285

Using the definitions

~0.(k, cv) = (k1k1/k2)~,‘(k,cv), I~(k, cv) = (k

1k1/k2 )I~(k,cv),

onehas

4ire~naf k~Ofa(X, t)/aP

k2 jdP cv—k~V+i4 , a=e,i,

XUb(k, cv) = ~4lrelel.nUb Jdx dP ~ (—i)”J~(a1k T)Jm(aik~v/cv0)

k (a/OP+ a1 a/op)fUb(~l~’,t)

xexp(ia,k.r) w—k.V—(n+m)cv0+i4 ‘ (3.7)

- e~n~w2f fa(X, t)

I~(k,cv)—i k2 JdPcvkV+i4 , a=e,i,

IUb(k, cv) = i ~e,elnabcv JdxdP ~ J~(a1k.r)J1(a,k.v/cv0)

abx (—i) exp(ia1.k.r) cv — k~V—(n+ m)cv0 + i4 - (3.8)

Here, we haveomitted the imaginary parts of I,~, becausethey do not makea contributionto thecorrelationfunctions.

If we take into accountthe conservationof the distribution functionsalongthe phasetrajectories,theneqs. (3.8) transformto

(~J~2)~= I~(k,cv) + I(k, cv) = 2~e~n~w2JdPfa(X, t)6(cv — k .V), a = e, i,

= IUb(k, cv) + jab(k, cv)

e1e1cv fk2 J dxdP~(—i)~Jn(aikr)Jm(aik•v/cvo)

xexp(iaI.k.r)fUb(~,t)6(cv —k.V—(n + m)w0). (3.9)

Accordingto the continuityequationthe quantities(~J~2)~ and(~,(0)2)~ are relatedby

/~ (0)2\cr — ‘~ /S,(O)2\o’\ P 1kw — 2 \ L 1kwcv

It is easyto seethat the calculationof ~ on the basisof eqs. (3.9) leadsto the sameresultasfollows from direct calculationsfor (~~(O)2)~,.

The aboveequationsfor the responsefunctionsandthe correlationtensorsarederivedwithoutusingan expansionin termsof atomicandthermalparameters.Therefore,they arerigorous for the chosen

286 Yu.L. Klimontovicheta!., Statistical theoryofplasma—molecularsystems

model. In the caseof a smallatomicparameter(uat -~ r0/A-~kr —‘ kv0!w) the generalrelationsmaybeabmuchsimplified, if one expandsx11 (k, cv) and I~(k,cv) in powersof ~ and /.L.th (/.Lth -~ VthIC’, VLh is the

thermalvelocity of a particle) and keepsonly a few termsof the expansion.Obviously, this procedurewill correspondto a multipole expansionof the molecularmicrocurrents.In particular, if we restrictourselvesto calculationsup to secondorder in the atomic parameterand neglect the higher-orderterms,the generalrelationslead us to the following result:

2

ab ~1dxdP 1

X1~(k,W) 2w p=±~i cv—k~V+/3w0+i4

x {(ô~ — /3 kkVJIwQ)[3k1(w — k . V) + kkV1]fUh(~,t)

— {[6k,(w — k . V) + kkVj][Vj(k~ r)2 — f3r

1w0k r]

—[vI—/3V1kv/wO](~kJkv—kkvJ)}mOfUh(~],t)IOPk} , (3.10)

jab 2 ___________________________i~ (k, cv) = 2ie~~n~~~ Jdx dP

w—k~V+/3w()+i4

x [1/,V1(k.r)2 + vu. — /3 I/v /c. v/cu

0 — [3Vw0rk~r] , (3.11)

wherew~m= 4~re~bnUb/m,eUb = ea =

For the longitudinal componentsit follows from eqs. (3.10), (3.11) that2

+ [mw0(k.r)2/k2jk O/OP}fQb(~,t)

Xab(k, cv) = —~~ f dx dP2w

0 ~±1i w—kV+/3w!)+ii2 2 (k~r)

2f~~(~,t)

‘ab(’~’ cv) = 1 eUbnUbw ~ dx dP2k f3=~I cv—k~V+/3w0+i4

Here,we usethe virial theoremwhichgives (v2) = w ~r2). If fa~(~,t) is an equilibrium function, then

T Taf~~(~,t) — — fab (~,t), (r1r1) = 2 (v1v~)= —

OPk — T mw~~

andthe generalrelationsare simplified to the following form:

2

ab ~ I dP ~— /3k1V/cv1~— ,BkJV,Iw()+ k2V1~/cv~(())

2w ~+1i w—kV+/3w0+i4

1/V.ab ieabnabT ~ fdP f~)(p) / — /3 + _~L2ik

2).l~ (k, cv) = 2m p=±1 ~ cv — k~V + /3w

0 + 14 — ~ w0 (00 cv()

In the caseof equilibrium distributions one can find the susceptibility tensorsand correlationfunctionsof microcurrentsin an explicit form,

k.k. / kk\(7... ~x~(’c~w) = —p- XL~k, cv) ~ ~2

1)~~(k, cv), (3.12)


where

~ cv) = (k~/k2)W(z~), = e, i,

f3cvo+cvW(zmp)

2w0~+1 cv+/3cv0

2

cv~,XT(K,cu) = —-—-y[1—W(z~)], o=e,i,

cv

=_~~(i+~) ~ 1—W(z~) cT=el, (3.13)2cv w0 ~ cv+/3cv0

k~= 4ire~n~/T,z~= w/ks~,s~= T~/m~, cv~= 4ire~n~/m~, u = e, i,

Zmp = (cv + /3w0)/ksm,5m = Tei/M~

z

W(z)=1—z e~2/2(JeY2/2dy — i(~/2)h/2),

0

and

(~j(O)~j(O)’ = k~k1(sJ~°~

2y+ ( k.k.\I J

lkw k2 °k2/ T 1kw’

where

1/2 ,2)2 ~ Tcv /~r\ K~ cv —w2/2k2s~(&IL )kw~.’~2) X—~~-—e , o=e,i,

2

x ~ -.~— ~ e_~~~0)2l21(~j, ~ = ci2w~~m p±i

1/2 cv2)2 ~ T / ~T J~T —w2/2k2s~

(~ )~w= 2~~2) ks~X —e

2

x -~~-~ e_~w0)2/2/c~, o=ei. (3.14)LI1~”m~

The responsefunctions of infinite plasma—molecularsystemsdefine the well-known dispersion relationsfor longitudinal,

TL(Ic,cv)—1+ ~ ~~(k,cv)=0, (3.15)ue,i,ei

288 Yu.L. Klimontovichet al. Statistical theory ofplasma—molecularsystems

andtransverse,

2ET(k,cu)—l+ ~ ~~(k,w)—~-, (3.16)

ue.i,ei (0

wavesin the systemsunderconsideration.Analysing theseequations in the transparencydomains,we are led to the conclusionthat the

molecularsubsystemmayaffect the spectraand dissipationboth of longitudinaland transversewaves.For instance,in the high-frequencydomain (w ~ kSe, cv ±w0~~‘ kSm) eq. (3.15) gives the followingresults:

= wl2k + 1112k

where

cvl

2k = ~i,2 + 3k2s~(w~e/w~

2)F(wi2)

2 1 2 2 2 1 2 2 2 2 2 21/2w1,2 = ~(w0+ ~

0pe + wpm)±~[(cv0+ wpe + Wpm) —

4wpewol

= — (~/8)h/2cvl2kF(wl2)(~~ ~ e L,2k/

2k + 2 W1,2k ~ I e ~i,2

0)2I2k25~) (3.17)(w2_w~)2

F(w)— 2 22 2 2 -

(cv — w~)+ WQWpm

The presenceof two branchesof the spectrumis a characteristicfeatureof this solution. The firstbranchmaybeconsideredas a modified (dueto the presenceof plasma)wave in molecularmatterandthe secondone as a modified (due to the presenceof molecules)plasmamode.

In the caseof atransversewavewith cv ±w0~~‘ kSm~w ~‘ kSe~the solution of eq. (3.16) is describedby

2 1 2 2 2 22 1 2 2 2 222 2 22 21/2wl,2k=2(wo+wpe+wpm+ck)±2[(cvo+wpe+wpm+ck) —4(wpe~ck)cvo]

2 2 2i cv 2 22/ cv w1/2 pe —WI 2~i2k s pm pe

= —(ir/8) cvl2kF(wl2k)L , e e ~ 2 2 — 2— ~1,2k w1~

+ ~ e_~1,2 0)

2/2k2~,, — — (i — — 2 ~pm 2))] . (3.18)2cvj

2kksm,3±1 kc wl2k w12~~—

We see that in the domainof anomalousdispersionof the responsefunctionsthe transversewavesmaybe dampedcollisionlesslydue to a resonanceinteractionwith the electrons.Besidesthat, collisionlessdissipationproducedby the interactionwith oscillatorelectronsmayalso takeplace.

If the responsefunctions are known, it is easyto calculatethe fluctuationalelectromagneticfieldgeneratedby fluctuationsources.The solution of eqs. (2.38) is as follows:

Yu.L. Klimontovich eta!., Statisticaltheoryofplasma—molecularsystems 289

~E(R,t) = J~J dk 8E exp[i(k . R — cut)], (3.19)2ir (2ir)

where

= — ~ A~(k,cv) ~ , A0(k, cv) = e,1(k, cv) — ~ — k1k1)

(3.20)s11(k, cv) = + ~ y~(k, cv), 8J~°j=

o=e,i,eI a~e,I,eI

Here, the responsetensors~ cv) are defined by eqs. (3.3) or (3.10).

3.1.2. Microscopicfluctuation spectra in plasma—molecularmatterUsing the general solution (3.19), (3.20) it is possible to calculate the fluctuationsof different

physicalquantities(current,charge,magneticfield, etc.). Statisticalaveragingof anappropriatebilinearcombinationwill give the relevantcorrelationfunctions of electrodynamicquantities.The statisticalaveragingshould be performedon the basisof the correlationfunctionsof Langevinsourcesgivenbyeqs. (3.4), (3.5).

The final resultof the calculationsfor the correlationfunctionsof an electric field is as follows:

(~E~(R,t) ~E1(R’,t’)) = J~J dk (8E1&E/)kW exp{i[k. (R — R’) — cvr]}, (3.21)21T (2~)

where

(~E~6EJ)kw = l6ir A~(k,cv)A~*(k,cv) ~ (bJ~0~~cv o=e,i,ei

= j~.2 A~(k,cv)A~,jk,cv) ~ (oJ~°~~J~°~)~W , (3.22)

cv 4(k, cv)I ue,i,ei

A.k(k, cv) = einmekjlAjn(lc, cv)Aim(k,cv), 4(k, cv) = det A11(k, cv),

earn is the completelyantisymmetricunit tensorof the third order.The condition 4(k, cv) =0 definesthe nontrivial solutionsof the field equationswhen&i~j =0, i.e.,

it representsthe dispersionrelationfor eigenexcitationsin homogeneousplasma—molecularmatter.Forthe particularcaseof an isotropic systemwith

s~1(k,cv) = EL(k, cv) + (~j— ~t~)eT(k, cv),

sucha dispersionrelation splits into the two aboverelationsfor longitudinal andtransversewaves,eqs.(3.15), (3.16).

If slowly dampingcollective excitationscan exist in the system[i.e., cvk ~ I I, where cvk and f~aredefinedby the equation4(k, cvk + iF~)= 0], it can be shownthat well-defined 5-type maximawill be

290 Yu.L. Klimontovich et a!., Statistical theoryof plasma—molecularsystems

presentin the spectraof the electromagneticfield fluctuations. In fact, following refs. [39,1101, onefinds that in the domain Re4(k, cv)~~‘ Im 4(k, w)~

161T3 A. (k w)A~(kcv)~- .

K~EI~EJ)kW Im 4(k,w) 5(Re4(k, cv)). (3.23)

Taking into accountthat

A11(k, wk)= e~e~Tr A(k, (.0k)

e. e1(k, (.0k)= A,1(k, wk)aJ/[A~J(k,wk)aIaJTr A(k, wk)]

wheree is the unit vector directedalong the electric field of the wave (polarizationvector),a is anarbitrary vector and

Im 4(k, wk) = ~iTr A(k, wk)e~eJ[E~(k, cuk) — e,1(k, wk)]

one finds the following result for the transparencydomain:

K~E~~EJ)kW = T(k, cv)e1e7Tr A(k, cv)~(Re4(k, cv))

8i~2 e~e~Tr A(k cv)

= ~ T~(k,cv) ~ Re 4(k, cv)/Ow 8(cv — cv~), (3.24)

where

(s) 4~e~’~e~~ j~~)aj7) ~

T (k, cv) = ~ e*e~[r~(k,cv) — r~1(k,cv)]

is the effective temperaturefor collective fluctuations(cf. refs. [39,110]).In the caseof an isotropic system

(~E~~EI)kW = —p- K~E~L)~W+ (~,— ~‘)(~E~)kW, (3.25)

where

2 v is r(0)2\o/~,-2 ~ l

6~T L.~,=e,i,ei \UJ L.T Ikw\0~~L.T/kw — 2 2

cv 4LT(~, cv)

4L(k, cv) = BL(k, w), 4T(k, cv) = ET(k, cv) — k2c21w2. (3.26)

In the transparencydomainseq. (3.24) yields

(~E~T)kW= TLT(k,w)~(Re4L.T(k, cv)). (3.27)

Yu.L. Klimontovichetal., Statistical theory of plasma—molecularsystems 291

The relations for the appropriateeffective temperaturecan easily be found substituting thepolarizationvectorof the transverse[eT = (a x k) lak] or longitudinal (eL = k/k) wave into the generalexpression.The final result is as follows:

21T V (0)2 ~TT(k, w) = — ~ (~‘T )kW~’ImET(k, w),

~ o=e,i,ei

(3.28)

TL(k,w)=~—~~ (~J(0)2YrII (k)2~°~ ~ (~p)kW!ImrL(k,w).

Ct) o=e,i.ei k o=e,i,ei

Naturally, in the caseof an equilibrium systemwith

V (~) (0) iwT *Li (6.1k &1~ )kw = ~ [s

11(k,w) — r,1(k, w)],o=e,i,ei

the effective temperaturesareequalto the equilibrium temperatureT.Comparingthe results obtainedabovewith the relevantrelationsfor a pure plasma [31,39], one

finds that they are formally the same.The only differencebetweenthem consistsin the fact that eqs.(3.21)—(3.28) containresponsefunctionsandcorrelationtensorsobtainedwith themolecularsubsystemtakeninto account.This is quite naturalbecausein the framework of the linear theory the additionalmolecularcontributionto the responsefunctionsand Langevin sourcesdoesnot changethe generalstructureof the result. However,analysisshowsthat in somecasesthe molecularsubsystemmay leadtonew peculiaritiesin the fluctuationspectra.For example,new resonancescorrespondingto a modifiedeigenmodespectrummayappear.In particular,the longitudinalfield fluctuationspectrumfor ~t0pe’ ~0pe~ w

0 is describedby the relation

= 4~2T 2 ~2~m 2 2 [~(w — V~e+ 3k2s~)+ ~(w + VWpe + 3k2s~)

(w — (Vpe) + WpeWpm

2 2 2 2

/ i W0+U) C,) C,)

+ ô(,~Ct)— + Wpm 2 pe + 3k2s~ pm pe 2

— ~~pe (co0 — ~~pe)

1 2 2 2 2/ I C00+w w Cu \1~ O~Wpm 2 pe +3k

2s~ 2 pe 2)]~ (3.29)Cu

0 — Wp~ (w0 — (ape)

i.e., additionalmaximanearthe oscillatoreigenfrequencyarepresent.Onthe otherhand,for ~0pe Cu

0,

IC1oi~kSm *

= 41T2T[5(w — V~°pe+ 3k2s~+ Wpmwpe) + ~(w+ V’~pe+ 3k2s~+ C°pmC~pe)

+ tS(w — + 3k2s~— CVpmWpe)+ ô(w + + 3k2s~— ~°pm~°pe)] (3.30)

Thespectraldensityof the totalchargefluctuation~Pkw= =e,i,ei ~P~, can becalculatedin asimilarway. The resultof the calculationsis as follows:

= 1 2 k~k1(~E~~j)kw (3.31)

(4ir)

292 Yu.L. Klimontovich eta!., Statistical theory of plasma—molecularsystems

It follows from (3.31) that the spectraof the electricchargedensity fluctuationsare alsocharacterizedby the collectivemaxima.

In the potential approximation [6E(R,t) = —grad6~(R,t)] the generalrelationsare simplified to

(0)2\o 22 ~ae,i,ei (6p lkw k

(6p )kw = e(k, cu)2 = ~— T(k, cv) 6(Ree(k, cv)), (3.32)

wheree(k,cv) is the dielectricresponsefunction, which definesthe dispersivepropertiesof the electricfluctuations[46],

kke(k, cv) = —~-~ r,

1(k, cv) = 1 + ~ x~(k,cv),o=e.iei

and

(0)2\ a27rcv ~ose,i,ei (6p /kwk,cv)_~~?_ ImE(k,cv) -

The appropriatecalculationsfor the cross-overcorrelationfunctionsof the electric chargedensityfluctuationsgive

1 ~ k,k1 ~ ~ (7(7

~/~POP lkw = 2 LI Yik Yji S*(6J(Q)61(Q)\(7 (3.33)/ /kW’cv oe,i,ei

where

‘Ira,

=

8aa”8ik — ~ w)A7~’(k,cv).

In the quasi-staticlimit one has

(0)2 a”

~6P )~w (3.34)‘c” r(k, w)~2-

Here, we use the notations y,,, as y~~(k,cv) = ,~e(k,cv) — ~ cv). For an isotropic systemeq.(3.34) is rigorous,but in othercasesit is valid for kc~ cv.

In the equilibrium casethe aboverelationscan be simplified. In particular,

2 k2T Im(—1/e(k,cv)),~\6P)kw =

2’ c, kT [e(k, cv) — x~(k,cv)]~~(k,cv)c6P lkW as ~6p 6p )kw = Im (3.35)2irw e(k, cv)

(6E~6EJ) = ~ [A~(k, cv) — A~J1(k,cv)].icv “

Yu. L. Klimontovich et a!., Statistical theory ofplasma—molecularsystems 293

Integrating(3.35) over cv one can find the relevantrelationsfor the spatial spectraldensities.Herethe following pointsshould be noted.The spatialspectraldensitiesarenondissipativethermodynamiccharacteristics.For this reason,theyshouldnot dependon the typeof relaxationprocessandtherefore,on the particularapproximationfor calculationsof the relaxationparameters(collision integrals).Forexample,eqs. (3.36) are obtainedin the collisionlessapproximation.They definethe spaceand timedistribution of the microscopicfluctuations. Naturally, calculating the relevant quantitieswithin theframework of large-scalefluctuationtheorywe shall obtainotherresults.However, in the equilibriumstate,theintegralsover cv which definethe staticcorrelations(thermodynamicfunctions)shouldremainin the previous form, becausethe approximation in calculating the non-dissipativecharacteristicsremainsunchanged.

The explicit form of such static correlationfunctionsmay be foundwith the aid of the Kramers—Kronig relations.The result is

12r / i \ ,2r / (I...

2 ~ ~ I I I 2 a ~ ~ I Xcr’.”~(6p )k~l~(k,0))’ (6p ~ e(k,0)

(3.36)1 \

(6Ei6EJ)k=47rT_t~l\ e(k,0))

where~0(k,0) is the partial contributionof aparticleof specieso to the static responsefunction,

~ x~(k,0), ~~(k,0)= 4~e~n~a=e,i,ei k T

(3.37)4ire1e1n . f

XeiQ~,0) = k2T ~‘ J dx dPfei(x, P) exp[—ik. r(a

1 — a1)].

In the caseof an equilibrium systemthe explicit form of x (k, 0) can be found,

~~(k,O)=k~/k2,

= (k~,/k2)(1 — ek2~S~2), ~ = ei,

where

2 22 4~en~ . 2 V 4ire,n~

1 2 TT ‘ cr=e,i; km=Lj ,,, ,t ~ mcv0

Thus, the dielectric responsefunction hasthe form

e(k,0)1+ + ~ (1_e~S~2)12),~

It follows from this relation that in the casek2( r2) ~ 1 (largedistancesR ~ (r2) 1/2)

e(k,0)1+k~/k2+cv~m/cv~.

294 Yu.L. Klimontovich eta!., Statistical theoryof plasma—molecularsystems

In the oppositecasek2(r2)~1(small distancesR~<(r2)’~2)

e(k,0)=1+ k~~,/k2+k~,/k2,

i.e., boundparticlesmay give contributionsto the screeningof a test particle.It should alsobe pointedout thatuseof the aboverelationsfor e(k, 0), as well as eq. (3.37), is not

restrictedto the collisionlessapproximation.Furthermore,the explicit form of the potential~ah(1~) doesnot influencethe final relations.In fact, usingthe continuity equationandthe generalrelations(2.39),it can be shown that in the caseof a homogeneousequilibrium systemwith arbitrary tab(r),

r(k, 0) = 1 ~ ~ J~r f dPJdP’ f~(P)f d(R — R’) exp[—ik. (R — R’)] W~(X,X’, r)

4~e1e~n~1Jdx dPf dx’ dP’ f drjd(R — R’) f,(x’, P’)

x exp[—ik~(R — R’)] exp[i(a1k r — a1k r’)]

~ [~ — 4 + °~~‘) ~ _iaik.v’] Wej(~,~‘, T). (3.38)

By simple transformationsthis relation leadsto eq. (3.37).Since wedo not usethe explicit form of the potentialwhenderiving eq. (3.38) it is obviousthat the

latter formula [as well as eq. (3.37)] can be applied to describea systemwith arbitrary cbah(T). Inparticularfor a systemof rigid dipoles,

e_~~T= 1 , r = r11 ~ = ‘ r + r0

fab(X, P) = ~-~---~ ~(r — r0) 21T e ~2mT6(pr)(

2~T) e~2MT.

In such a case

e(k,0) = 1 + k~~,Ik2+ (k~/k2)[1 — sin(kr

0)/kr0]

which gives the following result whenkr0 ~ 1:

2 2 22e(k, 0) = 1 + kD/k + kmro/6,

in accordancewith the resultsobtainedby the othermethods[71,1111.Using the relationsobtainedabove, it is possibleto investigatethe static form factors(6p

2) k and(6p2) . For the caseof harmonicdipoles

2 ei — k~1Tk

2 + k~e+ k~[1 — exp(—k2(r2)/2)](6p )k 4~ k2+k~+k~[1—exp(—k2(r2)I2)]

Yu.L. Klimontovich et a!., Statistical theoryof plasma—molecularsystems 295

/ 2 ci k~T (k2+k~)[1—exp(—k2(r2)/2)]ç&p )k 4ir k2+k~+k~,[1—exp(—k2(r2)/2)]

In the caseof rigid dipolesone has

k2 .T k2 + k~e+ k~[1— sin(kr0)/kr0]

/ 2’ e,i cic6P 1k = 4~i~k2 + k~+ k~,,[1 — sin(kr

0)/kr0]

/ 2 ci k~T (k2 + k2D)[1 —sin(kr

0)/kr0]~ )~ 4ir k

2+k~+k~[1—sin(kr0)/kr0]~

Let us calculate finally the spectraldistribution of the pair density fluctuations. Using the definition

6nei(R, t) = Jdx dP6Nei(~,t)

and eq. (2.26) for 6Nej(~, t), one obtains, after standardcalculations,the following result:

(6Nei(X,P) 6Nei(x’, P’))k~ = (6N~?~(x,P) 6N~?~(x,P))k~

4ir /a~(x’,P’) (6N~?~(x,P)6P~~)kW+ akW(x,P)(6p~6N~(x’,P’))kW)e(k,cv)

16~r2akW(x, P)a~(x’,P’) ‘~ (0)2 a

+ k4 e(k, cu)’2 (6p )kwo=e,i,ei

(6n~~)k~=JdxdPjdx’dP’ (6Nej(x,P)6Nei(x’,P’))kw. (3.39)

Here,

ak~(x,P) = ~inei ~ e1 f dx’ dP’ exp(ia1k r’)Wk~(x,P, x’, P’)k(~ + cx1 ~7)fei(~’, P’),

(8N~(x,P) 6p~°~)kw = nej ~ J dx’ dP’ exp(ia1k r’)

x [WkW(x,P, x’, P’)fej(X’, P’) + W’~(x’,P’, x, P)fei(X, F)],

WkW(x, F, x’, P’) = JdT exp(icvT) f d(R — R’) exp[—ik. (R — R’)] Wei(~,~“ ~Y

0

In the equilibrium caseeq. (3.39) yields

(6Nei(x,P) 6Nej(X’, P’))k~ = (&N~(x,F) 8N~(x’,P’))k~ — Im akW(x,P)akW(x’, F’)kcvT e(k,cv)

296 Yu.L. K!imontovich eta!., Statistical theoryof plasma—molecularsystems

After integratingover cv, one obtains

(6Nei(X, F) 6Nei(X’,~‘))k = n7j(6(P— P’)8(x — X’)fei(X, P)

— k2T:(k,O) ~ e,e

1 exp[ik. (a,r — a,.r’)] fej(X, P)f7(x’,

= ne(1— 4~Tflej~jIk2T), (3.40)

where~ei is the effective “electric” chargeof a boundpair,

= e,Jdx dPexp(ia,k.r) fei(X, P) = e1 f drexp(ia,k.r)fej(r).

In the caseof harmonicoscillators

~ei = e~exp(—a~k2(r2)/2mw~),

andin the caseof rigid dipoles

= e~sin(a1kr0)Ia1kr0

As follows from a comparisonwith the resultsobtainedfor plasmasystems[31],eqs. (3.40) coincidewith the static form factor for chargedparticleswith an effective electriccharge~er

3.1.3. Electromagneticradiation in plasma—molecularmatterIf one knowsthe correlationfunctionsof the electromagneticfields it is easyto calculatethe energy

characteristicsof the fluctuation fields. In particular,the Poyntingvector flow through a unit surfacewith normal n is defined by the relation

(Pa) = ~- n(6E X

where

(PW) = ~— n[(6E(R) X 613(R)) + c.c.]

c2 dk k.k. k.k. n.k.

= 4lTw I (2~r)~ - ~~i) + - LLjk. n(6E~6E1)kw + c.c.}. (3.41)

The contributionof collective processescan be expressedin termsof effective temperatures

Yu. L. Klimontovich et a!., Statistical theoryof plasma—molecularsystems 297

(P~) C0~= 2irc2 ~ (2 IT)3 (Ts(k, cv) Tr A(k, cv)

x [k. n - (e~. n)(e(7)* . k)] a Re4(k,cv)IOcvl+

wherethe summationis over all typesof collective excitations.In the isotropic case

k.k. / k.k.\A

11(k, cv) = —~ 4~.(k,cv) + — —~,~)4T(k~cv)EL(k, cv),

Tr A(k, cv) = 4T(k, cv)[4.p(k, cv) + 2sL(k, cv)].

If we take into accountthe transversefields only,

a Re4(k,:)/Ocv 6(cv — cv~)= c5(Re~T(1’,cv)),

coil = ~2 ~ dk TT(k, cv)(k. n)~(Re4T(k, cv)).

cv (2ir)

The radiation intensitywill be definedso that

1 coIl I

= j du1I~cosi~,

where t~is the angle betweenthe vectorsk andn. According to this definition, one has

= 1 ~ f dk TT(k, cv)k3~(k2— (cv2/c2) Re CT(k, cv)). (3.42)

(2ir) ~

If the systemis isotropic (see, for example, the high-frequencylimit), eq. (3.42) is reducedto

2cv Re CT(w)

= 3 2 TT((cvfc)VReET(cv), cv). (3.43)4ir c

In the equilibriumcasethiscoincideswith Kirchhoff’s law for the intensityof black-bodyradiationin aninfinite transparentmedium,

— cv2 Re ET(cv) T— 4IT3c2

The energydensityof the radiatedfields is describedby the Rayleigh—Jeanslaw,

= cv2[~ cv)]312 TT((w/c)VRCET(w), cv). (3.44)


Let us recall that eqs. (3.41)—(3.44) are obtained for an infinite medium. This meansthat thedimensionsof the systemunderconsiderationshouldbe muchlargerthanthe spatial scalesof dampingof the radiated wave. If this condition is not satisfied, i.e., if the system is transparentfor elec-tromagneticwavesof a given frequency,the radiationintensityshouldbe calculatedtaking into accountthe finite size of the system.The refractionof the wavesat the boundaryas well as wave transforma-tions shouldalsobe takeninto account.Finally, the correlationfunctionsof theLangevinsourcesin thecaseof a boundedmedium are alsoinfluencedby the boundary,which may makethemvery differentfrom thosein an infinite medium. Theseeffects are consideredbelow in sections3.2.3 and 3.3.3,

If refractionand transformationprocessescan be neglected,and the contributionof the additionalterms to the correlationfunctions is small, the radiation intensity may be estimatedon the basis ofelectromagneticfield energycalculationsin the far zone.

In such a case,the differentialcoefficient d2P/dcvdul, which definesthe powertransmittedthroughthe surfaceof thesysteminto a solid angledQ, is used,asa rule, to describethe radiationpropertiesofthe system. Calculationof this quantity on the basis of well-known relationsfor retardedpotentialsgives

dwdfl = 4~2c3(6J~W/(,,O, (3.45)

where12 is the unit vector in the direction of the solid angle.

3.1.4. Collision integralsfor an infinite plasma—molecularsystemAs was mentionedabove,the kinetic equationsfor the single-particledistributionfunctionsfa(X, t)

and the single-pairdistributionfunctionsfab(~C,t) maybe obtainedby averagingthe equationsfor themicroscopic phase densities. Our nextproblemis to representthe collision integrals1~(a’ = e, i, ci) inthe form of functionalsof single-particledistributionfunctions,supposingthat thereexistsa hierarchyof characteristictimesof evolutionfor statisticalprocesses.Sucharepresentationmay be obtainedif weexpressboth the fluctuationsof the microscopicphasedensitiesand the electromagneticfield throughthe sourcecurrents(i.e., in termsof fluctuationsin the systemwithout self-consistentinteractions)andthentakethe statisticalaverageof the relevantbilinear combinationsusingthe correlationfunctionsofLangevinsources.The final result is as follows:

‘U = — ~ I (2)~f ~ [(i — ~ — ~1]-~- {(6E, 6N~°~(P))k1,,

(6EI6Ek)kW [( k~V\ klVkl Ofa(X,tfl

— i(cv — k~V— i4) [~1 — — —~---jna ~, j~ a = e, i, (3.46)

Iah~f (2~)3I2~~dt[(1W(V+a/v))6zJ+W(V1+a/~)]

x exp(ia1k.r)(~ + a1 ~)[(6E1 6N~(x,P))k~+ (6E1 6Ek)kW

x ~ ei,(a~*(x,P) + [3~*(x, P) + y~(x,P) ~)n~bf~b~, ~]. (3.47)

Yu. L. Klimontovichet a!., Statistical theory ofplasma—molecularsystems 299

Here,

4ir

(6E16N~°~(F))kW= — —~ A~’(k,w)2lrnaVkfa(X,t)5(cv — ~. V), a = e,i,47T1 -1(6EJ 6N~j(x,F))k~= — ~ ~ (k, cv) ~ ei2lrnab

mcv0 ncv

X ~ J~(a1k. r)(_i)nJm(ai,k.v/cvo)(Vk + Uk - + rkk~v k~r~

n,m= —

XfUb(~, t)8(cv — k’V—(n + m)cv0), (3.48)

2l6IT —1 —1*

(6E1 6Ek)kw = T ~ (k, w)Akfl (k, cv)cv

X2IT[ ~ e~naJdPVnlYa(X,t)ô(cv—k.V)cr=e,i

+ ~ eletnabf dx dP ~ (— i)°J~(a1kr)Jm(atk~v/cu0)nm= —1,1

/ mcv0 nw0\x exp(ia~k~r)(V~+ a1u~)(~1’ + v. i— + r.

x fej(~’ t)5(cv — k~V— (n + m)cvo)], (3.49)

/ ~ — k.V/cv)+ k1V1/cva~~(x,F) = ~ (~Y~1n((1tk’~r)Jm(aik•v/cv0)(,.

n.m i(cv—k.V—(n+m)cv0+i4

— a1 (8,~k~v/cu — k1v1/w) — if3cv0(ô11 k~r/cv — k,r1/cv) \

2 ~ i(cv—k~V—(n+m—f3)cv0+i4)

f3~”~(x,F) = ~ ai(i)°Jn(aik~r) Jm(ai11. v/cv0)iin.m

~< (1 ~ 3,~(1—k.V/w) + k11’/w2 ~ i(cv—kV—(n+m—f3)cv0+i4)

— a1. (3~),k~v/cu — k,u1/cu)— if3cv0(ô1k~r/w — k,r1/cv)4 ~ i(cv—k~V—(n+m—2f3)cv0+i4)

— a1. k v/cu — k1v1/cv \2 i(cv—k.V—(n+m)cv0+i4))’

a1(1)-r~1 (x, F) = ~ (i)nJ~(aik. r) 1’m(0~i11. v/cu) —n.m mcv0

if3[8~1(1—k.V/cv)+k,V./cv]2 ~ i(cv—k.V—(n+m—[3)cv0+i4)

300 Yu.L. Klimontovich eta!., Statistical theoryofplasma—molecularsystems

a, (6,,, k~r/cv — k,r1/cv)+ i13(6,,1 k v/cu — k,v1/cv)

i(cu—kV—(n+m—2/3)cv0+i4)

— a1 cv0(6,1 k r/cv — k,r1Icv) 50

2 i(cv—kV—(n+m)cv0+i4) . ( -

Let us recall that, in orderto take into considerationthe retardationof the distributionfunctions,itis necessaryto usethe replacement

(6E/6EI)kW 1

i(cv — k.V—(n+ m)w0+i4)~i(cv — k~V—(n+ m)w0+i4)

/ ia/atX~1+i(cv_k.V_(n+m)cvo+i4))(6Ei6Ej)kW~

It shouldbe pointed out that the rangeof applicationof the aboveresultsis limited by our chosenmodelof atomicoscillators.Therestrictionof such a classicalmodelwhich we haveusedin the presentpaperis displayedin the structureof the collision integralsfor the distribution functionsof free, fa

(a = e, i), and bound,fab’ chargedparticles,eqs. (3.46) and (3.47). Namely, in theserelations,unlikethe correspondinggeneralresultsfor the quantumkinetic theory (seeeqs. (2.62), (2.64) in ref. [24]),the contributionsof chemicaltransformationsare absent.For this reasonthe condition of chemicalequilibrium (the Saha formula), which gives the relation betweenthe concentrationsof chargedandneutralparticles,doesnot follow from eqs. (3.46), (3.47).

If we are interestedin relaxationprocessesof an internaldegreeof freedom,it maybe possibletoneglectthe thermalmotion of the atomicoscillatorsandtakeinto accountquantitiesof first order in theatomicparameter.In such an approximationone has

e.a.8k / ~c?/e3t

/3(x,P)=~~ i(cv+/3w0+i4)~~cv+[3w0+i4

e1a1 v ~/36kJ ( ia/at ‘~

Yk~(X~P)=_2mw~ i(cv+[3w0+i4) ~

(6E1 6N~(x,P))k~= — ~ A~(k,cv) ~ (vk+ i[3wOrk)fUb(~, t)6(cv +

Usingthe antihermitianpropertyof the tensor(6E1 6E1) kw we areled to the following result for thecollision integralof a boundpair:

‘ab = ~ [Y~bpjfab(~, t)] + ~- [~w~r,mcvOfUb(~, t)] + D~Ô~b~t) + G~a~b(~t),

‘ (3.51)

where

Yu.L. Klimon:ovichetal, Statisticaltheory ofplasma—molecularsystems 301

D~= eab J (2)~[(bE, bEj)kw + J~(~coX~~,,(w))~ (bE,

ab eat, f dk LC~dw (bE,bEJ)kW m f dco ( ô * 1G~1= — 2 1 (2ir)3 Li 2iT 0) — ~o + 42 b J ~ ~ c°X~(0)))~ (bEe bEJ)kj

a a (3.52)

~ab = - 4lTeab I (2)~ImA~1(k,We), ~ 4i~e~~I (2f ReA~’(k,con),

47re~bnab 1Xab(~)= m (w2—w~)+2icoz1~

The generalexpressions(3.51)correspondto the resultobtainedin refs. [24,112]. At thesametime thecoefficientsD~and y~differ from therelevantquantitiesin refs.[24,112] by numericalfactors.Suchadifference appears,becausein the presentpaper D and y~’are given in the laboratoryreferencesystemwhile in refs. [24~,112] they are relatedto the referencesystem associatedwith an oscillatingparticle. Going from the laboratoryreferencesystemto an oscillating one, we obtain full agreementbetweenthe results.

As regardsthe collision integralsfor free chargedparticles,eq. (3.46),they representthegeneraliza-tion of Balescu—Lenardintegrals to the casewhen the total electromagneticinteraction and theinfluenceof themolecularsubsystemaretakeninto account.Up to first order in theplasmaparameterthe following expressionfor (bE, bE

1)~ should be substitutedinto eq. (3.46):

(bE, ~j)kw = ~T A~1(kw)A~*(k,co) . 21T(~ e~naJdP %‘V~6(co— k . V)fa(X, t)

+ ;ab f dx dPv~~ ~(v1 + i~w0r1)6(w— k . V + ~~Q)fab(~, t)).

In the caseof large phasevelocitiesof fluctuation excitations ü ~ kSa (~= Ta/ma,a = e, i) thecollision integral for free chargedparticlesmay be reducedto theFokker—Planckform,

‘a = - ~a(’~)fa(X, t)] + ~ [D~(P)fa(X, t)], (3.53)

where

D~(P)= e~f dk ~ (bEebEI)kk.v,(2ir)

e2 I k. 1 oD,~(P)


For isotropicdistributions the correlationtensor(bE, bEJ)k,,~is definedby the two terms (bE~T)k~,

[seeeq. (3.25)]. According to sucha representation

D~-8 eab f (~(bE~)kW0+3(bET)kW).

Using the relationsfor the spectraldensitiesof the correlationfunctionsof the electromagneticfieldin the region of collectivefluctuations,eqs. (3.29), (3.30), andmaking the appropriatecalculationswiththe aid of 6-functions,we obtain

D0~) _ Dab(c0~)ô,1

where

= 3eab(3 \/Re ~T(~O) TT((wO/c)VReET(wo), ~

+ ~ ~ ~Re ~L(~o) TL((w~I~wpese)VReEL(coo),Se 6\/~wpe

(3.54)2 2 2 2 2 2 2 2

Re ET(wo) = 1 — cope/coo — ~pm’4~o , Re EL(wo) = 1 — +

TT(k, co) and TL(k, co) are the effective temperaturesof the collective fluctuationsdefinedby eqs.(3.28). It shouldbe notedthat the longitudinal collectivefluctuations(andtherefore,the secondterminDab(c011)) exist only when the following inequality is satisfied:

ReEL(wo)~w~e/co~.

This inequality correspondsto the requirementof weak dampingof longitudinal waves.In the equilibrium case

D0b~0h) 2 e~w~Tx/Re ET(wo) (i + 6V~w~~~~Re :~::~)• (3.55)

When Re EL(coo)~ ~ the secondterm in parenthesesin eq. (3.54)maybe omittedandwe areled to the well-known result for D~c0I~[18,24]. At the same time, if the opposite conditionsaresatisfied, ReEL(wo) ~ ~ the contribution of the longitudinal fluctuations may be significant.Besides,due to the presenceof free chargedparticles the influenceof individual-particlefluctuationsbecomesimportant.This contributionto D~il~is given by the relation

~ab T dkk2 Im s(k,~e(k, w

0)J

wherekmjn is the upperlimit of the collective fluctuationdomain,kmaxis a wavenumberof the orderof

Yu.L. Klimontovichet al., Statisticaltheory ofplasma—molecularsystems 303

the inverse of the shortestdistancebetweenthe particles.For chargedparticles kmin kD (k~,=

~a=e,i 47re~na/Ta Eaej k~)and ~ -~1Ir~,-~3Ta/e~.As far the molecular componentis con-cerned, the quantity k~J—~hr0 (r0 is the effective size of an atom) is important for furtherconsiderationonly. EstimatingD”~p~on the basis of this schemegives

= 2 e~t,w~T(ir/2)112 ~pe ~ [ln(k~~Ik~) + 2 ‘~ (~b~)2] . (3.56)

3 rrc (00 S~ 4O~m ke

Becausefor a rarefied gas ~ w ~, the contributionof a molecularcollision (comparedto thecontribution of a boundparticle to thecollectivepart of D~u~)may be neglected.At the sametime, inthe caseof a relatively denseplasmasubsystemtheelectroncontributionmay be decisive.

It is also easyto calculatethe radiationfriction coefficient y ‘)‘. In the caseof an isotropic mediumonehas

ab abyij =7 ~

where

7ab = 7ab(coll) + 7~th(ip)

~ab(coll) = 2 eab 2 \/ReST(coo) (i + 6\/~(O~e~ (3.57)

ab(i.p.) — 2 eat, __E 11 Ik(e) 1k ‘~+ —~ ~- ~ 1— 3 ~ ~ L n~max e) 42 ~m ~ ke “ ~

which in thecaseof puremoleculesis in good agreementwith thewell-known result for the radiationfriction coefficient.

For equilibrium distributions

= D~ImT, i~co~’= G~w0IT,

and I~t,becomesequalto zero. Here, we have used the following representationfor the correlationfunction of the electromagneticfield in the equilibrium state[seeeqs. (3.36)]:

8irT -1(bE, bE1) = — Im nil (k, co),

and performedthe integrationover co in eqs. (3.52) usingthe Kramers—Kronigrelations.Thus,we seethat the presenceof a plasmasubsystemmayaffect appreciablythe kinetic coefficient

ab which definesthe relaxationof the internal motion in molecules.

Let us recall that eqs. (3.51)—(3.57) were obtained taking into account the dipole part of themolecularfields only, and do not include the correlationsof bound pairs at shortdistances.Let usconsiderthepossibility to describethe influenceof suchcorrelations.In order to do this, it is necessaryto representthe generalrelation (3.47) in the following equivalentform:

304 Yu.L. Klimontovichet a!., Statistical theoryofplasma—molecularsystems

4lTif dk fdw [7 k ~ k.‘ab~J (2~r)~J2

xA~1(k,w)fdPf[~ ~ (bNa(P’)bNab(X,P))kw

+ ~ e1(V~+ a,v~)(~+a~. (bNab(X’~P’) bNCb(X, P))kW], (3.58)

where

kk 22 kkA~1

0~(k,co) = —~ + (~— ~-4-)(~,— ~

In zeroth order in the thermal parameter,eq. (3.58) gives the following result for the partial

contributionof atom—atomcollisions:

47re~bI dk f dw ~ , a f , ,/ 1 a a 1 a2 a2 i a3 a3

‘nab ~(2~)~J ~ ~‘~ik (k, w)rk ~- j dx dP + — ~ ~—~+ ~ at3

x (bNab(X’, P’) bNab(X, ~))k~ . (3.59)

Here, wehaveappliedthe relationbJ~= — ico bP~(which is valid in the dipole approximationusedabove)and madethe replacementv’ —~ — iwr’ in the integrand.We havealsotakeninto accounttheretardationof the distributionfunctions.Obviously, if oneneglectsthe retardationandusesthe dipoleapproximationto describethe quantity (bNab(X’, P’) bNab(X, P)), eq. (3.59) transformsto the aboveeq. (3.51).

Let us try to take into account the correlationsat short distances.We shall assumethat suchcorrelationsare defined by the interactionsof pairs as a whole. According to this assumption,thefollowing approximaterelationfor the correlationfunctionsof the pair microscopicphasedensitycanbe written down:

(bNab(X’, P’) bNab(X, P))k~ (bNab(X’~P’) bNab(X~P))~ + n~bfab(~e,t)fab(~’, t)g~~,

where (bNab(X’, P’) bN5t,(X, p))~thP)is the spectrumof the microscopicphasedensitycalculatedin the

dipole approximation,~ is the spectral density of the pair correlation function g(R,r) at shortdistances.The presenceof the secondterm in the last equationleadsto anadditionalcollision integral:

blab 3 afab(~,t) ~ab I (2~)~I ~~I dx’ dP’ r’fab(~’,t)

— 2i eabflab afab(~l2,t) f dco a3 I d ‘dP’ ‘

3 ~ ap J ~ k0 ~ x r fab( , t)

— eab afat,(~,t) Pat,(R~t)g(0) + ~ eabnabwo afab(~,t) I dr g(r) aPab(R,t) , (3.60)


where

g(R)asg(R,0), Pat,(R, t)= eabnabJdxdPd~~(~,t).

It is easyto seethat the first termin (3.60)definesthepolarizationcontributionto theeffectiveLorentzfield and thesecondone that to theeffectivecoefficient of themolecularpolarizationdamping.In fact,combining eqs. (3.51) and (3.60) and performing a transition to the evolution equation for thepolarizationvector, one obtains

+ ab 81~ab+ ~20p = e~F~(E — ~irg(0)P~t,),

where7ab = yab(l + ~ab I g(R)dR).

Here, we take into accountthat in thechosenapproximation

jat,(R, t) as eabnabI dx dPVfab(~t,t) = (9Pab(R,t

)

ôJat,(R, t) — W20Pat,R,t) — eabnabE(R, t) = — g(O)P~t,(R,t) — 7.f~tat,(’~, t)

The last two equationsmay be obtainedby iterating the kinetic equationwith the weightsr and v,respectively.

The general relations obtained above enable us to investigate the influence of the molecularsubsystemon the kinetic coefficientsfor chargedparticlesandto estimatethe effective frequenciesofcollisions betweenelectronsand molecules.In particular,in the caseof a nonisothermalplasmathekinetic coefficientsfor electrons(up to Chandrasekhar-dominantterms[113])may be representedas

D,~(P)= 7T D~(P)+ (6,, —

D~(P)= ~ (a~,j e~naln ~a G(VI’thSa) + ea ,~2ab(v2) In Aa~)~ (3.61)

D~(P)= ~ (a~,i e~naln “a [cb(VI\hsa) - G(V/\I~5a)]

+ eabnab (v2)[~(k~2V2Ico~-1) _lflAa~]),

Fe(P) = —4rre~~( ~ (1+ me/ma)ln ~a G(V/~/~Sa)a=e,, mesa

+ ~ -~-~-~ [ln ~ab + ((02) /V2)~(k~2V2/w~—3)]). (3.62)

306 Yu.L. Klimontovichet al., Statistical theory ofplasma—molecularsystems

Herewe use the notation

~(x)= ~fe~2dy, G(x)

k(a) (ab)max . max (ab)= —~--— , a = e, 1; ~ab = O(kmaxVIW() — 1),

~0

(u2) = ~ I dx dP V2fab(~~t), ~ =

r~,1is the shortestdistancebetweenan electronand a particle of speciesa. For an electroncolliding

with a chargedparticle

~ = eeea~/3Te, a=e, i,

andin the caseof a collision with an atom r~ r0, where r0 is the effective size of an atom.The effective collision frequenciesmaybe estimatedon the basis of the following relations:

= —(aea/at)st/ea, Ea fdP ~ fa~, t), (~a) fdPv.F(P)f(x t).

Taking into accountthe structureof the quantity Fa(P) = ~~r=e,i,ei Fa(T(P), it becomespossibletoidentify the contributionsrelatedto the collisions with particlesof differentspecies,

= ~, ~ = _fdPV~Fa~(P)fa(X,t)/fdP ~maV2fa(X,t).=e i ci

In particular,for electron—ionandelectron—moleculecollisions one obtains

4 ~ e~e~n1

3 ‘~m~T~’2 ln A,

4 e~e~n~,m~ 2 1’~max(0) / U \ / w~ \13 T~’2TCI ~— (v)~ 2 exp~-2k~2s~)_E1~

2k~j)22)j, (3.63)

where

Ei(x)=f~_—dy.

As follows from theserelations, the frequencyof electron—ioncollisions coincideswith the resultobtainedon the basisof the Landaucollision integral.As regardselectron—atomcollisions,the effectivecollision frequencymaybe written in a form which is similar to Spitzer’s result [114],

= °e,ei

t1ei5e

Vu. L. Klimontovich et a!., Statistical theory of plasma—molecular systems 307

where

Ueei = /~ r212k(d1)2r2exp(_ 2 1 \ (2k~2r2)]oL max k(di)2 2) — 2Eimax r

0 max 0

If it is necessaryto investigatethe evolution of atom distributions (asawhole), we shouldconsiderthe quantitieswhichareproportionalto themomentumP on the basisof generalrelations.However, itis possible to consider one particular casethat enablesus to simplify the generalexpressionsforcollision integrals.This simplification is relatedto the quasi-staticapproximationfor the fluctuationsofelectromagneticfields (cIw0> L, whereL is the real dimensionof the system),i.e. to low-frequencyoscillators.

In order to obtain approximateexpressionsfor the collision integralscorrespondingto the quasi-static description,we should takebEkU = — ik b4kW. The final result is

‘a = ~ l6ir eaea~naJ ~ . — ~)faC~, t)fa(X’, t)1 6(k.V—k.V’)322 (2~.)3I~0 a Ik fa a _________

a’e,i ~I s(k, kV)12

I dk+ i ~ 16i e,e,e~nabj ~ I dx’ dP’ k a ~ e~’~J~a,kr’)Jm(aik~v/w

0)kaPn,m

it’ (2ir)[ k a k•V ~.9\

~< ~ô.P k•V’nm nm

x (—i)”~ a=e,i; (3.64)

I dk f‘at, = i ~16ir

2e,e,n~,t,J (2ir)3 J dx’ dP’ e~1~~1T’)1’,l”

rk (~ aXI—~ +a

L k2 ~i “ ~) ~ ~ e,e,J~(a~kr)Jm(a,k~v/w0)J~(a,k.r’)Jm(a,k~v’1w0)

,m1 ,~,1(_i)~1 k / a k~Vnm ~ ~ t)f0t,(~’,t)]

XIE(k,k•Vnm)~2(k•Vnm_k~Vl +~4)k2~t9PnmkV1n1m1 n1m1

8~nm

+ i ~ ~ 16~r2e,e,~e~n~I (dk ~ I dP’ ~2~)a=e,i 1,!’

Ik (ô axl—. +aL k2 ~P ~~—) ~ J,,(a,k r) ~m(°~i”~ v/w

0)

(—i)~ k / a k’Vnm aX s(k,k~Vnm)

2(k~Vnm— k•V’ +i4) ~ — kV’ ~)fa~(~, t)fa(X’, t)]

ía a— ~ F,(R + a

1r) ~ + a, ~~)fa~(~C,t), (3.65)

where

308 Yu.L. Kiimontovich et al.. Statistical theory ofplasma—molecularsystems

a a mco0 a a—‘ kV =kV+(n+m)co~,

aPnm a~ kv ap mw0kr ar

F1(R + a,r) = 4~ie1f dk ~ e, ~ Jn(ai~r) Jm(aik~v/w0)(i)~ 2 k exp(ia1k.r)(2ir) i’ n.m k r(k, 0)

.Idk~ k= —4~rie1J L e1 2 exp[ik~r(a — a1.)]

(2ir) i k r(k,0)

a f dk exp(ik’r)= ~ei~eab~~ ~ I (2~ k

2s(k,0) (3.66)

Let usnote that the quantitiesF~(R+ a1r) arethe forcesactingon a particleof species1 (i.e. on one

of theparticlesboundin pairs) dueto the presenceof a screeningchargecloud inducedby a boundpairsituatedat R. Equation(3.66) maybe transformedto

~ F1(R + air)(~ + a, ~)fah(~~ t) = 41re~h~ I (2)~~ k afUh(7C, t

)

acb~(r)k~afQh(~t)— ar ap (3.67)

2 dk exp(ik.r)~ab(T) = ~

4~eab f 3 2(2ir) k r(k, 0)

It is easyto see that in the caseof equilibrium distributions the first terms in ‘ab’ as well as thecollision integralsfor free electronsandions ~ becomeequalto zero. At the sametime, the last termin

‘ab’ which takesinto accountstaticpolarizationforces,is not equalto zero. The presenceof thisterm iscausedby theparticleinteractions,whichmaylead to a changedatomicdistributionfunction. In fact, inthe generalcaseof stationaryequilibrium the distributionfunction for atomsshould be calculatedonthe basis of the following kinetic equation:

(~. - [c~t,fr)- ~ ~)fab(~) = a~(r) afflh(~C) (3.68)

where 4~b(r)= —e~t,Ir. The solution of eq. (3.68) is as follows:

fat,(~)= f~(P)f~(p)exp[- ~ah(T) IT]/f drexp[- ~~h(r) IT], (3.69)

where

~ab(r)~ab(T)+ ~b2Idkexp(ikr) (i e(k~)) (3.70)

In the caseof rarefied systems(k~(r2)~ 1) the contributionof the secondterm to eq. (3.70) can be

neglected.Let us formulatethe basicpointsof the aboveresultsbeforemakingfurthercalculations.We seethat


in order to calculatethe collision integralswhich arepresentin thekineticequationsfor electrons,ionsand boundpairs, it is necessaryto know the correlationfunctions(or their spectraldensities)of themicroscopic(collisionless)fluctuations.The kinetic equationswith the collision integralscalculatedonthe basisof the above approachenableus to define different kinetic coefficientswhich give relationsbetweenthe responseof differentphysicalquantitiesand therelevantgeneralizedforces.This approachto the calculationof the kinetic coefficientsmay be calledthe kinetic one.

Naturally, it includesthehydrodynamicdescription,too. Namely,takinga transitionfrom thekineticequationsto hydrodynamic ones, we obtain expressionsfor the appropriatekinetic coefficients:viscosity, heatconductionanddiffusion coefficients.In suchcases,for instancethe viscosity coefficientwill give a relation betweenthe momentumflow and velocity derivatives,and the heat conductioncoefficient betweenthe heatflow and the temperaturegradient.This methodof descriptionmay becalledthe hydrodynamicone.

It is a well-known fact that thereis alsoanotherapproachto thecalculationofthekineticcoefficientsboth for the kinetic and hydrodynamic levels of description. This approachmay be called thefluctuational one. It wasproposedby Kubo, GreenandMori and then wasfurtherdevelopedby manyotherauthors(see,for instance,ref. [115]).In all casessuchan approachis basedon the inversionoffluctuation—dissipationrelations.

It is important that the fluctuation—dissipationrelations can be formulated in the framework ofdifferent levelsof description,in particular,in thecaseof thecollisionlessapproximation.An exampleof such relations is given by eq. (2.58), which expressesthe correlation function of the electricmicrocurrentsand the collisionlesssusceptibility.This type of relationis well known in the theory ofplasmasandgases.However,theirpractical usefulnessis not so greatbecausethe rangeof applicationof thecollisionlessapproximationis restrictedby the relevanttemporaland spatial scales.

Much interest hasrecently arisen in applicationof the fluctuational approachto calculationsofdissipativecharacteristicsboth on the kinetic and hydrodynamiclevelsof description[i.e. throughthespectraldensitiesof large-scale(kineticor hydrodynamic)fluctuations].Naturally, the resultsobtainedin the frameworkof suchan approachwill also includethecharacteristicscalculatedin thecollisionlessapproximation.An example of suchcalculationswill be given in section5, where the correlationfunctionsof bremsstrahlungfields are usedto describedissipativeparameters.

3.2. Half-space

3.2.1. Electromagneticfields in a semi-infiniteplasma—molecularsystemLet the plasma—molecularsystemunderconsiderationoccupythe half-spaceZ >0. The external

region(Z <0) is occupiedby a dielectricwith dielectricconstant~. We shall assumethat particlesof allkinds arereflectedspecularlyfrom theboundaryZ = 0. In this casethe boundaryconditions(2.31)and(2.32) for the transition probabilitiesare of the form

~“a(~, X’, ~ = Wa(Xt, X’, r)[~0 ‘ ~~ab(~’ ~“ T)[~0 = 14Tat,(~2t,a”, T)Izo

The solutionsof eqs. (2.27) and(2.29) supplementedwith theseboundaryconditionsandthe initialconditions (2.28) and (2.30) are

Wa(X,X’, r) = 6(R1 — R~— V1r)5(P1 — P~)

x [8(Z — Z’ — V~r)6(P5— P~)+ 8(Z + Z’ — V~r)6(P~+ P~)]exp(—i~T), (3.71)

310 Yu.L. Klimontovichet a!., Statistical theoryof plasma—molecularsystems

Wat,(~,~“, T) = 6(R1 — R~— V1 T)6(P1 — PI)6(r±— r~cosw0r — (p1Imw0)sin w0r)

x 3(p1 — p~cosw~r+ mw0r1sin w0r)

x [6(Z — Z’ — V~r)6(P2— P~)6(z— z’ cosw~r— (p~Imco0) sin w0r)

x 6(p~— p~cosw0r + mw0zsin w0r)

+ 6(Z + Z’ — V~r)6(P~+ P~)6(z+ f cosw0r + (p~Imw0)sin w0r)

x8(p~+p~cosw

0r—mw0zsin (00T)]exp(IXT). (3.72)

Comparingthesesolutionswith eqs. (3.1) and (3.2),we seethatthe presenceof a boundaryresultsin additional termswhich describethe reflectionof particles from the boundarysurface.Taking intoaccountthe explicit form of the transitionprobabilityfor an infinite mediumit is possibleto rewrite eqs.(3.71) and (3.72) in the following form:

~‘a(’~’ X’, r) = W’~°~(X,X’, r) + W~°~(X,X’t, r) , a= e, i

Wat,(~,~“, T) = W~(~’,~“, r) + ~ ~ r)

where ~ as(x’~,Xit), xit (nt, pit) X’~ (R’~,pit), Rit (Ri, —Z’), P’t (Pt, —P~);w~°~(x,X’, T) and W~?](~,~‘, r) are the transitionprobabilities in an infinite medium, which aredescribedby eqs. (3.1), (3.2). Taking also into accountthat

W~°~(X,X’t, r) = W~°~(Xt,X’, r) , ~ ~ r) = W~(~It,~‘, i-)

one can see that the abovesolutionssatisfy the boundaryconditions.Substitutingeqs. (3.71) and (3.72) into the generalrelationsfor the susceptibilitytensors(2.30) and

Fourier transformingthe time andtransversecoordinates,one finds [54]

~ Z’, k1, co) = f ~ {exp[ik~(Z— Z’)l P~(k,co) + ~ exp[ik~(Z+ Z’)] P~(k,w)},

(3.73)

where

P7(k, i0) = G~(k,co) ±iF~(k,co) a/aZ’, (3.74)

G~(k,w) 4ire~na I I’~afa(X, t)/aPk8kJ(~— . V

1) + kIkV~

= 2 idP . a=e,i,

F~(k,co) co co—k• +6kj~’Z+ V6kz~

G~(k,co)‘1 ~ 4~eleinab~ (~i)~Jn(ajk~r)Jm(a,k~vIwo)

F~”(k,co) ti’ W n.m

V,+vmco0/(k.v)+r1nw0/(k.r) / a a ‘\

X w—k.V—(n+m)co~+iz1 ~~a)

Yu.L. Klimoniovich ci al., Statistical theory ofplasma—molecularsystems 311

t5kJ(0) — k1 .~f — cr,,k1 .v1)+ kjkI’ + a,.klkuJ,x

+ a,,v5) + ~kZ(V~ +

s11, jx,y, s1—1, j=z.

The correlationfunctionsof the fluctuationcurrentscalculatedon the basisof eqs. (3.71), (3.72) are

(bJ~°~(Z)bJ5°~(Z’))~~= I “~z {exp[ik~(Z— Z’)] + e~,exp[ik~(Z+ Z’)J}(bJ~°~bJ~°~)~~(3.75)

where (bJ~°~bJ~,°~) ~ is the spectraldensityof thesourcecorrelationfunction for an infinite mediumgiven by eq. (3.6).

Usingthe generalrelationsfor the dielectricpermittivity tensor,it is possibleto calculatethe electricinduction,

bDIkW(Z) = J dZ’ e~1(Z,Z’, k~,co) bEJkU(Z’),

where

e,1(Z,Z’, k1, co) = 6,16(Z— Z’) + ~ ~y~(Z,Z’, k1, co).oe,i,eI

It is easy to see that the structure of bDk~(Z)can be simplified if one usesthe well-knownprocedureof specularcontinuationof electricfields bE(Z) into the regionZ <0 [116],

= n,

Using the explicit form of the susceptibility tensorsgiven by eqs. (3.73), (3.74) and the symmetry

propertiesof the fields, one obtains

bD,k~(Z)= f ~r J dZ’ exp[ik~(Z— Z’)] e~1(k,co) bE)k~(Z’), (3.76)

where e~1(k,co) = — Eo~ejet ~ co) is the Fouriercomponentof the dielectricpermittivity tensorfor the relevantinfinite medium.

Notice that the electric inductiondescribedby eq. (3.76) hasjustthe samesymmetrypropertiesastheelectric field. This enablesone to extendthe induction to the regionZ <0 and to performFouriertransformationwith respectto Z. The result is

= e,1(k,co) &EJk~

Using thenthe Maxwell—Lorentzequations(2.24) and taking into accountthe symmetryproperties

312 Yu.L. Klimontovichet a!., Statistical theory of plasma—molecularsystems

of fields and induction, we obtain a set of algebraicequationsfor the fluctuation fields in the (k, co)

representation,

A,1(k,w) bEIkW — 4iri (bJ~L— e2,1bBJkW(+0)) , (3.77)

where

bBJkW(+0)= urn bBIkW(Z), bJ~= ~

= e,i

bJ~°~is theFouriercomponentof thecurrentbJ~°~(R,t) givenin the region Z >0and thenextendedto the region Z <0 in a specularmanner.The tensorA~1(k,co) is defined by eq. (3.20).

Solving eq. (2.72) together with the boundary conditions for the electromagneticfields (thetangentialcomponentsof the electricandmagneticfields areequalat the boundary),we areled to thefollowing final resultdescribingthe distributionof fluctuation microfields [55,94]:

J (dk~I2ir)exp(ik~Z)bEk~, Z >0,

bE(R, t) = I ~ f dk1 exp[i(k1 . R1 — cot)] (3.78)21T (2w) exp(—ikZZ)bEk~, Z<0,

where

= — ~ A~(k,co)(bJ~+ ~ (~j~+ k~k1k)L1(k co)

e 4iri I z —1 (0)= — j -i—— A~

1(k, co) bJjkia

ck ck~k.L,1(k1, ~.O)r 6~j+ —i- S,1(k1,co) + ± ~ S1~(k1,co),co k5w

2ic I dkS.1(k1,w)—j -~—~A,j’(k,w),i,j=x,y,

= L~(k1,w)bEk~, i, j=X, y,

bEzk U = k1~L~’(k1,co) bE k ~/kZ [(w2/c2)e—k~]v2, Im lc~>0. (3.79)

It follows from eqs. (3.78) and (3.79) that in a semi-infinite system both surface and volumeeigenexcitations(longitudinal and transverse)can exist. As will be shownlater on, surfaceexcitationsmaygive a significant contributionto the spectraldensitiesof electromagneticfluctuations.

The generalexpressionof the dispersionrelation for surfaceexcitationsis definedby the conditionthat therebe a nontrivial solution for bJ~?j= 0 and detA,

1(k, co) ~ 0. The result is

det L,1(k1, co) as L(k1, co) = 0. (3.80)

Yu.L. Klimontovichci a!., Statistical theory ofplasma—molecularsystems 313

This equationdefinesthecomplexfrequencyw(k1)= + i1 for differentbranchesof surfacewavesfor given valuesof k1. In the transparencydomain [ReL(k1, co) ~ urnL(k1, w)I] their spectrumis asfollows:

Im L(k1, co)ReL(k1,(°k1)0, ~ = — a ReL(k

1,w)/ôcow=w~

In thequasi-staticlimit (k1clco~ 1) theseequationsyield

k1~ f dk5 — Im f dk~Ik2e(k,cok)

1 + —~— Re j k2r(k, (Ok) 0, “k.t = — (a/awk)Re f dk2lk

2s(k,~k)

Forisotropic systemseq. (3.80)splits into two equationsfor p (E~~ 0, B~= 0) and s (E5 =0, B~~ 0)

polarizedexcitations,

L~(k1,w)0, La(k±,04=0, (3.81)

where

~ fdk5f k~ k~L~(k1,w) 1+ ~-;:;~‘-j -~- ~L(k, co) + ET(k, (0)— k2c2/w2)

L~(k±,co) =1+ ~ J dk2 2 2 2’ ~ = [k~ — (co2/c2)fl”2.

IT CT(k,o))kc/co

Let us consideras an examplethe high-frequencysurfaceexcitations~ ~ k±Se,0) ± (O~~ 1’5m) inisotropic plasma—molecularmatter.In this case

k2ReL~(k±,co) 1+ _E() (K~— (k~— W4S(0))/3W~e)1/2)~

/ 8 \1/2 ~k±5eW~eImL~(k

1,U)~=—~—) 3IT (0

x (Ie(k, )I~ + ~ ~2 ~2 2 Ie(k, w)12 ~)a

5e 0)pe ~ p=±i (w+I3coo)

2 2 2 1/2K

5[k1(o) Ic )s(w)]

It follows from theseequationsthat in theapproximationof a “cold” systemco and k~arerelatedby

2 — 2 22 ~ sls(w)I

0’~pe ~0pmk1(w)= —i- ,. ~. , s(co) 1 — 2 — 2 2

C ~W~S co

If the frequencyco tendsto ~12 definedby the equation ~(~)I= ~, i.e.,

314 Yu.L. K!imontovich et al., Statistical theory of plasma—molecular systems

2 2 2 2 2 2 21/2

—2 1 ( 2 0)pm+C0pe~\ 1 Fl 2 0)pm+0)pe~4~pe~O1

~ 1~ )±~L~U)0+1~ ) — i+~ ]

the wave becomesquasi-static.In this case it is necessaryto take into accountthermaleffects. Theresultof the appropriatecalculationsis as follows:

2 -2 1 F 2 2 1 ((~~e+ (0~m)2 2 2 2

0)1,2k1

0) +2(1+~)L0)p 0)pm± 1+~ +wo(wpmwpe)

x (~ k1S;wpe - ~2w2)

~1.2 c k1(3.82)

(2 \1/2 [(1+ )(k~ - w~)- ‘°~m](°~j- w~)1 r k±Se 2 22 2 2(1 + s)(wk — ~ +

0)pm0)O

2 2 2( 1 S~0)pmWkj _______________~ ~ Wk)uZ~~ ~e ~ 4~~ 1 (k~ + pWO)uE( k k)~

The volume wavesin the caseof a specularlyreflecting boundaryare describedby the relationsobtainedfor an infinite medium.

In order to define the averagevaluesof physical quantitiesand their correlation functions, oneshould know the spectraldensityof the correlationfunctionsof fluctuation sourcescalculatedwith thespecularextensionof a currentto the region Z <0 takeninto account.As maybeshownon the basisofeq. (3.75), the correlationfunctionsare describedby the relations

(bJ~°~bJ~°~)~= 2IT[8(k~— k~)+ e.6(k~+ k~)](bJ~°~bJ~°~)U, (3.83)

where the quantities (bJ~°~bJ~°’) ~ are the spectral densities of the correlation function of thefluctuation sourcesfor an infinite medium definedby eqs. (3.6).

3.2.2. Electromagneticfluctuationsin a semi-infiniteplasma—molecularsystemIf the solutionof the problemof electromagneticfield excitationsby given externalcurrentsand the

statisticalpropertiesof the sourcesare known, it is not difficult to calculatethe correlationfunctionsboth of the electromagneticfield and electricchargedensity fluctuations. In particular,

Idw I dk Idk Idk’(bE,(R,t) bE

1(R’, t’)) = J 2IT J (21T)2 J ~ J 2ir

x exp{i[k1 . (R1 — RI) + k~Z— k~Z’— w(t — t’)]}(bE,k bElk)k~

(3.84)

where

(bEIkbEfk)kW=2ir[6(k~ — k~)+ E.6(k + k)](bE, bE1)~~

+ (bE~kbEIk)~W+ (bE~kbEJk)~U (3.85)


(bEt bE1)~’~= 16’~r2A~’(kw)Al*(k co) ~ (bJ~°~bJ~°~)~

0Ct) oe,j,ei

(bE~kbEJk.)~W= E~1(k2,k) + E(k~,ks),

E11(k5,k~)= i 64IT2c2k~A~’(k,w)Ai~*(k,w)A7~*(kl,co)

x ~ + k±~k±~,Ik~)L~*(k1, co) Z . (bJ~°~~

o.=e,I,ei

(bE,kbElk;)W = 64ITc~lk5I2A~’(k ~)A~l*(kl ~)(~k’ + k±kk±m/k~)

x (~+ kink±iIk:2)~mt{~ (bJ~°~bJ~)~~},o=e,I,el

çbm,{T~p(1~,w)} = L~(k1,0))L~*(ki, ~I dk5A~,’(k,~)ni*(k W)Ta~(k,w).

Let us note that in the caseunderconsiderationthe correlationfunction for the electromagneticfields (3.84), (3.85) includesthe traditionalvolume term,aswell astheadditionalcorrelationfunctionsproducedby theboundary,which take into accounttheexistenceof collective surfacefluctuations.

In fact, in thedomainwherethesolutionof theequationL(k1, co) =0 exists,the surfacepart of thecorrelationfunction (bElk bE/k ) ~, is proportionalto ö(ReL(k1, w)), i.e., the spectrumis character-ized by well-pronouncedmaximarelatedto theexcitationof fluctuation-likesurfacewaves.In this case

‘i’mi{ ~ , (bJ~°~bJ~0))~~}= ~- Ii~HTr~~1(k±,w)uT(k±,co)5(ReL(k1, co))o=e,I,e, c

5~5m5n*5k[~~nk(hhi,co) + S~(k1,w)]ss{k[kS(k co) + k1~S~(k1,0))] i~I2[sflk(kI, co) + Skfl(k±,w)]}’

where

41T 5~Sk~ dk5A~(k,~‘)~k’ (k w) Eu.e.j.ei (b,1~°~bJ~°~)~~

T(k1, w) 10) 5~Skf dk5fl~(k,w)A~*(k,o))[s~(k,co) — e~1(k,co)] (3.86)

is the effective temperatureof thecollective surfaceexcitations[55],

as s1(k1,~k) = {~1(k1,k) I[~1(k~,wk)a~alTr ~1(k1, wk1)1}al

is thepolarizationvectorfor the projectionof the electric field on the boundarysurface,

~1(k1,co) = Tr L~1(k1,co) — L~1(k1,co).

In the quasi-staticlimit (k1c~ w) s ask11k1, i~= k1,

316 Yu. L. Klimontovich et al., Statistical theory of plasma—molecular systems

— ITW k±ik±nk T(k1,w)6(ReL(k1, co)),— ~ ~2 I

where

‘ (O)2’a I

~cre,i,ei f (dk5/k4)çbp lkWIIe(k, w)12

T(k1, co) =2rno J(dk5/k

2)ImE(k, )/ls(k, ~)J2 (3.87)

and L(k1, co) is definedby

L(k1,w)=1+~-~fdk~ 1IT Jk

2s(k,w)

In the caseof an isotropic medium the generalrelationsmay be simplified. For such a systemthevolume part of the correlationfunction is describedby just the sameequationsas in the caseof aninfinite medium [seeeqs. (3.25), (3.26)].The surfacetermsof the correlationfunctionsareas follows:

64 IT2C2k5 I (6~— k1~k11/k~)~a~e,i,ei(b ~(0)2\i7

~T /kwE~1(k5,k~)=i 4~ L ~T(”~’ w)I

2L~(k’,w)L(k1, w)

2- k2k’w s / ~ k

11 — k~k~/k’2

+ c2~k~L(k1,w) ~ki2s~(ki,co) + 4~(k’,co) )

x (kik~E,y=ejet (bJ(0)2)a (k — k~k~Ik2)~,r=e,i,ei (bJ~2~L kw+ li

k2lEL(k, ~I2 ~T(k, ~)J2 kU)]

(O)2~a

s 641TC4Ik~l2r (6~— k1~k11Ik~) 2 I dk2 ~ae,i,ej (bJT 1kw

‘2(bEik~bEik;)k1w ~6 L4T(k, co)4(k’, w)lL~(k±,)l k~T(k, )i

___________ _________ ~T /kw__________ ~L 1kw _________

+ L~(k±,)12 I dk~ ~ /k~(b ,(0)2\u k2 (b ,(O)2\u

EL(k, ~)I + ~ ~(k,o’e,j,ei ~ 2

/ k~k~ k11—k~k~Ik

2 k’k2_______ __________ ________ k

11—k;k~/k’2X ~k2sL(k,co) + ~T(k, co) )(kl2E*(kl, co) + )]. (3.88)

T(k, w)

In the transparencydomainfor surfacewaves (if such wavesexist),

~ 32Ik~2c4r (6t — k11k11/k~)

(bEik~bElk~)k±w= ~ ~ w)~*(ki ~T~(k±,w)6(ReL~(k±,co))

/ k~k~ k1, — ktk~/k2)( k;k~ k

11 — k,~k~Ik’2

+ ~k2eL(k, co) + ~T(k, co) k’2s~(k’,co) + A*(k# co) )T

x T~(k±,w)3(ReL~(k1,w)], (3.89)

Yu.L. Klimontovichci a!., Statistical theoryofplasma—molecularsystems 317

where

21T Ecy=e.i,ei f dk ‘bJ’°~2’~I~ (k, 03)12T /kw I TT,(k

1,o4=—(0 J dk5 Im ST(k, w)II4~(k,w)12

.1. \UJL /kw z\ T /kw~

~ Idkz (k2 j~y(O)2\~ k2’bJ~°~2’~2ir o~e,i,ei k I~L(1~,(0)12 + k2I~T(k, w)121

T~(k.1.,w)— (3.90)

I dk5 (k~ Im �L(k, co) k~Im ~T(k,co)~2 Ie(k, ~)I2 + ~ 4~(k 2)

are the effective temperaturesfor p- and s-polarizedcollective fluctuations with eigenfrequenciesdeterminedby eqs. (3.81).

It is also easyto calculatethe spectraldensitiesof the correlationfunctionsof thechargedensityfluctuations. The result is

lie, U e, 0’’ V(bPk0zbP~)kJ.w —2IT[8(k~ — k~)+ ô(k5 + k~)jçop op 1k~

+ (bp~ bpk0)k~~Sw+ (bp~bp~)~~, (3.91)

where

k.k’o”’VS I a’. ‘vs(bp~bpk.)k~= —~-—--~Xu~~”~w)x1, (k, w)(bEkkbE,k)kW

l6irk.k’ kk’‘I a’ ___

—

4lTi0) Xli *(kI w)I~(k5,k~)+ 4iriw ~ w)l~*(k~,k5),k.k.e, a’S I J a’.(bp~OPk~lkw= 2 X1k(.J~’0))X~~*(hdt, W)(bE kk bE,k,)~~, (3.92)l6IT

_________ (0)~a- — 16IT2K

5 fl~l*fki w)(~+ k±jkjkIk2)L~*(kI,w)A~*(k,w)(bJ~°~bJm 1kw’

3 ~j !~,C,)

and the volume partof the correlationfunction (3.91) hasthe sameform as in the caseof an infinitemedium,eq. (3.33). The spectraldensityfor fluctuationsof the total chargedensitymay be calculatedby summingthe cross-overcorrelations,

(bpkbpk,)kW = ~ (bp~bp~)~~0,0’

In the isotropic casethegeneralrelationsarereducedto

P iku7~(k, (0) * (b (O)2s a”(bp0 bp0’)~‘~= .. y0.0.(k,co)0” Ie(k, o)12

U” VS(bp~ bpk~)k~= J00(k5,k~)+ J:0(k~, k5),

318 Yu.L, Klimontovich et a!., Statistical theory of plasma—molecular systems

(0)2’a4i~’k~ XL (k’, co) ~,, y0~(k,w)(bp lkw1au(kz, k~) = — __________ ________ __________________

k~L(k1, co) et(k’, co) Ie(k, w)12

—2i,~ a a’S

4E ~ co) X~(k’,co)~opk~bpk)ki_w_ ITlk~l2 L~(k±,w)I2CL(k,w) e~(k,w)

“L /k”w ___________________ \OJT /kwx ~ dk” ( k~(b i(0)2\a” k”2 ‘~ y(0)2\a”a” ~k”~ eL(k, ~)I2+ ~ klT(k”, w)l2)~

A further simplification of the general relations is possible in the caseof equilibrium systems.Furthermore,in the quasi-staticlimit (kc ~‘ w) the correlationfunctionsmaybepresentedin the form offluctuation—dissipationrelationsfor semi-infinite systemsof interactingparticles [94].For example,

Tk2(bp~bp~.)k = — Im[(1 — x~(k,w)Is(k,w))x~(k,w)][6(k~ — k~)+ 6(k~+ k~)]

+ 2k1~TIm( x~(k,w)X~(k’,co) )

ITO) \ e(k,w)s(k’, w)L(k1,co)(3.93)

Tk2

(bpk bpk~)k = — Im[— 1/r(k, w)][6(k~ — k) + 6(k~+ k~)]z ~ (0

+ 2k1,~T Im( [s(k, co) — 1][e(k’, w) — 1] ~ITO) \ e(k, w)s(k’, co)L(k1, co) /

On the basis of these relations,we can calculate the static form factors for semi-infinite plasma—molecularsystems.Using the Kramers—Kronigrelations,we obtain

(bp~bp~)ki_ = 2 ( — e(k,0) )x~(k~o)[6(k~— k~)+ 6(k2+ k~)J

+ Tk1g x~(k,0)x~(k’,0)IT e(k,0)e(k’,0)L(k1,0)

(3.94)

Tk2 / 1 ) Tk

1~ [e(k, 0) — 1][s(k’, 0) — 1](bpk~bpk;)k~= —~--- (~1— e(k 0) [6(k~— k~)+ 6(k~+ k~)]+ IT E(k, 0)s(k’,0)L(k1, 0)

Making a Fourier transformation,it is possibleto find the spatialcorrelationfunction. For example,

e~ne ______ ______________________1 — e~‘~) k~exp(—kDIR — R’!)

(bP(R,t)bP(R’,t))=___~(1+Zi+Zm)[(1+ ~2 4ir6(R—R’)—41TE0

2 ~ dk1k1 ~k~+k~—k1~/e0 J0(k1lR1—RIj)~kDJ Vk~+k~~k~+k~+k1~/E0

x exp[—(Z + Z’)~k~+ k~]]. (3.95)

Yu.L. Klimontovichci a!., Statistical theoryofplasma—molecularsystems 319

Here,we usethe following notations:

1, R-R’I4\,R?~=VTImw~,

1+O)~mIW~, R—R’l~’\f~?5,

~ lR—R’HV~?5,Zm= Zj=IejIIIeeI, (3.95a)

0,

~ IR—R’I<\/7,’?5,k~,=

(k~+k~)Ie0, lR—R’l~\I~.

As mentionedabove,the relationsconcerningthe static form factorsarestill valid in thecaseof rigiddipoles. This is also true in the caseof a semi-infinite system. It is easy to show that the spatialcorrelationfunction(3.95) is valid for a systemof rigid dipoleswhenone usesthe following quantities:

1, R—R’l<r0, ~ R—R’l<r0,Zm=

1+k~,,r~I6,R—R’I~’r0, 0,(3 .95b)

k~,= k~+k~+k~,IR—R’l<ro,(k~+k~)Ie0, IR—R’I~’ro.

It follows from the aboverelationsthatthe influenceof aboundarymayberathersignificant evenforstaticcorrelations.In particular,it is possibleto showthat the integraltermin eq. (3.95)(just this termdescribesthecontributionof surfaceeffects)hasthe asymptoticexpression

e2n ~‘(1+Z.+Z )

(bp(R,t)bp(R’,t))— 3 I ~ exp[—kD(Z+Z’)],2ITE

0 R1 — R1

which is dominantfor kDIR±— Rh ~‘ 1, i.e. at distancesgreaterthanthe Debye radius [117].In thecaseof dynamicfluctuationsthe influenceof theboundarymaybe muchamplified. In fact in

the caseof an isotropic systemin the domainof volume excitationsthe generalrelationsyield

(bp(R)bp(R’))~ ~ ~ e~nek2oTL(ko, w) (sin k

0R... + sin koR÷) (3.96)

which differs from the correspondingresult for an infinite systemby the presenceof the secondterm.Herewe usethe notation

k~= ~ e(03)=1— — 2’ R+={(R~—RI)2+(Z±Z’)}~2, (3.97)

3~e~pe ~

and TL(k, w) is definedby eqs. (3.28).

320 Yu.L. K!imontovichet al., Statistical theoryof plasma—molecularsystems

Eq. (3.96) is very close to the correspondingresult for a semi-infinite plasma [52,53]. This isexplainedby the fact that the contributionof the collective volume fluctuationsto the spatialchargecorrelationsis determinedby the electronmotion. In sucha casethe molecularsubsystemchangesthedielectric responseonly.

In a similar way wecan estimatethecontributionof thecollective surfacefluctuationsto the spatialcorrelationfunction [94],

2-32Tk

(bp(R)bp(R’)~_~4\f~ ~~?__ P J-0’ J0(k10lR1— RIl)cb(Z)~(Z’), (3.98)

z e0)pe e

where

whl exp(—~k~0—k~~,Z) ~2~(Z) = ~(Z) — 2 2 2 2 k10 = ,—~ [e +

6~pe~e Vk±okD V3E5eWpe

and T0(k1, co) is definedby eqs. (3.90).

Let us recall that the approximaterelationsare valid in the vicinity of eigenfrequencieswhich aredeterminedby the equationse(w) = 0 and s(co)+ ~= 0.

3.2.3. Intensityofspontaneousemission.Kirchhoff’s law. Effectivetemperaturesofspontaneousemission

Using thegeneralrelationsfor randomfields in anexternalmediumand thecorrelationfunctionsofsources,it is possibleto show that the intensity of spontaneousemission from the half-spaceof aplasma—molecularsystemmay be representedby Kirchhoffs law in the Levin—Rytov form [35,95],

= ~I~~(1— R~)+ ~I~~(1— R,), ~ = 32 ~ (3.99)41T c

I~’arethe intensitiesof the black-bodyradiationwith effectivetemperatures

T — 41T k11k11Ik~)~11{E~(bJ~°~bJ~°)~~}

p — iW (k±jk±jIk~)4jj{8a(k, co) — CaiI(1

1Ci w)}

T — 41T (~—k1~k11/k~)411{E0 (bJ~°~bJ~)~~} 3 100

— iw (~— k11k11Ik~)cb11{e0(k,co) — Eap(l~,w)} ( .

k1=~V~’sini~cos~,~ k~=~V~’cos~,

~ are the angles which give the direction of the solid angle. ~ are the energy reflectivitycoefficients,

R~= (1 — r1~)(1+ r00) + r,~r~,I2+ 4(c2k~i~w2)lr~~l2

(1 — r,,)(l — r0~)— ~

Yu.L. Klinwntovich et al., Statisticaltheory of plasma—molecularsystems 321

R~= (1 — r~~)(1+ r~~)+ + 4(w2i/c2k~)Ir~~I21(1 — r~~)(l— r~~)— ~

cue ±j ck~ 2rpp = — ~ k~ S.

1(k1,cv) , ~ = — — (ô,~— k11k11/k1)S~1(k1,cv)

ck~ kl,k±k cvi k±Ikjk~ = — — eZkI k~ Sq(k1,cv), r~,= ——i- e~1 k~ S,1(k1,cv).

The explicit form of the functional is given by eqs. (3.22).Notice that the generalrelation for the intensity (3.99) is obtainedwithout the assumptionabout

thermodynamicequilibrium in any subsystemof particles.This enablesone to useeqs. (3.99),(3.100)to study the radiation from nonequilibrium plasma—molecularsystems. In the particular caseofequilibrium T~= T, = T and the radiation intensity reducesto the well-known Levin—Rytov formula[78,79]

1= w:2T(1_R) R=~(R~+R~).

4ir c

Using the above relationsit is easyto describe,as particularcases,the radiationintensitiesfor aplasmahalf-space(Xej = 0) and semi-infinitemolecularmatter (Xe = X~= 0).

3.2.4. Kinetic equationsfor a semi-infinite plasma—molecularsystemThe calculations of the collision integrals for semi-infinite systems on the basis of the field

distributions(3.74)and thecorrelationfunctions(3.80)canbe performedin thesamewayasin thecaseof an infinite medium. The result is as follows:

— e~f dcv f dk f dk~1(1 k~V~ k.V1a n~J2ITJ (21T)

3J 2ir L~ cv Jil cv

x exp[i(k~— k~)Z]{(~EJkN~~))+

~ }, a=e,i, (3.101)

‘ab = — ~J~J~ J~‘I~~ ei[(i — (V + a,v))5,j — (V1 + aivi)] exp(ia,k~r)

x (~-+ a1 exp[i(k~— k)Z][(~EJk bN~,e(x,P))k + (bE/k bEkk)kW

~ ei~(a~2”(x,P) + ~~*(x, P) + y(V)*(x P) ~)k~=k; ~ab fab(~’, t)] , (3.102)

where the quantitiesa, I~and y as before are defined by eqs. (3.50) and the correlationfunctions(bE/k &Ekk)kW by eqs. (3.22). For the mixed correlation functions (SE/k 8N~(P))kUand

322 Yu.L. Klimontovichet al., Statistical theoryof plasma—molecularsystems

(0)

(6EJk,~1Nabk~(X, P))k~ the following relationsshould be used:4iri -1

(3E ~N~°~(P))k~= ~ ‘~jk (k, W)2lTflafa(X, t)eajk~ ak~

{ö(w — k V)2ir[5(k~— k~)+ ~k~(’~z + k~)]Vk

— 4ic2k~

~ (~m+ klkk~m/k~)L~(k±,w)A~’(k’,w)V~(w— k’ V)}, a = e, j,2 (3.103)

~E3 ~ (x, P))k~ = — ~ A~(k,w) ~ el.

2lrnabk~ abk

mwox ~ (_i)~[Jfl(al~k.r)Jm(alk~vIwo)(Vk+ + r~k•r!

n ,m

x 8(w — kV—(n + m)con) 21T[e5(k~— k~)+ Ekô(kZ + k~)]

4ic2k~— 2 (~m+ kIkk±mIk~)L~(kl,w)A~’(k’,to)a)

x Jn(t ik’•T)Jm( ,~k’ o)o)(V~+ + 1.v r.—Jô(w--k’~V—(n+m)w0)‘kv ‘kr!

If the motion of pairs(asa whole)maybeneglectedandonerestrictsoneselfto a descriptionup to firstorderin the atomic parameter,the molecularcollision integral(3.103) is reducedto

abab ] + [~~11 (Z)rjmwOfab(~~’,t)]‘ab = ~ [y~1(Z)Pjfab(c?~’,t) ~

2

+ D~(Z)Ô fat~(~’t) + G,”~,”(Z) ~2fab(~’ t) (3.104)~9p.~p. op

1 Or1

whereab abab(Z) ab + ~y~1b(Z), ~ (Z) = +

(3.105)

D~’(Z)= D~+ ~D~(Z), G~”(Z)= G~+

Herethe quantitiesy~7’,~ D~’andG~coincidewith the correspondingkinetic coefficientsfor aninfinite medium,eq. (3.52), and the additionaltermsare as follows:

ab ________ ________4i~-e~bJ dk r dk’I —i [Im A~1(k,w

0)~. 2i~(k~+ k~)3mw0 (2ir) J 2ii~

+ 2 Im ~A~1(k,k’, to

0)] exp[i(k~ — k~)Z],

w,1 (Z) — 41Te~b I dk I dk’ab _____ _____

— mw0 J (2)3J —~[ReA~1(k,w

0)e1.2ir6(k~+k~)

+ 2 ReM~’(k,k’, ~~)]exp[i(k~— k~)Z],


dk 1dk~eab J (2ir)3 J -i-— [(8E

1&EI)~’~O2ir~1~(k2+ k)

+ (8E,k ~EJk,)~~SU + (~E~k~EJk)~ ~]exp[i(k~— k~)Z],

dk I dk ~ [(SE,ÔEJ)~’W 21Ts15(k~+ k~)

8G(Z) = — ~b ~(2i7~~J 2ir

+ <bEtk &EJk,)~SU+ (~Efk ~EJk)~W] exp[i(k~ — k)Z], (3.106)

where

2ic2k

~A~1(k,k’, (U) = — Z A~(k,°~)(~m+ k±kk±mIk~)L~(k±,w)A~1(k’,o).

The explicit form of the termswhich define the spectraldensityof the field correlation(3.22) are

~ 16i~(~E~8EJ) kw = ~ A~1(k,o)A~(k,w)(0

x21T( ~ e~naJdPV,V,~(w—kV)fa(X,t)- ae,i

+ ~eabnab ~ Jdx dPv1v~~(w— k V + f

3~~O)fab(~’,

321T2(~E ~E )VS =ik~ jk~ k

1w 2 “ik (k, w) ~A~(k’, k, w)

~ e~naJdPVkVmö(o_k.V)fa(X,t)a=e,i

+ ~e~bnab ~i JdXdPVkUm~(W— k V + I~O)fab(~,t))

+c.c.(i~j,k~~k),

(8EIk~EIk.)~W= ~- Jdk~A~(k,k”, w) 3A_l*(kl k”, (i))

(0

x ( ~e~naJdP VkVmS(W — k V)fa(X, t)ae,i

+ Jdx dP VkVmfab(~~,t)8(w — k V + f3w~)). (3.107)

The collision integralsfor free chargedparticles‘a (a e, i) may be representedin the Fokker—Planckform


2

‘a = — [F~(Z, P)fa(P)1 + ~ ~ [D~(Z, P)fa(P)]’ (3.108)

where

2 ~ dw I dk I dk~(?iEIk ?JEIk)kWD~,(Z,P) 2iea J ~ J (2i~)3J 2~- k’ V— — ~ exp[i(k~— k~)Z1,

~dw I dk I dk’ 1 .~9D~(Z,P)F~(Z,P) = ea j 2~i (21T)~J ~ (~E~~N~°~(P))k~+ 2 oP

1

For isotropic distributionsthegeneralrelationis simplified, which enablesus to estimatethekineticcoefficients,

aby~(Z) 2e~b Im dk k2( e~1(0) 6~— e~1(0)

~~ab(Z)J = ~ ~mw~{Re}~i SL(k, ~ — ~T(k, ~

/ e.(2kZ) ô11e(2kZ)— e11(2kZ))

)

II +E~(k~w~) 4T(k, w~)

/ e~(k1,Z) e~(k1,Z) \—i~Jdk1kI~L(k w~)+ L~(k1,we))]’ (3.109)

0

wheresin x

e~1(x)= ~e1(x) + ~18~5e~(x), e~(x) ~ (~— cosx I

sinx 2\ sinx 2( xe(x)=— e(x)= 1—--~J—-——+--~cosx,2

CL)2 2 ~exp[i(kZT—kT)ZI,

c

e~(k1,Z) = ô~e~(k1,Z) + ~ Z),

~jkZT~ exp(ikZTZ) — exp(ikZLZ) 2

e~(k1,Z)= 2k~ ET(WO) ~L(~O)

I exp(ikZTZ) — exp(ikZLZ) 2e~(k1,w) =

k~ ST(w0) EL(W0)

~/ k~ kZT ~L~(k1,~ = 1 + ~ ~kZLeL(w0) + ~T(~0)~ L~(k1,~ = 1 +

kZT

______ 2 4 2_______ 2 ~0 —kZL T ~k

2 — ~ k2 = ~ ET(CV0), k~= 2 2 EL(%), k~= EL,T 1tJ~ T 3(0peSe

Yu.L. Klimontovkhet a!., Statistical theoryof plasma—molecularsystems 325

These relations describe the contributions of different collective processes (surfaceonesincluded)tothekinetic coefficients.For example,

ab(colI)(Z) = 2 eLZb VST(t0Q)({611 + ~eJ[8~Je(2kTZ)— e11(2kTZ)]}

+ ~ + 3sje,j(2kjZ)])

+ 2e~bk — k e~(k10,Z)~ 1~K5( ±~)IOReL~(k10,w0)It9k10I

where k10 satisfies the equation

Re L~(k10,w)= 0, ~5(k10)= — g (U~Ic2.

In the “cold” plasmaapproximationwith w~~ k±Se,

2 2 —2,2 ~0 — ~T(~0) —2i, ~. ~0 ____________

= S -, K I,K10) = -

C ST((U0)+6 z c IET(wo)+sIone has

ab(coII)(Z) = 2 ~b VET(wo){~~J+ ~eJ[~~Je(2kTZ)— e~J(2kTZ)]}

e~w~ 4(w~) (_.,,~ w0 ___________

+ 2 — — 5/2 — exp~ ~ — __________

m c IET(w0)+6I EE—ST(wO)] \ c Vs+ST(wo)

~(~~ + ~iz~iz

It is obvious that in the case of purely molecularmatterfor ~T(~) 1 thecontributionof collectivesurfaceexcitationswill be absent.However,thepresenceof boundarieswill be displayedby thesecondterms in the kinetic coefficients.

As mentioned,the collisional contributionto the kinetic coefficients of the volume terms will besignificant only for Z < 1 Ikmax~This means that whenkmaxZ> 1 the collisionalpart of the coefficientshasthe sameform asin the case of an infinite medium.

Let us reproducethe estimatesfor the kinetic coefficients. These calculations are relatively simple,but very lengthy. Therefore, we show the final result only:

D7~°11~(Z)= ~ e~o~\/BT(wo) (TT(kT, w~){8~+ ~eJ[~IJe(2kTZ)— efJ(2kTZ)]}

+ ~ £ TL(kL, wo)[Sq + 3sJe~,(2kLZ)J)

2e2 e~’.(ko~Z)+ —~ k

10i5(k10)T~(k10,w0) ~ ReL(k10, wo)IOk±01


In the equilibrium case

D~0hl)(Z) = Tmy~~~°~(Z), G~0~(Z) = T ~W~~o~(Z)/(Vo

We alsopresentthe resultrelatedto the quasi-staticapproximation.Such an approximationmaybeuseful for a descriptionof systemswith a low oscillatoreigenfrequencyw~.The final result is

dk

u e i ei 1 / a1.,~kr)(k. ~ + a1~kI~= i ~ e1~e1~1 (2 )3 exp(i

xfdx’ dP’ ~ s(k, kV ~)G~(k, ~‘)‘2~ )i (k. V~nm— k V~nm+ U)nm n1,m1

0 k•V~,=,~ k~ ,xexp(iai~k.r’)(k.OP~~m(k)— k.V;nm OP~nm(k))fa(~~t)fa(~’,t)

dk+ i ~ e/ffe/~fff ~ exp(ia~k~. r)(k~.~ + a1,~k

~ (2i~)

xfdx’dP’~ ~ ‘2animi)I (k. V~nm— k V~nm+ U)n,m n1,m1

o k~Vunmk , a )f~(~~t)f~(~’,t)exp(—2ik~Z)x exp(ia1~kr’)(k. OPffflm(k) — k• V~~nm 0Pu~nm(k)

dk Idk~— i ~ e/ffe/,~1 (2~ J ~ exp{i[~1~k’. r — (k~— k~)Z}}

o~’ 1’,!”

~IT \ OP

n,m n1,m1 1,/i

)(k~ a — k1’~3, 0G”~(k’ ~ . ________ nm k’• ) exp(ia~k’r’)k) k’V’ or (k’)’

~ { 2 n1m1 ‘ ~nm( a’n1m1k E(k k’•V’ )~

2S*(kk’~V )L*(k k’V’~nm)(kV~nm—k.V,:,nm +U)o’n

1,n1 ‘ on1m~ I~ a

Gh1~(k,~‘)(k. ~ (k’ k•Vunm k~anm~) — k V’~ OPa~nm(k))exp(ia1~kr’)nl

+ k’21s(k,k~V~nm)~2S(k’,k• Va~nm)L(ki,kVa )(k•Vanm— k•V~njm

1~

G”°’ (k”, ~)(k 0 k•Vornm k”•

— ~ r dk~ nlml 0P~m(k)— k”~V~nm OP~.nm(k”))exp(ia1~k”r’)IT J k”

2 k’2~(k’ck”.V,:,nm)L(k1, k”.V;nm)1

2s(k’ k”•V’ )s*(k k.V,:~nm)‘ an

1m1

1 }f(~,t)fu(~’,t)+~Ia, ~=e,i,ei. (3.110)xk•V — ‘-“~V’ +U aanm A

Yu.L. Kllmontovich eta!., Statisticaltheory ofplasma—molecularsystems 327

Here, in orderto havea uniform representationof ‘a (a = e, i) andI~,we introducedthefollowingnotations:

0, cr=e,i, , o=e,i, 0, o~e,i,= ela = k~Vffnm = k• V +

a1, r=ei, e1, o=ei, (m+n)w0, cr=ei,

0, a-=e,i,0 0

OPunm(k) = ~p + tflW~-~- + ~ —~-- -~- r = eikp Op mw0 kr Or’

A ~ cr = e, i,-,lcr ii ~7~5\ -nT11nm~’.”~ — —i- e!,,,

(—i)~J~(a1k.r)Jm(a1k~vIw0), o = ei,

f,~(X,t)~(x),ue,i,

~ t)= X’as(R,P’),fei(~,t), cr=ei,

—F~(R).0f~(X, t)/OP, a- = e, i,

= —~ F,(R + a1r) ~ + a~~)fei(~~ t), a- = ei.

The quantitiesFa(R) andF1(R + air), which arepresentin therelationfor ~ arethe forcesactingon a particle of speciesa (or species1) from a chargeinducedby chargedparticlesor a bound pair,

Fa(R) = —e~f (2IT)~k2s(k,0)(exp(2ikzZ)— irL(k

1,0) exp(ik5Z)Jdk~

F1(R + a,r) = e1 e1 f (2~)~k2s~,0)(exp[ik. r(a

1 — a1.)] + exp[i(k. (a,r — ai,rt) + 2k5Z)]

— exp[i(a1k. r + k~Z)]ITL~1,O)f dk~exp[i(a1k’-r +k~Z)]) (3.111)

Using theserepresentationsfor Fa(R) andF,(R + a1r), the expressionfor ~ can be written in thefollowing form [98,99]:

Ofa(X, t) OUa(R) Ofa(X, t)

= OR OP a=e,i,

~‘ab = —>~F,(R+ a,r)-(~+ a1 fab(~

12’t)

= (c9Uab(R~r). + [~~(r)+ Uab(R,r)]~~ t), (3.112)

328 Yu.L. Klimontovich et al., Statistical theoryof plasma—molecularsystems

where

Ua(R) e~U(R)= ~e~(R, R), Uah(R, r) = ~ e,e1ço~(R+ a1r, R +

(3i13)

~(R, R’) = ~‘(R,R’) — q~0(R, R’) -

ço(R, R’) and ~0(R,R’) are the potentialsof a unit charge in a semi-infinite and infinite medium,respectively[1181,

(2~)~k2~o)(exp(_ik.R’)+exp(_ik~.R’)

2k ~ dk’exp(—ik’.R’)— ITL(k

1,O) I Zk,2(k,o) )expik.R,

çc~~(R,R’)Jdk41T exp[ik(R—R’)].

(2IT) k s(k, 0)

The generalstructureof the collision integralsshowsthe strong influenceof the boundaryon theequilibrium distributions.In fact, if the Gibbsdistributionfor oscillatorsandthe Maxwell distributionsfor free particlesareusedas equilibrium distributions,only the first threetermsbecomeequalto zero.At the same time, the term related to the polarizationforces is not equalto zero. This showsthatspatiallyhomogeneousequilibriumdistributionsmaybeconsideredonly as the simplestapproximation.The correctionsto suchan approximationshouldbedeterminedon the basisof the kinetic equationsfora stationary state, which in the caseunder considerationare as follows:

/ a a\ OU(R) Of(X)~V•~+KE(R))~~)fa(X)= ~

[v. ~ + v. - ~b(r)1 ~- + ~ ~ + air)) . (~+ a~~)]fah~

— (OUab(R,r) a O[t.~(r) + Uab(R, r)1 0 3 114

\. OR Or ~Jfab( ) L )

The self-consistentfield ~E(R)) = —gradK 4(R)~ satisfies the Poissonequation,

div (E(R)) = 4IT(a~,i f dP eanafa(X) + I dx dPelnabfah(~!)).

The solution of eqs. (3.113), (3.114)has the following form (seerefs. ~95—99]):

fa(X) =f~(P) exp{-[eaK~(Z)~+ Ua(Z)IIT},(3.115)

fab(~) = f~(P)f~(p)f~(r)exp[_ (~e~~(Z + a~z)~+ Uab(Z, r)) IT],

Yu.L. Klimontovich et al., Statisticaltheoryofplasma—molecularsystems 329

where

~0) — exp[—~ab(r)IT]Jab(T) —f dr exp[—4ab(r)IT]

— 2 ( dk exp(ikr) ~ 1~ab(r)~ab(~)

4~abJ (2IT)3 k2 ~e(k,0) _1)~

= eana JdR’ ~(R,R’) exp[—Ua(Z)IT].

Here, in the above calculations we have used the linearized equations (3.114) with respect to{exp[—U~(Z)IT]— 1} and {exp[~Uab(Z,r)IT] — 1) just as for a semi-infinite plasma[97].

Thus,eqs. (3.115) enableus to representthe distributionsboth of free andboundchargedparticlesas well as the macroscopicself-consistentpotential, in termsof a screenedpotential.

Eqs. (3.115) canbe simplified by linearizing themwith respectto Ua(Z) and Uab(Z, r). Then

fa(~V)=f~°~(P)[1+ aa(Z)] ‘ fab(~’)=f ~(P)f~?(p)f~(r)[1 + aOb(Z, r)] , (3.116)

where

aa(Z)= — [ea(~(Z)) + Ua(Z)], aOb(Z, r) = — ~ (~e1(ç~(Z+ a1z)) + Uab(Z,

(~(Z))= —~ ~~~~eJco(R,R1)U(ZP)dRl, ~=>. k~e~/k~e~.

Integratingfab(~’) over thevariablesr andp, onecan find the distributionof the centresof massofthe pairs,

fab(X) = Jdx fab(~)= f~(P)[i + aab(Z)1, (3.117)

where

(~) U

aab(Z)=aab (Z)+aab(Z),

a~(Z)= — ~ Jdrf~(r)~ e1( cb(Z +

a~,(Z)= — ~ JdrJ~°,,~(r) Uab(Z, r).

The quantitiesin eqs. (3.115)—(3.117) which dependon r can be expandedin terms of the ratio(rIR). Takinginto accountthe expansionof the aboveequationsup to order (rIR)

2, which correspondsto the dipole approximationfor the molecularfield description,one obtains


Uah(Z,r)= 2e,~ lim (r~VR)(r~VR)pS(R,R’),R —~R

(3.118)2

kDeeEo fA I dR p(R,R’)U(Z’),v7T J

V

wherep(R, R’) is the screenedpotentialof a unit chargein a semi-infiniteplasmawith the backgrounddielectricconstante~= 1 + Xah [117,118],

cD(R, R’) = ~R, R’) + -~- I dk1 8(k1)J0(kJR1 — R~j)exp[—a(Z + Z’)],E0 a

exp(—kD~R— R’~)ç’0(R, R’) — r0~R— R’~

a~k~+k~, 8(k )eoaEkI~

____________ 4 (0) 2 D47re~bnab ~ Jd - - k

2Xab= rrT 3 fab(~~), kD=—.

0

In the caseunder consideration

aSb(Z, r) = — ~ [ea(r~VR)(1 + ~ mb — ma (r.V))K~(Z))+ Uab(Z, r)],2 m~,+ ma

aah(Z) = — Xab (1 mb — ma ~K~z) + V(Z)),4ITflab ea mb + ma

2aV(Z)= lim ,

R—~ROR, OR, -

Taking into accountthe explicit form of the potential p~(R,R’), we may find additionalterms in the

homogeneousdistributions in the dipole approximation,

k1 8(k1) - -2aZ kDZ)

K~R(~(z)~=~ a 4k~+3k~(kDe -2ae

2e I k ____

____________ 2aZ —kaa(R) aa(Z) = — ~— j dk ~ (~~ k~6(ki) (~De —

2a e DZ) + 6(k1) e2~),a ea 4k~+3k~

aab(R, r) aab(Z,r) = ea2b 1dk

1 k1~(k1)~2~~ k~z~2 — mb — ma2T60 ~ eab 4k~+3kD mb + ma 2

( 1 ~ — ma ‘\ —2aZ1_________ 2 2 —2aZl— — azjemh+ma / ]—(~k~r~+az)e j,


aab(R)=aab(Z)= 8IT~OflabJdk1k±6(k1)(~ea rnb+rna 4k~+3~(~DeZ_2ae2~)

-

1(2k~+ k~)e2az). (3.119)

If one knows the distribution function of pairs, the averageddipole momentof moleculescan becalculated,

(d(R)) = eab I dPdx Tfab(~) = - ~ Jdr ,~(r)(~ e~(q5(R + air)) + Uab(R, r)).

In thedipole approximationthis yields

(d(Z)) = e5~Xabeab Jdk±k1 ~8~1)2 (e

2~ ekDZ)ITSOflab 4k±+3kD

It shows that in the dipole approximationthe quantity (d) is non-vanishingonly for systemswithspatially separatedchargesae(Z)~a~(Z),i.e., when ~ s~0.

If therearefree chargedparticlesof two kindswith ee = i’ ‘~e= n,, then

a — I —2aZaa(Z)—~y—jdk±6(ki)—e

aab(Z, r) = — ~ Jdk1 ~(k1) ~ (~k~r~+ a

2z2)e2az, (3.120)

aab(Z) = — Xah Jdk1 8(k1) ~ (2k~+ k~)e

2az.a

On the otherhand,if the systemis a purely molecularone,i.e., na = 0 (a = e, i), thefollowing relationsare satisfied:

aab(Z, r) = — eabÔ ~ (~r~+ z2), aab(Z) = — Xab8 ~, (3.121)8Te

0Z l6lTnabeo Z

where8 = ~ — ~)I(e~ + i’).Thus,in the caseconsideredthe perturbationof the moleculardensitydueto theboundarydecreases

as the inversecubeof the distancefrom the boundary(a similar dependencewaspreviouslyobtainedfor the modelof a rigid dipole in ref. [119]),and the sign of theadditional termis determinedby theratio betweeng andr,,~.Theinfinite valuesofaab(Z, r) and aa,,(Z) as Z—*0, g areconnectedwiththedivergenceof the integrals(3.119)atthe upperlimit of integrationandcan beremovedby correctlychoosingtheupperlimit of integrationkmax~ CO. When~< e~the divergencecan also be removedbydiscardingthe linear approximation,following a proceduresimilar to that proposedin ref. [1201forcalculatingthe equilibrium distributions in a semi-infiniteplasma.

332 Yu.L. Klimontovichet al., Statisticaltheory of plasma—molecularsystems

It follows from the aboveequationsthat for the model of rigid dipoles

— 3 Xab6 1+cos2O

aab(Z, r) = aab(Z,0) = — A~J 1Tn,~E

0 L

where0 is the angle betweenthe radiusvector r andthe Oz axis. Consequently,as distinct from theisotropic distribution of moleculesin an infinite medium,in the presenceof a boundarythereis ananisotropyof fab(Z, r) in planesperpendicularto the boundarysurface.In particular,the moleculesarepreferentiallyorientednormal to the boundarywhene~< ~, andparallel to the boundarywhen t~> ~.

This is so becausewhen s0>~ the particlesare repelledfrom the boundaryirrespectiveof the sign ofthe charge [F5(Z)>0, OF5(Z)IaZ<0), so that the resulting torque Km = ~ r1 x F1(r1) acts on thedipole [it can be evaluatedin the dipole approximationKm = e~bz(eXrY— e~rJ(OF2(Z)IaZ) /4] in suchaway that a boundpair is locatedparallelto the boundary.HereF~(Z)= —0U(Z)IOZ is the force actingon a unit chargenearthe boundary,and U(Z) is defined by eq. (3.113). On the otherhand,when

<s, F~(Z)<0, 0F2(Z)IOZ>0, and the boundpairs are oriented perpendicularto the boundary.Just this type of distributionminimizes Uab(Z, r).

If we discardthe linear approximationthe normalizeddistributionfab(R) can be written as

fab(~) — ( 3 Xab 6 cos2 0

fab(~’)~O__ir’2 — exp\_ 16 4’TTflab E0 Z

3 -

As mentionedabove,we thus removethe divergenceas Z—* 0, ~ <

However, in the generalcaseit is necessaryto restrictthe upperlimit of integration.This is relatedto the moleculardistributions as well as the distribution of free chargedparticles.For the latter ones~ = 1/r~,,—3TI2e~(a = e,i). In the caseof a rigid dipole r~ can be estimatedon the basis of the

(ab) (ab) 2 (ab)3 . (ab) —

equationU(rmjn)=3T, where U(rmin)—=d 6/4rmjn ~ i.e., rm,n ~XaJ6Ih16’7Tt1ab~0Notice, finally, anotherimportantpoint which follows from the generalequations(3.115). This is

concernedwith the following representationfor the moleculardistribution:

fab(~) =f~(P)f~(p)f~(R,r) exp(-~e1K~(R+ air)) IT),

where

exp[—~ab(R,r)/T]

fab(R,T) Jdrexp[—cbab(R,r)IT]

~ab(R, r) = cbab(T) + Uab(R, r) = cbab(r) - [~b(r) - ~r)] + Uab(R, r).

This enablesus to interpret the quantity 4ab(R,r) as the effective interaction potential for particlesbound in pairs, which takesinto accountthe presenceof a boundedplasma—molecularmedium.

~It may be shown that in the case of real systems r ~ K r2) 1/2 r

0, kDrO ~ 1, kmro ~1. Then4’ab(T) — ‘~abfr) =‘O.

Using also the dipole approximation,which is valid at distancesZ ~‘ Kr2) 1/2 one can representthe

effective potential in the following form:

1 V 2 2~ w,r1

i=x,y,z

Yu. L. Klimontovicha a!., Statisticaltheory ofplasma—molecularsystems 333

where

w~(R) w~= eab R~-aROR,OR~~ R’).

Thus,the presenceof a boundaryleadsto a renormalizationof the eigenfrequencyof the oscillators.In particular, for a semi-infinite system

w5(Z) = w~(Z)= w~[1 + ~T(Z)], w~(Z)= w~[1+

where

Xab‘TTflabEO a

8ITnQbso fdk1 ~(k±)k±ae2az.

If thereis no plasmasubsystem(ne = = 0),

zl~(Z)= 2LIT(Z) = 32irn~s~

Equations(3.119) give the possibility to calculatein detail the spatialdistributionboth of free andbound particles,as well as the orientationaldistributions of pairs. A discussionof the results ofnumericalcalculationscan be found in ref. [99]. Here, we reproducesome of them for a systemcontaining a plasmasubsystem,as well as a subsystemof rigid dipoleswith the parameters:n~,=1018 cm3, lie = 1014cm3, T = 1040 K, r

0 = 0.59A, eel = 0.65Ie~I.In the caseof a hydrogen-likeplasma(e~= — e1) in contactwith a mediumwith a dielectric constant

equalto unity, we observea decreasingdensityof particlesof both kindsnearthe boundary(curve1,fig. 3.1) (calculationsshow that sucha behaviouris also observedfor thecase~< e~,wheres~= 1 + Xab

is the dielectric constantof the medium in which the plasmais situated),and the spatial profiles ofelectronsand ions are the same.If ~> s~,the densityof particlesnear the boundaryincreaseswithincreasing~1The densityprofiles of electronsand ions still coincide(fig. 3.1). This behaviourcanbeexplainedby the fact that in the caseof a hydrogen-likeplasmathe self-consistentpotentialis equaltozero and the density profiles are determinedby the potentialsUe(Z) = U,(Z) = e~U(Z), proportionalfor Z < 1 IkD to thequantity ~ — i) /(~~+ ~), whosesign determinesthecharacterof thebehaviourofUa(Z) nearthe boundary.The density profile of the moleculeshasqualitatively the samecharacter.

If theplasmaconsistsof severalkinds of ions, or the ions aresingly charged,the densityprofiles ofelectronsand ions do not coincide;this results in a self-consistentpotential (4(Z)), and the characterof the spatial distributions of chargedparticleswill be determinedby a superpositionof (~(Z))andU(Z) (figs. 3.2, 3.3). The presenceof multiply ionized atoms turns out to be most important here.Calculationsshow (fig. 3.3) that the presenceof only 1% of ions with a fivefold chargerelativeto theconcentrationofhydrogen-likeions leadsto a majorchangein thespatialdistributionsbothof electronsand ions (for thesakeof comparison,thedashedline in fig. 3.3 illustratesthedistributionof electrons

334 Yu.L. K!imontovichet al., Statistical theory ofplasma—molecularsystems

1 - 1.5 ~cL(Z) 100

0.5 2~ ~

~0.5~1 KDZ 1.5 ~ K~Z

-0.5 -0.5

Fig. 3.1. Dependenceof the densityprofiles of chargedparticleson the dielectric constantof an externalmedium in thecaseof a hydrogen-likeplasma. 1, ~ 1; 2, i 1.1; 3, L~=10; 4, ~= 100.

0.1 I- ~fJz)2

0 3K0Z

—0.1

—0.3

-0.5

Fig. 3.2. Densitydistributionsof electrons(1) and ions (2) in the caseof a doubly ionizedplasma.

Yu.L. Klimontovichet a!., Statistical theoryofplasma—molecu!arsystems 335

0.1

2 3K~Z

Fig. 3.3. Distributions of electronsand ions in the caseof a three-componentplasma.1, Electrons;2, singly chargedions; 3, ions with a fivefoldcharge,i=1, n,(Z,5)/n(Z,=’l)O.Ol.

and ions for the caseof a hydrogen-likeplasma).The dependenceof the electrondistributionon the ioncompositionis presentedin fig. 3.4.

We nowconsiderthe resultof an analysisof the orientationaldistributionof the molecules.Figures3.5—3.9 representthe functionsAfab = [fab(Z, 0) — fab(Z, 1T12)] 1’fab(Z, IT!

2) with different assumptionsconcerningthe plasmacomposition and the dielectric permittivity of the externalmedium. Sincefor(4(Z)) = 0, thenormalizeddistributionAfab(Z, 0) is an evenfunctionof thevariable 0 = 0 — iT!2, figs.3.5 and 3.6 representthe functions Wab(Z,0) only in the region 0� 0 � irI2.

Figures3.5 and3.6 correspondto ahydrogen-likeplasmasubsystemwith i = 1 (fig. 3.5) and~= 1.01

2 jKDZ

—1.5

Fig. 3.4. Electrondensity profile for multiply ionized plasma.1, Electronplasmain neutralizingbackground;2, Z, = 1; 3, Z, = 2; 4, Z, = 5.


0 5~/4 5~/29 0 5~74 — — 9~/293~~7 2~-~

~,, II -— ~.— , /

— ,I 3 , I2—’ ii

- - - 1 -0.5

~3f 2

/

-~ - -1//

/ 3.

Fig. 3.5. Orientationaldistributionof dipoles in plasma—molecular(solid curves)andmolecular(dashedcurves)systemsin contactwith avacuumatdifferent distancesfrom the boundary.(a) 1, Z = 5 A; 2, Z = 7A; 3, Z = bA; (b) 1, Z = 30 A; 2, Z = 50A; 3, Z = 100 A; z =

(fig. 3.6). The dashedlinesshowthe functions lXfab(Z, 0) for purely molecularmatter.As is seenfromthe figures, in the caseof a systemin contactwith vacuum,orientationof the dipolesparallel to theboundaryis preferred(asmentionedin ref. [119])andthe presenceof aplasmasubsystemdoesnot leadto qualitativedifferencesin the orientationaldistributionof the dipoles.This is so becausefor ~ = 1,

~> 1, the presenceof a plasmasubsystemcannotchangethe characterof the distributioninasmuchas

the quantity

6(k1) = (e0\/k~+ - ~k1)I(E0\/k~ + + ~k1)

10 -~f,,~1011

\ 0.5-~ \ \

\\\

\ \ 0.252

5~ ~

-..

\\ ._~_ .. \

0 -

Fig. 3.6. Samefunctions asin fig. 3.5 for systemsin contact with a dielectric medium. (a) 1, Z=200A; 2, Z=250A; (b) 1, Z=SOOA: 2.Z=750A; 3, z=i000A;~=i.0i, Z~=1.

Yu.L. Klimontovichetal., Statisticaltheoryofplasma—molecularsystems 337

determiningthesign of thepotential Uab(Z, r) and its derivativesremainsin this casepositivedefinite.The situation,however,can changewhen~> t~(fig. 3.6). In this casefor tie = n1 =0 (dottedlines)

5 <0 andorientationnormalto theboundarybecomespreferred,in keepingwith the formulafor Km.Thepresenceof a plasmadoesnot changethequalitativecharacterof the functionconsideredfor smallZ (curve 1, fig. 3 .6a), whenthe dominantcontributionto Uab is givenby the regionk ~ kD, in which5(k1)<0. As Z increases(curve2, fig. 3.6aandcurves1—3, fig. 3.6b),the contributionof theplasmasubsystembecomesimportant (kmax S k~)and the distributions I~fab arethe sameas for ~‘ = 1.

If the plasmasubsystemis not hydrogen-like,a self-consistentmacroscopicfield leadsto additionalfeaturesin theorientationaldistribution of dipoles evenfor e~= ~ (fig. 3.7). As is seenfrom the figure,only for very small Z (of the orderof a few dipole sizes) doesthe characterof the distributionremainthesameasfor (4(Z)) = 0 (the dashedline in fig. 3.7 showsthe functionsI~fab(Z,0) in thecaseof ahydrogen-likeplasma),i.e., the interactionof a pairwith a chargeinducedby it is dominant(curves1,2, fig. 3.7a). However,as Z increases,the influenceof a field that orients the dipole normal to theboundarybecomesmajor. As a result,at large distancesthe distributionbecomespractically antisym-metric with respectto the point 0 = ir/2 (curves 1—4, fig. 3 .7b), i.e., the contribution of the termsconnectedwith Uab(Z, r) becomesvanishingly small.

A similar situationis also observedfor i> s~(figs. 3.8, 3.9),but here,becausethe distributionofthe field (E5(Z)) hasan alternatingsign, a reorientationof dipoles is observedasZ increases.Thus,for example,for i= 1.01 (fig. 3.8) thedipolesin the regionof small Z (fig. 3.8a)areorientednormaltotheboundary,thedirections0 = 0 and0 = ir beingpracticallyequallyprobable(curves1—3), andas Zincreasesthedirections0 = ~ (curves1—4, fig. 3.8b andcurves1, 2, fig. 3.8c) or 0 = 0 (curve4, fig. 3.8cand curves1—4, fig. 3.8d) becomepreferred.

If ~ r~(fig. 3.9), the regionZ, in which thedirections0 = 0 and 0 = IT would be equallyprobable,is muchreduced,i.e., underthis condition the dominantphysicalmechanismis theorientationof thedipoles in the self-consistentfield (E~(Z)),but not the actionof the rotationalmoment.

a 2 ~~i108 b

0.3 1

30.1 1

0 2~1~f4 ,‘ /2 ~ 3~/c9

19 4—0.1 / ___________________

/ C

1:: ~/4 ~72~4~8

Fig. 3.7. Orientationaldistributionsof dipoles in plasma—molecularmatterin thecaseof multiply charged(Z1 = 3, solidcurves) andsingly charged(Z1 = 1, dashedcurves)ionsat different distancesfrom theboundary.(a) 1, Z= iA; 2, z= 10 A; 3, Z= 100A; (b) 1, Z= i0

3 A; 2, Z= 2 x i03 A;3, Z5 x iO~A; 4, Z= io4A.


~m~10~ 2.5 uI75

if 2

I 1”/ 1.5 \50 I I /

I II / 1 \ 3

I I .

I 1 0.5 ~. ,‘

-) \ I, 3,\~

0 I~74 ~j2 3c/4 ?~9

‘5 ~ ~ /3_. -0.5a b0 ~ 572 3574 ~ir8

~m~108 1.5 ~f3~108

0.754

13 20.5 1

0.50.25 4

1 2 00 - - — - -. ~74 572 357~~8

~ ~/2 3~74 9r9

-15

Fig. 3.8. Samefunctions asin fig. 3.7 for ~=1.01. (a) 1, ZioA; 2, Z=15A; 3, Z=20A; (b) 1, Z35A; 2, z=SOA;3, Z75A; (c) 1,Z = 100 A; 2, Z = 350A; 3, Z= 600A; 4, Z= 800A; (d) 1, Z = i03 A; 2, Z = 2 X i03 A; 3, Z = 5 x i03 A; 4, Z= iO~A.

3.3. Plasma—molecularlayer

3.3.1. Electromagneticfields in a plasma—molecularlayerIf plasma—molecularmatter occupiesthe region (_oo ~ X, V ~ ~, 0< Z ~ L) and both free and

bound chargedparticlesare specularlyreflectedfrom the boundary,the boundaryconditions for thetransitionprobabilitiesare as follows:

Wa(R, P, X’, T)~zoL = Wa(R, pt, x’, r)IZOL(3.122)

Wab(T, p, R, P, ~“, T)~zoL = Wab(Tt, ~ R, pt, ~ T)1z0L

Yu,L. Klimontovich eta!,, Statistical theoryofplasma—molecularsystems 339

• ~fm~106 1 1.5 ~fl06

3 .. 1

1’/ 0.5

2 I‘ / 1 —-

1 C’~~—574 572 3574 ~9

• ~—‘ 3,‘ —0.5

1,3

0 ~514 572 35r/4 5’~ -i 1

2

-1 a -1.5 bFig. 3.9. Samefunctionsasin fig. 3.7 for i=10. (a)!, Z35A; 2, Z50A; 3, Z=75A; (b) 1, Z’100A; 2, Zi03A; 3, Z=5xilJ3A.

It is easyto seethat the following solutionsof the equationsfor the transitionprobabilities (2.27),(2.29) satisfy theboundaryconditions (3.122):

W~(X,X’, r) = ~ [5(Z — Z’ + 2nL — V~’r)5(P~— P~)+ 5(Z + Z’ + 2nL — V~T)5(P~+ P~)]

x 5(R1 — RI — V1’r)5(P1 — P1), (3.123)

Wab(~1’,~‘, T) ~ [5(Z — Z’ + 2nL — V~I-)5(P5+ P~)5(z— z’ cosco0r — (p~Imo.0)sin co0r)

x 5(p~— p~cosw~r+ moi0z’ sin w0r)

+ 5(Z+ Z’ + 2nL — V~r)8(P5+ P~)5(z+ z’ coso0r + (p~Imw0)sin o0r)

x5(~~+ p~cosw

0r — mw0z’sin a~r)]

x 5(R1 — RI — V1r)8(P1 — PI)S(r± — rI cos — (p~/mwo) sin (00T)

x S(p1 — p~, cos co0r + mw0r~sin ~r). (3.124)

These results can be expressedin termsof thecorrespondingtransition probabilitiesfor an infinitemedium,

Wa(X,X’, i-) ~ [Wa~°~(X,X~,r) + W~°~(X,X~,r)],

Wab(~, a”, T) = ~ [W~(~, ~, r) + ~ ~, r)],

where

~ X~(R~,P’), R~=(RI,Z’—2nL).


Using the above transition probabilities and the general relations (2.39), (3.43) for the susceptibilitytensors and correlation functions of Langevin sources, the following result for the latter quantities canbe found [541:

~ Z’, k1, w) = ~ m~ p[m~~ — Z’)] P~~(km,w)

+ ~, exp[i(mi-r/L)(Z + Z’)jP~(km,w)} , (3.125)

J~°~(Z)~ = ~ m~ {exp[i(mIT/L)(Z — Z’)I

+ r~exp[i(mITIL)(Z + Z’)1}(~J~°~~j~O)y , (3.126)(0) (0) ~r

where km (k1, mir!L) and the quantities P11 (km,w) and (~iJ1 ~J1 ~ are determined by eqs.(3.74) and (3.6), as before.

Our next problem is to calculate the electric induction

aD~kW(Z) dZ’ r~1(Z,Z’, k1, w) ~EJk(Z’),

andto solve the basic set of equationsfor the fluctuation microfields. Suchcalculationsare simplifiedsignificantly, if one extendsthe electric fields and currents to the region — L < Z <0 in a specularmanner.In the caseunder consideration

~DtkW(Z)= dZ’ m~(mZ_ Z’)] Ejj(km, w)~EJkW(Z’).

If we thenextendall quantitiesto the region — ~ Z ~ ~ periodicallywith a period equalto 2L andperform a Fourier transformation,we are led again to an algebraicrelation betweenthe Fouriercomponentsof the inductionand electric fields,

~Di(km,w) = Eij(km, w) ~EJkW

The basic set of equations in this case is reduced to

Aij(km, c~)~ikmW = — ~ — ~- e flaB/k (+0) — (—1)~~BIk(L — o)]},

where~Bjk ~(L — 0) = limZ...L ~BJk01(Z).

Taking into accountthe boundaryconditionsfor the electromagneticfields, one obtains[56,94]

~E(R,t) ~f ~ exp[—i(wt— k1 .R1)] x ~ exp(—ik~Z), Z<0,

x E ~ exp[i(mITIL)Z], 0~Z~L,

X ~ exp[ik~(Z— L)1, Z> L1 (3.127)

Yu.L. Klimontovichet a!., Statistical theory ofplasma—molecularsystems 341

where

41T1 1= — — ~ A~J’(k,,,,k~,w) ~j(,°) (3.128)jk

3w ‘

11i71(lCm, k3, w) 2L5mnA~~j1(km,o) + ~A~1(km,k~,w),

a a (a)~ ~

0) a±1

a (a)—1~ y,y,~A

11 (km,kn,W),a± 1

L~(k1,w) = 5’ + + ck,1k, s~(k,,w),~ (k,, w) k~~&)

2ic ~ ~A~i(km,0)), y~=l±(—l)m (a±1),

s~(k,,w) = —

(I) m~oo2L

+ _~6Ee+ += ‘{[L~

1(k,, w)] jk1w _[L~(k,, ~)]~i ~E~}ik1w 2

41T1~E~W = — “~ m�-oo Ym A~1(k w) &I~,

~J~W is the Fouriercomponentof thecurrent~J°~(R,t) = ~r=e,i,ei&t°’~°~(R,t), which is given in theregion0 � Z ~ L andextendedspecularlyto the region— L � Z � 0 andthenperiodicallyto the region— <Z <x. The spectraldensityof the correlationfunctionsof the microcurrentsin the caseunderconsiderationwill be asfollows [54]:

(3J(O) 3~,(O) ~ (2IT)3t5(k1— kI)5(w — w’)2L(Smn + ejSmn)(bJ~°~ J~m”~ (3.129)

iICmW jk,w’/

3.3.2. Fluctuations of the electric charge densityand electromagneticfieldsThe correlationfunctionsof the electromagneticfield fluctuationscan be calculatedby a statistical

averageof the fluctuation fields (3.127) on the basis of eq. (3.129).The final result is as follows:

dwldk, 1~ t) ~E1(R’,t’)) I 2i~ J (2ir)2 m.n (2L)2

x exp{i[k, . (R1 — RI) + kzmZ— k5~Z’— w(t — t’)]} (6EIk &EJ~)~W, (3.130)

where

(~E~k&EIk)k,W =2L(tSmn+ ~i5m,_n)(8Ei &E1)~~

S VS+ (~EjkzmbElk 1kw + (~E~k &E/k)~W . (3.131)

zm

Here,as above,the quantity (~E~bE1)~ is definedby eq. (3.22) and the otherquantitiesare

(bE,k bEIk)VS = E (k k5~)+ E(k5~,kzm),k1w if Sm’

342 Vu.L. Klimontovichet al., Statistical theoryofplasma—molecularsystems

Eij(kzm,k53) = ~2 A~’(km,w) bA~*(k km~w) ~ (bJ~°~bJ~°~~,

0) oe,I,el

(bE~kbEJk)L = ~ bA~’(k~, k,, W)bAflh*(kfl, k1, 0)) ~ (aJ~°~bJ~°~)~WCi) I—~~ oe,l,el

32ir V V a a a (a)—1

T L~ L4 YmYnYibhhjk (km,ki,W)Lw a±1/=—

X bA~~1~K(k~,k

1, w) ~ (bJ~°~bJ~°~)~. (3.132)11e,i,ei

It follows from the aboverelationsthat the correlationfunctionswill becharacterizedby resonancesnearthe frequenciesthatare determinedby the dispersionrelations,

ReLI(km,wk)=0, m=0,±1,.,., (3.133)

and

Re L~(k,, ~k) = 0. (3.134)

The resonancesdeterminedby eq. (3.133) arerelatedto the excitationand interferenceof volumewaves, while eqs. (3.134) determinethe conditions for (symmetricand antisymmetric)surfacewaveexcitations.The effective temperaturesrelatedto thesecollective excitationscan be introducedjustinthe sameway as the effective temperaturesT(k, w) and T(k,,w) [seeeq. (3.86)]. In particular, theeffective temperaturesfor collective surfacefluctuationscan be written in the following form:

—

41T ~ y~’A~(k1,)A

1~(k w) ~y=e,i,ei (aJ~°~bJ~°~)~,1(k1, w) — ~ s~*s~~ y~A~(k1,w)A~l*(k w)[e~(k1,w) — e~1(k1,w)1

where

~(a) = ~ ~ Tr ~(a)~

is the unit vectorin the direction of the electricfield projectionon theboundarysurface,andw is thesolution of eq. (3.134).Using this definition of the effectivetemperatures,the essentialsurfacepart ofthe correlationfunctionsof the electromagneticfield fluctuationscan bedescribedby the sameequationas in the caseof a semi-infinite system[seeeqs. (3.85)1with the following functional:

~ ~(a) ~ ITLW

3 ~jTr~ yay~T(a)(k w)

s~)*s~)[S~)(k,, w) + S~(k,,w)]s~*s~5(ReL~(k,, w))

s~s~{k,I[k,kS~~(k,,w) + k,3S~?*(k1,w)] — ~ w) + S~~~(k1,w)]}

~7~(k1, w) = 3~1TrL~7~(k,,w) — L~7~(k,,w).


The spectraldensityof the correlationfunction of the electric chargedensityis as follows:

ia.. if~. if,%V~ °. ‘YPk Pk~~1kj,w_

2L(Sm,fl+Om,_fl)~I0P~ )k~WZn,

+ (bp~ bp~)~.TS~+ (8p~ p0~)~w . (3.135)kzm k

2Zm

Here

mY” CT’if”*~ bP~’)~= kmikmj ~ yi~ (kn,, w)y~~ (km,w)(8J~3J(O)~”1 /kmw’

0) if”

~‘ ‘vs kmiknj ‘~ ~ if’*_______ 1*

(bp~ bPkznlkj.w = —2 2 Li Yik (km, w)x1~(k3, w) 8A~(k3, km, 0))(3J~O)bJ~°~)°~’1 kn,O)CL) if”

+c.c. (o~o~’,man),

/ if ~ (7’ s~ ~ g if’*~bpk 0Pk5)k1w = 2 Xik(km, w)x1~ (ku, w) ~ bA~i

1(km,k,, w) bA~~*(k~,k1, w)

Lw 1=—’”

x ~ (8J~°~~j(c~\if” (3.136)i~ k1 /kjwif”

In the caseof an isotropic systemeqs. (3.136) yield

(0)2’ u”/~, if 0’~ Vcop °P ~ (bp )kn,w‘2ISL(km,0))I

In. if if’ ‘VSY)Pkzm bpk )kw = Jifif~(kzm,k5~)+ J(Y,if(kZfl, kzm)

2LlSk,xL (ku, co) Eif~.Yifif~(km,W)(bp(0)2) w y~y~

Jifif~(kzm,kzn) k~k~,e~(km,w) SL(h1m,0))~2 a L~(k,,w)’

a a(~u ~ ‘s 2~’k~X~L(km,~ (kn,W) YmYn

zmSPklk,w = I~5I

2L 6L(km, w)s~(k~,co) a L~(k,,~)~2

2s (0)2 \ if”

y7 k2, (bp~°~lkjw k51 (UT /k~w ‘\

if•’ i~ ~ �jk1,w)I w 14T(kl, ~)~2). (3.137)2+ 2

Here

— ,2LC ~“ a( “±

1~zmL~(k1,w)’1+-~--—L Ym’ 2

k5L m-~ \kmEL(km,co) + k~T(km,

If the systemis in an equilibrium state with a temperatureT, eqs. (3.137) can be simplified.Specifically, this is related to the quasi-static fluctuations (kc ~‘ co). In this case


(bp~ bp~’)kW = (8~+ ~mn) Im{[1 — Xif(km, 0))~(km, 0))lXa(km,w)}

Tk1r ‘

7Xcr(”m,~) X(7(’1fl,~) _______

+ Im(\ ) , (3.138)ITO) E(km,w) e(k~,w) a L~(k,,w)

where

2L m=_ookm~(km,~)

Using the Kramers—Kronigrelations,one obtainsthe following resultsfor the static form factors:

(bpLbpL)k1= ~Tk~(8mn+8m~n) Xff(km~0)[1Xif(km,0)~(km,0)1

+ Tk,r Xff(j1rn,O) x~(k~,0)~, Y”nY~IT E(km,0) r(k~,0) a L~(k1,o)~ (3.139)

Onthe basisof eq. (3.139) it is easyto calculatethe spatialcorrelationfunction. In particular,for thefluctuationsof the total chargedensity for distances R — R’~~ (r

2)”2 (in the caseof rigid dipoles— R’~~ r

0) one obtains

(bp(R)bp(R’)) = ~ {(i - I ~ ~R)4IT5(R - R’) - k~exp(-k~~R-R’~)

2 f dk, k, (~2— ~2)J

0(k1~R1—RI~)(2 2

i f3 /3 +p~+2f3pcothf3L \

+ ~ {exp[—/3(2nL — Z — Z’)~+ exp[—/3(2nL + Z + Z’)l})

— k2 f dk,k, (p2 — ~2)J

0(k, R, — R~j)

Dj /3 /32+~2+2/L/3cothf3L

x ~ {exp[—$(2nL — Z + Z’)l + exp[—$(2nL + Z — Z’)]}}.

Here, /3 = (k2, + ~ j.t = k,c1e0and the quantities~ and k~are definedby eqs. (3.95a,b). For

distancesR, — RI ~“ k~’,the abovecorrelationfunctiondecreasesas const.I R — R’ ~, as in the caseofa semi-infinite system.Such a behaviouris determinedby the presenceof boundaries[117].

Let us consider, finally, the contribution of the collective fluctuations to the spatial correlationfunction with the thermalmotion of anelectrontakeninto account.The volume partof the correlationsin the caseunder considerationis describedby a relation similar to that for pure plasmasystems,


(bp(R)bp(R’)) = e~n~~4 TL(ko, co) k~~ J0(\Jk~— (mITIL)2~R,— Rh)

~ 0)pe m=0

x [cos(mITIL)(Z — Z’) + cos(mirlL)(Z + Z’)],

whereTL(k, co) and k0 are describedby eqs. (3.28) and (3.97), respectively,andN = [k0LIIT].

The contributionof the collectivesurfacefluctuationsto the spatialcorrelationfunction can alsobecalculated.In the domainwheresurfacewavesof the spatialchargeexist, the following result for thesurfacepart of the correlationfunction can be written down, if retardationeffects are neglected:

(3P(R)bp(R’))~=. T~~(k1a(W),W)

xJ(k (w)R —R )

.La .1. .1. ô ReL~(k,,w)/akjjkk(w)

Here,

= 5(Z) + 2 2 2 2 [exp(_Vk~a(w) - k~Z)6sewpeVkj.a(W)— k0

+ cosh(Vk~a(w)— k~Z)({~Ot~}(~Vk~a(W)— k~L)—

T~(k,,w) = ~ w)L=k,

and the quantity k±a(W)is definedby the dispersionrelations(3.134) with

L;~(k,,co) = 1 + e(w) ({cothj(lk L) — \/k2

1—k~{~°~}~LVk~— k~)).

In the limit k,L 4 1, ~ — k~L4 1, theseequationsreduceto

L~(k,,w)—~1+r(w)k,L’ L(k,,w)—~1,

which give k,+(w)~2iIIe(w)lL.Taking into accountthe explicit form of thesequantities,one obtains

(bp(R)bp(R’))~e~ne~ ~ J0(k,+(w)IR,—Rh).S (w)kDL

3.3.3. IntensityofspontaneousemissionThe calculationsof the intensityof the emitted electromagneticradiation can be performedon the

basisof the generalrelations(3.127)—(3.129)usingthe definitionintroducedin section3 1.3. Calculat-

346 Vu.L. Klimontovich etal., Statistical theory of plasma—molecularsystems

ing Poynting’svectorfor the emittedfields for Z <0 or Z> L, one finds [56,95]

‘2’Ow’~p~2’Ow’~s’ ‘Ow 32 ~ (3.140)4ir C

The effective temperaturesusedabove are defined by eqs. (3.100); however, in the caseunderconsiderationthe following functional q~1{ Ta~(km, cu)} should be used:

øjj{Ta~(km,w)} = > ~J~{Ta~(km, w)},

~/3~{Ta~(km,w)} = [L~(k,, w)11[L~(k,,w)11*

x m~amm,0)~1*m,0))Ta~m,0)Y (3.141)

The quantities I~are the energy absorption coefficients for p- and s-polarizedplanewavesincidentupon the plasma—molecularlayer from outside,

= ~ = 1 — ~ (1 — r~~)(1+ r~)+ r~prs~2+ ~a±1 2 a±1 (1 — r~~)(l— r~~)— rsprps~

— a — 1 ~ (1 — r~)(1 + r~)+ ~ + 4(s~’w2Ic2k2)~r’”~2— Li Es — I — — Li a a a a 2

2 a±1 (1 r~~)(1— r55)— r~~r5~

— (3.142)= — -~ k11k1, S~(k,,w), r~= — ~ (~— k~~kl))() w),

r5 = — eSk] k~k,k S~(k,,w), r~= — -~ e~ k~klk S~(k,,w).

It is quite natural that in the caseof thermodynamicequilibrium at a temperatureT the effectivetemperaturesbecomeequal to T and eq. (3.140) is reducedto Kirchhoff’s law in the Levin—Rytovform,

32 T~ff,4ir C

where ~ is the effective absorption cross section for a plasma—molecular layer,

I~ff-2(I;+1~).

3.3.4. Collision integrals and equilibrium distributionsfor a plasma—molecularlayerUsing the general relations(2.25), (2.26), (3.127), (3.128) and the definition (2.17), we havethe

following relationsfor the collision integrals:

Yu.L. Kllmontovich etal., Statisticaltheory ofplasma—molecularsystems 347

ea f dk, fdw 1 ‘ç~‘a = — na J (2IT)2 J 21T (2L)2 .~ [(1 — km VIw)5,

1 — kmiVjlW]

(bE/k bEkk1n)k1

+~ ((Ikb~Z(P))k,W i(w—kV—u

Ofa(X,t)\x [(1 — k~VIw)Sflk — knnVklW]fla ~ ) exp[i (m — n)ITZIL], (3,143)

f dk, 1 dw 1~1ab J (2IT)2 i 2ir (2L)

2

x — ‘(V + a1 kmi (V, + aivj)] exp{i[aikm r + (m — n)irZIL]}N v))5,1——

/ ° ~E(8E1kZmBN~°~(x,P))~w+ (bE1~n,bEkkzn)kiwX ~ + a1

+ /3(l)*(,~ ~ (l’)* ~ 1x ~ el,(a~*(x, P) — F) — + Ylk (x, F) ~ t)],1’ op1

km = (k,, mirlL), (3.144)

wherea~,/3~ andy~aredefinedby eqs. (3.50), the correlationfunction (bE~k bEik)kW by eqs.(3.131) and (3.132) and

(bEjkzm bN~ (P))kw = — ~ A~(k,,,,k,, co)cuL

x 2~e~n~V,,S(w— !c1 . V) . 2L(51~+ SkSl,_n)fa(X’, t),

(bE/kmbNGbk (x, F))k~ = — 7~ ~ A7~(k~,Ice, co)

X e1. .

2ITflb ~ (—i)mJ~(a1,k,r)J~,(a1,k~. v/cu)

1’ n,m= — =

x [V,,~+ Uk mco0l(k,. v) + r,, nw0/(k1 r)]5(w — k1 . V— (n + m)w0)

x 2L(81,,+ ~k

51,-n)fab(~’ t).

In the framework of perturbationtheory up to first order in the atomicparameterand to zerothorder in the thermalparameter,eqs. (3.104)—(3.107)for the molecularcollision integralsstill remainvalid after the following replacement:

dk 1 ~ I dk~—~ ~ ~,, k2 ~ fliT

—~— k’—*—.2~Li, J 21T 2L ,, L ‘ ~ LIT 2Lm

Equations (3.108) for the collision integrals of free chargedparticles are also valid with thisreplacement.

348 Yu.L. Klimontovich eta!., Statistical theory ofplasma—molecularsystems

In the caseof an isotropic distribution the kinetic coefficientsof the type (3.105) can be found inexplicit form. Here, it shouldbe pointedout that in the caseof a layerthe contributionof the volumecorrelationsto the kinetic coefficients is qualitatively different from that in the casesof an infinite or asemi-infinite medium. For example,

2 NL 4 NT 2ab(coII) ________ _________

2 2 + ~ e~-~-~)[i + ~ cos(2mirZ/L)]. (3.145)‘/~~ (Z)= Ireab ( ~ L _____

mw0L m—NL

6~pe~e m-NT 2c

Here,

i.L I zm T I zme,’1 1k

2 k2 1k~T k2~ ~,, + ~ 5,ZSiZ , e~,= — ~ -~— ~ — ~ 8iz8fz

kILT = (k2 — k2 )112, NLT= [kLTLIir], and the quantitieskLT are definedby eqs. (3.109).L,T zmIn the limit L —~ ~ the sum over the discrete wave number m shouldbe replacedby an integration,

NLT1 fdk

— ~i 2ii~2L m-NLT

Taking into accountsuch a replacement,it is easyto see that eq. (3.145) transforms to the first twotermsin the radiationfriction coefficient for a semi-infinitesystem[seeeq. (3.109)1.

The general relations (3.143) and (3.144) enable us to describe the collision integrals in the potentialapproximation.The final result in this limit hasa form similar to eq. (3.110), in which the followingformal replacementsshould be made:

I dk1 (‘dk’ 1 ‘cc’ 1 ‘~ a a I dk” 1J~—~--~—~,~

IT 2Lm J2IT

L(k,, cu)—~L(a)(k1, w), k”+km, k’—~k~,k”—*k,,

and in the relations for bI~the following forces Fa(R) and F1(R + a1r) should be used:

a a

Fa(R)=_e~j dk, 1 4IT~km _____________

(2IT)2 2L m k~,iE(km,0) (e2Ikzmz— e’~~~ e YnYmL a ~ k~E(k~,0)La(k

1,0))’

I dk1 14’TTlkm (eik”n1_u1 + eikm

1r~a.r’~)IeF1(R + a1r) = —e1i:;i e, J (2IT)

2 2L m k~E(km,0)

a a

YnYm ik,,ra

1, inirZIL

— e’~’~’ elm JL ~ k~e(k~,o)r(k1,0) C e )Fa(R)= ~OUa(R)IdR,

~.F1(R+a1r) / 0 0.~—+al—)fb(~,t)

= (OUab(R~r)

\ OR + ~- [ç6~(r)+ Uab(R,T)1~~)fab(~,tY (3.146)

Yu.L. Klimontovich eta!., Statistical theory ofplasma—molecularsystems 349

Here,the quantitiesUa (R) and Uab(R, r) arethe self-interactionpotentials,which maybe expressedintermsof the screenedpotential,

Ua(R) = e~,q,,(R,R’) = e~U(R), Uab(R,r) = ~ e,e1,ço5(R+ a1r, R +

(3.147)co~(R,R’)~o(R,R’)—ço0(R,R’),

where

‘R R” = ~ dk1 141T ( ~tkn,’R’ +

‘ J (2i~)~2L m- k~S(km,0)~,e e

— 2 a 1 e~~~”~’)e~m~R, (3.148)L a±in~cokn~(kn,0)L (k

1,0)

R R’ — f dk 4ir ik’(R-R’)(2IT)3 k2s(k,0)e

It is easy to show that in the caseof an equilibrium systemthe distribution function should, asbefore,be calculatedon the basisof eqs. (3.113),(3.114).However,in thecaseof a layerit is necessaryto use the relevant screenedpotentials (3.148), in order to calculate the self-interactionpotentialUa(R) Ua(Z) and Uab(R,r) E Uab(Z, r). The final result for the distributionfunctionsis of the form[98]

fa(X)=f~°~P)exp{-[ea(~(Z))+ U0(Z)]/T)(~JdZexp{-[ea(~(Z))+ Ua(Z)]IT}),

0 (3.149)

fab(~)=f~(P)f~?](p)j~°~](r)exp(_~’e1(~(Z+ a~z))+ UQb(Z, r))

x [~f dZfdrexp(_ ~1e,(~(Z+ a,z)) + Uab(Z~r)) 1(Q)~]

where

(~(Z)) a~iieanafdR’~o(R,R’)(exp[_U~(Zo)IT]_ ~IdZeXp[_Ua(Z)IT]).

Thus,we have general relationswhich describethe profiles and the orientationaldistribution ofparticlesof differentkinds in systemswith plane-parallelboundaries.

It should be noted that the identity of the general structureof eqs. (3.115) and (3.149) up tonormalization constantsis not incidental. This is relatedto the fact that the representationof thedistributionfunctionsthrough the self-interactionpotentialsUa(R) and Uab(R, r) andthe self-consistentpotential (4(R)) is generaland doesnot dependupon the shapeof the boundarysurface.In otherwords, in the caseof boundedplasma—molecularsystemswith an arbitraryshapeof the boundary,the

350 Vu.L. Klimontovicheta!., Statistical theoryof plasma—molecularsystems

equilibrium distribution functionsare of the form

~(3.150)

- / U (R r)+~e(4(R+a,r))fat,(f’) = f~,~t,)(P)f~,.°~( p)f~,~t,)(r) exp~— ab T

11 1 1 / Uab(R,r) + ~ e,(q~(R+ a,r)) ~ -~X [~ j dR j drexp~— T )fat, (r)j

where

(cb(R))=).eanafdR’w(R,R’)(exp[—Ua(R’)IT]—1~fdRexp[—Ua(R)IT]), (3.151)

Ua(R) and Uab(R,r) are defined by eq. (3.147) and the screenedpotential ~(R, R’) satisfies thefollowing equation:

V2~(R,R’) — k~ço(R,R’) — ~ J dr4ITe1e1f~°t,~(r)ço(R+ a1r — a1r’, R’) = —4IT8(R—

1,!’ (3.152)

togetherwith the boundaryconditionsfrom macroscopicelectrodynamics,

p(R,R’)Is~(R,R’)~s, ~

Here ~(R, R’) is the potential of a unit charge (situatedat point R’) in an external medium withdielectricconstant~‘, S is the surfaceof the system.The potential~(R, R’) shouldbe calculatedon thebasis of the PoissonequationV

2~(R,R’) = 0. The proof of relations(3.150), (3.151) is given in refs.[98,99].

Linearizing eqs. (3.150), (3.151) with respectto Ua(R) and Uab(R,r), it is easyto obtain generalresultsfor a linear perturbationof the equilibrium distributionin boundedplasma—molecularsystemswith an arbitrary surface,

fa(X) = f~°~(P)[1+ aa(R)],

aa(R) = - ~ (ea(~(R))+ Ua(R) - J dR [ea(~b(R))+ Ua(R)1),

fat,(~)= f~(P)f~(p)f ~(r)[1+ aab(R, r)]

aab(R, r) = — ~ e1( q5(R + a1r)) + Uab(R, r) — f dRJ dr f~°t,~(r) Uat,(R,

(4(R)) = —~ ~ ço(R,R’)(U(R’)_~JdRU(R)).

Yu.L. Klimontovich etal., Statistical theory ofplasma—molecularsystems 351

4. Large-scalefluctuationsin plasma—molecularsystems

4.1. Equations for the smoothedmicroscopicphasedensities

Let us return to the basiceqs. (2.19) and(2.20) for the microscopicphasedensitiesaveragedovertheGibbsensemble,(N0(X, t)) and (N0b(~‘,t)). Nowit is necessaryto specifytheconceptof a “Gibbsensemble”.The incompletenessin describingthesystemin theframeworkof a nonstationaryensemblewill be consideredas the result of averagingover a physically infinitesimal time interval ‘Tph or therelevant volume V~h(sometimesit is useful to averagesimultaneouslyover ‘Tph and

11,h). Let usintroducethe following notationsfor the functionsaveragedover the Gibbs ensemble:

~‘a(X,t) (Na(X,t))7 , Nab(Il~’,t) (Nat,(~’,t))7

It is natural that thesefunctions are random. The equationsfor them coincide with eqs. (2.19) and(2.20) up to thereplacement~ —* < > ,.~,;thereforethe basicequationsfor thesmoothedfunctionsareof the form

La]c’ia(X, t) + Fa(R, V, t) ONa(X,t)IOP = nala{Na(X, t), N~t,(~’,t)) , (4.1)

Lab1~ab(fe,t) + ~ F1(R + a1r, V + a1v,t)’ (~+ a1 ~)ñab(~’, t)

= ~ab’ab{”ab(~” t), Na(X,t)} , (4.2)

Fa(R,V, t) = ea[E(R, t) + (V/c) x B(R, t)]

However, it is very importantthat now the explicit form of the collision inte~,ra1sis known, i.e., theyare expressedin termsof the smoothedmicroscopicphasedensitiesNa and N0t,,

Ia{Na(X, t), “~‘ab(~”t)} = ta{ fa(X, t)—* n,~Na(X,t), fab(~” t)—.~b”~0b(~” t)}

‘ab{~’ab(~’ t), Na(X, t)) = ‘ab{ fab(~” t)—. ~ab~ab(’~” t), fa(X)~ ~11Va(X’, 1))

where

I,~{f~~(X,t), fat,(~’t)} ‘ 1ab 1’ab{fab(~’ t), fa(X, t))

arecollision integralsof thetype(2.19),whosefunctionaldependenceon f0 (X, t) (a = e, i) andfat,(a” t)

is determinedby the microscopicfluctuationspectra[seeeqs. (3.46)—(3.50), (3.101)—(3.103),(3.143)and (3.144)]. Thus, eq~s.(4.1) and (4.2) togetherwith the equationsfor the smoothed(over TPh)electromagneticfields E(R,t) and B(R, t)*) form a closedsetof equations.

The averagedvaluesof the smoothedmicroscopicphasedensitiessatisfy the following equations:

~ ThefieldsE(R, t) andB(R, t) satisfy equationsof thetype(2.3), (2.4), (2.6) in whichthedensitiesN~(X,t) andN~5(~’,t)arereplacedby therelevantsmoothedquantitiesN~(X, t) andN~5(~’,t).


- - o(N(x t))

La(Na(X, t)) + (Fa(R, V, t))~

= ~a1a{~~a’(1Va(X, t)), flat,1(Nat,(~, t))} + ~ t)), ~at,’(1”~at,(~’ t))),

Lat,(~at,(~, t)) + ~ (F1(R + a1r, V + a1v, t)) (~+ a~~)(~a~(~’ t))

= flab1ab{flab(Nab(~, t)), fla’(Na(X, t))} + flat,~t,{flat,

1(Nat,(~, t)), flal (1Va(X, t))},

where

~{n~’ (!~a(X,t)), n~t,’(I~(~,t))} = _-~- (bF(R, v, t) (0/OF) bJ~a(X,t)),

~t,{t,(Nat,(~’ t)), n1(ñ~(X,t))}

= — -~— ~ (bi~1(R+ a1r, V + a1v, t). (0/OP+ a1 OIOp) bNat,(~,t))

are integral operatorstaking into accountthe influenceof the deviationof the smoothedfields fromtheir averagevalues (E(R,t)) and (B(R, t)) on theevolutionof thesingle-particledistributionfunctionf~(X, t) = n,~

1(Na(X, t)), fat,(~‘t) = n~(Nat,(~,t)). These operators play the role of collision integralsin the ordinary kinetic equationswhich do not take into account the contribution of large-scalefluctuations.

As regardsthe large-scalefluctuations,they satisfy the equations

La bNa(X,t) + bF0(R,V, t) [naOfa(X,t) lOP + 0 bNa(X,t) lOP]

= naIa{n~1Na(X, t), fl~’Nat,(~,t)} — ‘~a1a{ fa(X, t), fat,(~’ t)} , (4.3)

‘~abbNat,(~,t) + ~ 3F1(R + a1r, V + a,v, t)’ (~+ a1 ~)[uia~fa~(~, t) + bNat,(~,t)J

= ~at,1at,{’~aTat,(~’t), n~’Na(X,t)} — flat,iat,{fat,(~’ t), L(X, t)}. (4.4)

Omitting slowly varying quantities,aswell as the termsresponsiblefor the influenceof the averagefields on the fluctuations,and linearizing eqs. (4.3), (4.4), we obtain

(~- + ~ ~) 8Na(X~t) = ‘~a bla — bF(R, V, t)fla• Ofa(X, t) , (4.5)

/0 0 0 04~at,(T) 0\I~,—+V~—+v~——— ._)bJ%J (ft)

Of OR Or Or 0~ ab

= flab blat, — ~ bF1(R + a1r, V + a1v, t). (_~_+ a1 ~)‘za~fa~(~, t). (4.6)

Here,bla, bl0t, are the linearized (in bNa and bNab) parts of the operators


Ia{fl;1[(Na)+bNa],n~b1[(Nab)+bNab1}—Ia{fl:1(Na),fl:b1(Nab)}

Iab{fl~b1[(Nab) + bN0t,], ~a[(1V’a) + bNa]) — Iab{n:b’(Nab), ~~a

1(Na)}

Thegeneralform of themolecularcollision integralsmayberepresentedasthe sumof two operators[108],‘ab = 1~+ ,t, which describethe changesof the distributionin the internal (relative)variables(r, p) and in thevariablesof thepair centreof mass (R, P) as a result of collisions.

Sincein thegeneralcasethe collision integralsobtainedin theprevioussectionarevery lengthy, weshall usebelow for the operator~ a collision term of the type (3.51) calculatedin the dipoleapproximation,

= D~Op1ap1 + ~

Here, we neglectthecontributionsrelatedto the oscillatorfrequencyshift ~0)ab andthemixeddiffusionin p and r space.For the samereasonthe collision term I aswell asthe collision integralsfor freechargedparticles ‘a will be replacedby the model Bhatnagar—Gross—Krookcollision integral [100],which enablesone to take into accountqualitatively the influenceof a collision on the large-scalefluctuation spectra,

I~”{n;1Na, ~2a~1Na} = — E Paa~[Na(X,t) — fla(R,t)4aa(X, t)], a = e, j,

a’~e,i,ei

where

tt)aa(K, t) = (2”7TTaa,ma)~312exp[—rna(V— Ua)2/2Taa],

fla(R, t)=JdP Na(X, t), UaanUa(R,t)= fla(R, t) I dPVNa(X,t),

T00 Taa~(R,t) = [maTa,(R,t) + maTa(R,t)]/(ma + ma),

Ta(R,t) = t) JdP (V— Ua)2Na(X,t), 1~Tat,(X,t) = Jdx I~t,(~,t),

~ is theeffectivefrequencyof collisionsbetweenparticlesof speciesa anda’. Furtherwe shall assumethat i~,, is proportionalto thedensityof particlesof speciesa’. The BGK collision integrallinearizedina small perturbationhasthe form

~a’=e,i,ei

1 1 — F n rn V• bU ,(R,t)= — — ~ h1aa1~1’~a(A’,t) — Lbuz,,(R, t) + a a T

~0 a’=e,i,ei aa’

n /rnV2 3\/ rn m~ \1 0

+ T~ ~ 2Taa~— ~Ama +“m, bTa(R~t) + ma +arna, bTa(R, t))]~$~0~(P)

+ [f0(X, t) — ~~~P)] 8fla~(R,t) } , (4.7)

354 Vu.L. Klimontovichet a!., Statistical theory ofplasma—molecularsystems

bfla(R, t) = f dP bNa(X, t), bUa(R, t) = -~- I dPV bNa(X, t),~a ~

bTa(R, t) = -~- f dPV2 bNa(X,1)— bfla(R, t) Ta~

~ —3/2= (2ITrnalaa,) exp(~—maV2l2Taa,).aa

Thus, the equationsfor large-scalefluctuationsare as follows:

Ofa(X, t)Lab]~a(X,t)+bFa(R,V, t). ~p = ~ (bl~+bi~’?), (4.8)

a =e,i,ei

10Lat, bNat,(~’,t) + ~ bF,(R + a

1r, V + a1v,t)~~- + a1 ~ t)

~ (bI~i]a~+bl~~), (4.9)a’=e,i,ei

where

rnbI(I),=_Jaa,IdP~{5(P_PF~_th(O~(P)[1+ a) ‘aa ma + ma~

3 (maV2 —

1)(mahl’2 —

3T 3T , T jj}bNa(R,P,t), a~ei, (4.10)aa’ aa aa

(P)I

ImV’V’ m9~(II) dP’i a + aWaa~= “aa’~Paa’ L Taa~ ma + ma3 (maV

2 —

1~maV’2 T

0, \ /fa(X, t) — i)1~b~,(R,P’, t), a �ei, (4.11)X~\3T 3T~0~ T)~q5(°)(p’~aa’ aa ,aa’= D~02 aN0t,(~~’,t) 0 ab0~0 + ~ [yjj p1 bN~,(~t’~t)j.Here, insteadof Ta(R,t), Taa~(R,t) it is necessaryto usethe unperturbedtemperaturesT~,Taa~=

(maTa,+ ma,Ta)/(ma+ ma’).As regardsthe analogyof the BGK integralsfor boundparticles(a = ei), their explicit form depends

on the relationbetweenthe relaxationtimes for internalmotions(r1~~)and for theatomicdistributionsasa whole (r~).In particular,for y~~>~“‘ab,athe collision integralsbI~,’~?may be obtainedon the basisof eqs. (4.10) and (4.11) by the following replacement:

JdP’ bNa(R,P’, t)—~f dP’ dx’ bNab(T , p’, R,F’,(0)

— P’)—* 5(P— P’)8(x — x’) ‘ °~a’(~~)~”~ a’(’~)f~(’)fat, (p)

Vu.L. Klimontovichet al., Statistical theory of plasma—molecularsystems 355

When theoppositeconditionis satisfied,y~’~ ~ the quantitiesbI~’~,for a’ = e, i areobtainedfrom eq. (4.11) by multiplying it by 5(x — x’). The quantitybI~7,~b,as well asthe collision termsbI~a(a’ = e, i, ei) follow from eqs. (4.10) and (4.11) after substitution of bNab(X,R,P’, t) insteadofbNa(R,F’, t).

4.2. Transitionprobabilities in collisional systems

In the caseof a three-componentsystem(electrons,ions, andmolecules),which is consideredin thepresentpaper,eqs. (4.8) and (4.9) representa set of threecoupledintegro—differentialequations.

In spiteof the abovesimplificationstheseequationsarestill rathercomplicated.Therefore,hereweconsidersomelimiting caseswhenit is possibleto find the explicit form of their solutions.

In particular, if the densitiesof the chargedparticlesna (a = e, i) are much less than that of theneutralonesnab’ thecontributionof themolecularcollisionswith chargedparticlesto theequationforbNab can be neglected(i.e., one may take 1”ab,a = 0, a = e, i). On the otherhand, in sucha caseit ispossibleto restrictoneselfto a descriptionwith no collisionsbetweenchargedparticles(‘“‘ee = = ~“‘ei=

= 0). This meansthat in eq. (4.13) for a = e, i the termswith bI~?shouldbe omitted. As a result,theequationsfor bNa (a = e, i) and bNab becomeindependentand their solutioncanbewritten in theform of eqs. (2.25) and (2.26). However, in the case under considerationthe Green functionWa(X, X’, r) satisfiesthe following equation[108]:

LaI~a(X,X’, r) = — ~a{I~(X,X’, T)

23rnV rnV T m -

a(P)J(h1~”[1~(3j _1)(3~ T) ]W(RPXT)} a=e,i,

a,ab a,ab a,ab rn0(4.12)

where paanpa,ab~

For pair-boundparticleswith ~t, ~ 7~bone obtains

L0t,1~0t,(~,~‘, T) = biFP{ W0t,(~,~‘ r)} — tab{14cb&~, ~‘, T) — ~ab(P)fab(T)fab(P)

xfdxdP[1+~ (~ç~_i)(”2 _i)+ ~ (4.13)

Here, 1”’ab t”ab,ab’ 4’ab(’~) ~t~ab,ab(1’) In theoppositecase~ab ~‘ ~ insteadof eq. (4.13)the followingequationshouldbe used:

Lab*ab(~C,~‘, r) = biFP{ ~‘at,(~” ~‘ ~)} — Pab{ 14Tab(~’a”, T) — ~ab(’~)

x f dP”[l + ~ ~ — 1)(3T — i) + T]~~’ab(” p, R, F”, a”, T)}. (4.14)

The so-called “simple BGK model” is frequentlyusedin plasmatheory. Accordingto this model,

356 Vu.L. Klimonto~ichet al., Statistical theory ofplasma—molecularsystems

thecontributionof hydrodynamicvelocity andtemperaturefluctuationsaremuchless than the relative

contributionof numberdensityfluctuations.In thiscasethe following equationfor Wa(X, X’, T) is used:

La%~a(X,X’, r) = — pa(W0(X,X’, T) — c~a(P)JdF” Wa(R, F”, X’, r)). (4.15)

Notice that this equationcoincideswith that for the transition probability of a Brownian particle.Indeed,for Brownian particlesm~~ M, ~B ~ nab andthus on the right-handside of the equationforWB(X, X’, T) which follows from eqs. (4.9)—(4.11),one should keepthe operator

= — W8(X,X’, r) — q5~(P)f dF” WB(R, P”, X’,

i.e., we areled to eq. (4.15).

4.3. Dielectric responseand the correlation offluctuation sourcesin systemswith collisions

In the caseunderconsiderationthe currentsandchargesin the subsystemsof free chargedparticlesare still describedby equationsof the type (2.4), in which, however,the smoothedmicroscopicphasedensitiesare substituted.This leadsto the following relation for the dielectric susceptibilitytensor:

X~(R,R’, cu) = — 4~ah1aJdX’ JdP~ JdT e~T~a(X,X’, r)

I / . , o ~ . , a j Ofa(R, p’, t)+ iV o~~)— iV1 -,~7j ~, , a = e, 1, (4.16)

whereWa(X,X’, r) should be calculatedon the basis of eqs. (4.12) or (4.15).The general relationsfor the correlationfunction of the microcurrents in the subsystemsof free

chargedparticlesare also valid,

(bJ~°~(R,t) bJ~°~(R’,t’)~= e~naf dPf dP’ V,V

x [W’a(X, X’, T)fa(X’, t’) + ~4Ta(X’, X, ~T)fa(X, t)1 . (4.17)

As regardsthe susceptibilitytensorand the correlationfunctions for the molecularsubsystem,thepresenceof the Fokker—Planckoperatoron the right-handsides of eqs. (4.13), (4.14) makesit verydifficult to find their solutions,which, in turn, makes it impossibleto usethe generalrelationsfor

~~‘(R, R’, cu) and (bJ~°~(R,t) bJ~°~(R’,t’)~” [see,eqs. (2.39) and (2.43)]. However, keepingin mindthat for further electrodynamiccalculationsone needsto know only the first two momentsof themolecularGreenfunction,

~‘0b(X, X’, r) Wab(X,~‘, T) = Jdx Wat,(~, ~“ r),

w~°(X,~‘, T) = f dx UiWab(~l, ~“ r)

Vu.L. Klimontovichet al., Statistical theoryof plasma—molecularsystems 357

the theory developedin section2 canbe generalizedto the caseof collisional systems,if onecalculatesthe abovemomentsWab(X,X’, T) and w~(X, a”, r).

The appropriateequationsfor the momentscan easilybefoundby integratingeqs. (4.13) or (4.14)with the relevantweights. In particular,for Wab(X,X’, T) one obtains[108]

Lab1~’ab(X,X’, r) = — bat,{I4Tat,C1t~,X’, r)

3 MV2 MV”2 MV V” —

— 9~ab(1~)JdP” + ~ (~-~_ — 3T — ) + Tab ~ F”, X’, T)}. (4.18)

Let uspoint out that thisequationfollows from eq. (4.13),aswell asfrom eq. (4.14),andthereforeit doesnot dependon the relation between~ab andy~’.

When derivingthe equationfor w~°(X,~‘, T) oneis led to a hierarchyof equationsfor the momentsof different orders. However, we shall not look for a general solution of the problemand restrictourselvesto calculatingw~°(X,a”, r) in the dipole approximation,becausejust this approximationisrelevantto the replacementof the collision integralsby the Fokker—Planckoperatorin eqs. (3.13),(3.14). In thedipole approximationthe equationfor w~’~(X,a”, r) is simplified appreciablyand takesthe following form when i’~t,~ .y~th:

w~°(X,a”, T) = w~(X,~‘, r) = (~ + V + k~b)p~(X,a”, r),

(~ + V~ + ~ + V ~~pd(x ~‘, T)

+ ~v~”(~ + v~~)p~’)(X, ~“, r) + cu~p~(X,a”, r) = 0. (4.19)

In the oppositecase ~th ~7ab for ‘r> lIi’~, we should take Vat, = 0 in eqs. (4.19). In fact, when

r> 1 IVab the atomic velocity distributionbecomesMaxwellian and therefore the contributionof thesecondterm on the right-handside of eq. (4.14) is negligible.This enablesone to omit quantitiesproportional to ~‘�th in eq. (4.19). Following refs. [18,24], in our further calculationswe restrictourselvesto an analysisof the lattercase.

Equations(4.19) should be supplementedwith the initial conditions that follow from eq. (2.29),

Wab(X,X’, 0) = 5(X — X’), w~(X,~“, 0) = v’S(X — X’), p~(X,~“, 0) = r’S(X — X’).(4.20)

The boundaryconditions for the momentsof the molecularGreenfunction also follow from theboundaryconditionsfor W0t,(~’,a”, r)

~TOb(K, X’, r)15 = VVab(X’ X’, r)~, n . w~~(X,a”, ~ = —n w(d)(it, X’, r)~

(4.21)p(d)(x ~“, r)I~= —n p(d)(it X’, r)I~

358 Yu.L. Klirnontovich et a!., Statistical theory ofplasma—molecularsystems

Here, we introduced the notations Xt = (R, Pt), pt = (F1, —Pa), n is the normal to the surfacedirectedinto the system.

In termsof w~(X,~‘, r) the following results for thesusceptibilitytensorsandcorrelationfunctionsof the molecularmicrocurrentsareobtained:

bJat,(R,t) bJ~(R,t) — ‘~ab~bJdT f dFf d~”w~(X,~‘, r)

- Of (~~“t’)x [bE(R’,t’) + (V’lc) x bB(R’, t’)] ab’ , (4.22)

(bJ~°~(R,t) bJ~°~(R’,t~)~b= flabe~bJdPJdP’(J dx’ vw~(X,~‘, T)f~~(~ t’)

+ f dx v,w~(X’,~, —r)f~t,(~,t)). (4.23)

Let us recall that in contrastto the generalrelations(2.39), (2.43) the quantitiesdefinedby eqs.(4.22), (4.23) are calculatedin the framework of the dipole approximationup to zerothorder in thethermalparameter.

It is easyto find the correlationfunction of the moleculardensityfluctuations,

(bn~°~(R,t) bn~°~(R’,t~)yb = nab f dFf dP’ [Wab(X, X’, T)fab(X’, t’)

+ WOb(X’,X, ~‘r)fab(X, t)]. (4.24)

It follows from eq. (4.22)that in thestationarycasethehigh-frequencymolecularsusceptibilitycanbe written as

~ R’, cu) = 4ITi ~ab fdPf dx’ dP’ J dr e~wTw~(X,~‘, r) Ofab(~’, t’) (4.25)

For the spectraldensitiesof the correlationfunctionsof molecularsourcesone obtains from eq.(4.23)

(bJ~°~(R)bJ~°~(R’))~= jab(R R’, cu) + jab*(R~R, cu),

I~(R,R’, cu) = e~t,n0bf dFfdx’ dF’ f dr e~Tw~(X,~‘, T)fat,(~’, t’).

Vu.L. Klimontovicheta!., Statisticaltheory ofplasma—molecularsystems 359

5. Influence of large-scalefluctuations on the electromagneticprocessesin plasma—molecularmatter

5.1. Infinite plasma—molecularmatter

5.1.1. Fluctuationpropagators,dielectric responseand correlationfunctionsof externalcurrentsIn thecaseofinfinite plasma—molecularmatterthesolutionofeqs.(4.12),(4.15)caneasilybe found

by Fourier transformation[108],Ia(X, X’, r) = f dk f ~ exp{i[k. (R — R’) — wr]}W~(P, F’), (5.1)

(2ir) 2ir

where

w pp’ — iS(F—F’) — ~‘a4’a(1~)akw(’ W~k’V+~i/a (w—k~V+i~)(w—k.V’+i~~)

2rnV~ ~ + ~ata( ~ — 1)I~k~(P’)], (5.2)

a , (a)—1 aIIkW(F)iT1/ (k,cu)J1, t1,2,

(a) . fdFIT11 (k~w)l_1V0j -~-40(F),

IT~(k,w) _~~VaflataJ 2

(a) . fdPV

1T2i(k~W)l—lVaj ~~4’a(’°),

iT~(k, cu) = 1— ~ j ~ç ~i(~ ~,—

= 1, j~.a)= V’2I3s~ n~= M/(rna + M) , ta = Ta!Ta at,’ t1a = CU — k~V+lVa

In thecaseof a “simple BGK model” the fluctuationpropagatorsaredescribedby eq. (4.15), whichleadsto the following propagators:

W F F’ — iS(F—F’) — ~a~I~a(F)akw( ‘ cu—k’V+iz~ (w—k~V+i~)(w—kV’+i~)

/ . ~a(’~’) \_1X(\1—1V J dF . ) . (5.3)

0 w—kV+tii~

It is easyto showthat whenI~+ ~VaI> k8a thespectralfunctions(5.2) and (5.3)areclose to eachotherandcoincidein the high-frequencylimit [IT~j 1, 174a2) 0, I~ (1 — ta) — 0]. This justifies the“simple BGK model”.


The solutionof eq. (4.18) can be found in an analogousway. In this casethe spectraldensityis asfollows:

i8(PP’) “ab~ab(’~)Wabkw(P~P) W~k•V+iPat,— (w—k.V+i~t,)(w—k.V’+i~t,)

[Jab (F’) + ~‘ Jab(p~) + ab 1X 1kw Sab 2 — 1)I3k~(P)] , (5.4)

where

ab D’ (ab)—1 yabl.kW(1 ) = ~ (k, w)~ , i = 1,2,3

ab ____________________________ (ab) —Ik ab (F’) kx(kxV’) 1ab‘2kw(’) = k ‘2kw — 2 ‘Tkw ‘ ‘Tkw(’~) [ir11 (k, w)]k

5ab

ab,ab 1 ab1kw ‘ ~

2kw = k~V’lkSab “3kw = V’2/3s~b

(ab) __________________IT

11 (k,W)=1_~Vabf~4~ab(P)=w+iut,W(Zt,)W+11)ab

~ab

(ab) ~‘ab f ~ k . V4~at,(P)= hhlabZab W(Zat,)IT12 (k~w)=—~———~‘nab

CU +

(ab)

11”ab f dF 2 2 lUabir13 (k, w) = — —~-— j ~— (V ISab — 3)4’at,(~) = _________

ab 2(w + iP~b) Ri — Z~b)W(Zab) — 11,(at,) (ab)

‘iT21 (k, w) = ‘iT12 (k, w)

(ab) =1 lPab I dFIT22 (k, w) i — 22 J ~ (k . V)

2~ab(P)= 1 + ~I’ab Z~bW(Zah)~kSat, ab W+llbab

(ab) ~~‘ab f dP 2 2 ____________ir23 (k, w) = —

2kS ab ~ ~— (V ISat, — 3)k~VQ5at,(F)= — 2(w + ‘~‘ab) Zab{(1 — Za~b)W(Zah) — 1]ab

(ab) ‘~“ab 1 dP 2 1V~b {2[1 — W(Zat,)1— Z~bW(Zab)}

IT31 (k,w)1_—~—J ~~~V4at,(P)=1~ 3(w+iVb)3Sat, ab

(ab) ~ab f -~ V2k . V(~at,(P)= [1 — 2W(Z~~)+ ZabW(Zab)],IT

32 (k, w) = — 3ks~t,~ ~ab 3(w+

I dF V2

(ab)IT

33 (k, w) = 1 — ~ j ~— —i-- (V~ls~t,—

‘~‘ab {2[1 — Wab(Zab)] — Z~bW(Zab)— Zat,[1 + Z~bW(Zat,)1},—1 6(W+~Vat,)

Zat, = (CU + j i”,,t,) /kSab , Sab = (T0t,lM)’ /2 (5.5)


Finally, thesolutionof the setof eqs.(4.18)with thescalarradiationfriction coefficient given by eq.(3.57) is asfollows:

1 dk Idww~(X,a”, r) = J (2)~J ~ exp{i[k. (R — R’) — wr]}w~j(P, x’, F’), (5.6)

where

w~(P, x’, P’) —r’ô(P — P’) — (w — k V)[v’ — ir’(w — k . V + iy~’)Jö(P — F’)(d) _________________________________________________________2 2 ab (5.7)(w—kV) —wo+iy (w—k~V)

Takinginto accountthestructureofthepropagators,thesusceptibilitytensorsandcorrelationfunctionsof themicrocurrentscanbe written as

4~ie~n~f dPf dP’ V1[(w — k V’)~Jk+ kkVj]W~kW(P,P’) af~(X’,t)2 o• = e, 1,(U

= — 4iTie~1n~1 _______________ ~fei(~’,t)JdPJdP’dx’w~(P,x’,P) ~, , o~=ei,

(5.8)

(~j~O)~ = I~(k,w) + IcT*(k w),

I~(k,w) = e~n~JdPf dP’ VVW~kW(P,P’)f~(X’,1), = e, i,

= e~n~1f dPJdP’ dx’ w~(P,x’, P’)vf~~(~”,t), r = ei. (5.9)

Equation (4.24) gives the following result for the spectral distribution of the molecular densityfluctuations:

/ 2’ab~ ~ (5.10)

In the particularcaseof an isotropic systemeqs.(5.8) reduceto

~ w) (ö~1— k,k1Ik2)~~(k,w) + (k~k

1/k2)x~(k,),

2

~ w) = —i —~ I dPJdP’ (v_!~!k) W~k~(P,P’)•df~(X’,t) o=e,i,2w J

mJd’jfei(’X’,t)(w_k.V)2_~+iye1(w_k.V)~ r=ei, (5.11)


-2o.’. ~~pCr _________ _

= — —~— f dPf dP’ WifkW(P, P’)k. of~(X’, _— e,1,(5.12)

2 J fei(X,t)~pm dP uei.

In the caseunderconsiderationeqs. (5.9) for the correlationfunction of the microcurrentsyield

if

(~j~O)~j(°)~ =j 1kw ~. — k~k1Ik2)J~°~2)~~+ (kjkjIk2)(~J~°~2)kw

where

2 if — effnif 1dPf dP’ (v— ~ k)(V’ — k~V’k)~ ~kw

2J

X ~ P’)fif(X’, t) + c.c. , a- = e,i,(5.13)

2e~1n~wf f ((k x V)

2/k2)fei(~~t)1 2 dP dx +c.c., a-=e,i,

(w —k.v)2 — ~ +~~‘(~— k-V)2

)2 ~ eifnif f(~ ~w = ~ dPJ dP’ (k . V)(k . V’)fif(X’, t) WUkW(P,P’) + c.c., a- = e,i,(5.14)

= ~ie~ineiwJdP I dx [(k. V)2/k21fei(~,t) ci = ei.2 2 . eiJ (w—k.V) —w0+iy (w—k.V)

In the limits ~yel~ k~(V~and ~ei ~ k(V), the susceptibility tensorand correlation functions ofLangevin sourcesrelated to the molecularsubsystemcan be simplified. In particular, in the limityei~I((v)eqs. (5.11)—(5.14)give

2ei ‘. el, ~ Wpm Ic. ,(O)2~ei = / Tw

el~XL,T~K,w) = X (Pa) = — 2 2 el ‘ ~

0~L,T lkw \~J(O)2)eI = Im X (~u)). (5.15)~Wo+1Y W w 2~

This resultcoincideswith the relevantcalculationsperformedin refs. [18,24] for the caseof a uniformlybroadenedmolecularline.

In the oppositecasey ~ k(V), i.e. for the caseof Doppler broadening,one has

XL,T~’ w) ~ I dP fei(~)J (w—k~V)2—w~+iZ1’

(5.16)

- 2 ~ f dP fei(~) 2 + CC

\ JL,T~kw1Wpm

4~ (w—kV)2—w

0+i~

theseequationsarein agreementwith the resultsobtainedin section 3.1 [seeeqs. (3.13), (3.14)].It is easyto showthat radiationfriction andcollisionaldissipationresult in changedelectromagnetic

wave damping rates. In particular, if the condition ~ei ~ k(V~is satisfied, the damping rate of

Yu.L. Klimontovich eta!., Statistical theory ofplasma—molecularsystems 363

high-frequencylongitudinal wavespropagatingin plasma—molecularmatterwill be describedby therelation

2 22 1/2 2 2~1,2(~1,2 — ~O) 1/ IT\ ~1,2 ke 2 2 2

11e t’~pe

‘?kl 2 = — 2 2 2 2 2 R~) ---— —~ exp(—w1212k5e) + ~

(w12 — w~) + ~pm~O 0 ~e k L ~1,2

ei 2

+ 7 ~12~pm 12[(w~2— w~)2+ ei22}]

insteadof thedampingrate given by eqs.(3.17b). At the sametime, thespectrumof the longitudinalwavesis still describedby eqs. (3.17a).

In thecaseof transversewaveswith thespectrum(3.18a) thewavedampingrates(3.18b) transformto

2 22 2 ei 2

— ~~1,2@~1.2 — cv~) (~e~pe 7 ~pm~l22 22 2 2 3 + 2 22 ei22 -2[(w12 — ~-~) + ~pm~o] w12 (1.2 — °~o)+7 ~1,2

5.1.2. Large-scalefluctuation spectraand electromagneticradiationIf one knows the responsefunctions and the correlationpropertiesof the fluctuationsources,it is

possibleto calculatethe fluctuation microfieldsand, finally, to describethe fluctuation spectrafor allinterestingphysicalquantities. Becausethe generalrelations given in sections3.1.1 and 3.1.2 werefound without using the explicit form of the responsefunctions and the correlationfunctionsof thefluctuationsources,it is obviousthat theserelationsarealso valid for the caseoflarge-scalefluctuationsif one useseqs. (5.8) and (5.9) for ~ a) and (&i~°~&i~°~)~.In particular the generaleqs.(3.19)—(336),(3.41)—(3.44)etc.can be usedto describelarge-scalefluctuationswith an accuracyup tothe replacementof e.1(k, w) and (8J~°~~j~°) ) ~ by the relevantquantitiescalculatedwith collisionstakeninto account.Using theserelationstogetherwith eqs. (5.8), (5.9),onecan calculatethe spectralas well as spaceand time distributionsof fluctuationsfor the different systemsunderconsideration.

In particular,it is interestingto investigatetheoppositecaseof thecollisionlessapproximation,i.e.the caseof largecollision frequencies(~ ~ ü, ~ ~ kSa).This spectralregiondefinesthespaceandtimedistribution of fluctuations for intervalsr ~ 1 /Va~ R — R’~~ ~a~a (a = e, i, ei). Let us consider,forexample,thechargedensityfluctuationsandrelatedparticledensityfluctuations.As mentionedabove,we usefor this purposethegeneralrelations(3.34), obtaining

1 2 ~

E(k, w)~

+ IXe(1’, ~ + (6p~°~2)J}

(5.17)

= e(k,w)12 {I1 + Xe(k, w) + ~ ~

+ 1x1(k, w)~

2[(8p~°~2)~+ (~p~°~2):1}

where

364 Yu.L. Klimontovichet aL, Statistical theory ofplasma—molecularsystems

(0)2 if 2(6p )kw=eifnifjdPfdPWifkw(P,P)fif(P)+c.c., a-=e,i,

= — ie~inej1 dP 1 dx (k V)2fei(X, P) + ~ a- = ci, (5.18)el,

w J J (w—kV)2—w~+iy~w—icV)

e(k, w) = 1 + ~ife,i.ei xif(k, w) andXif(k, w) = ~ w) are given by eqs. (5.12).In the equilibrium casefor y ~ ks~

kT Im{[1 — xif(k, w)I~(k,w)]x~(k,w)}, (5.19)

=

with

k~(w+ivif)W(zif)Xif(”,~) ~ w+ivifW(zif) a-=e,i,

2— ~pm

—— 2 2 -ci , cr—el.~ ‘°o~7 ~

In the caseof interest,when ~ v~,w~k<<Sa/Pa eqs. (5.18), (5.19) reduceto

2 ci 2~ 2\2 ci(0)2~ei — .~eeinejyk2( v2) 2eeinejy k ç v y

P lkw — 2 2 2 ei2 2 4 , (5.20)(w —we) +7 (U (U

0

2 a 2e~naDak2

(~p(~)~kw = W+lk2DaI2 , a=e,i, (5.21)

___________ ab maTab+ MTa~k~Da Da=•~_JdPV2cba(P)=_~_, Taab= ma+M

xa(k,w)— - 2w + ik Da

Substitutingeqs. (5.20), (5.21) into the generaleq. (5.17) we areled to the following resultfor thespectralregion k~ ka:

2 k2k~DeDi / k2D T. k~De \K~P2)w= 2neee + ik2DA~2~ (k~D~+ k~De)2+ Te (k~D~+ k~D~)2)~ (5.22)

where

k~DeDi j~2= 47re~na ~2 \‘ ~2 2DA=k_~Di+k_e2De~ a TaE ‘ a’ e

0—1+2.0 a=e,i w0

Yu.L. K!imontovich eta!., Statistical theory ofplasma—molecularsystems 365

On the otherhand, in the spectralregionk~ ka~we have

(&P2)~~ ~ 2 = (~P)kw’ a = e, i. (5.23)w+ik DaI

The latter relation meansthat in the caseunder considerationthe chargedensity fluctuationsarerepresented,in fact, by thesourcefluctuations,i.e. by fluctuationsin asystemwithout electromagneticinteractions.This is quite natural, becausefor distancesIR — R’l ~ A~= lIkD the particles are notaffectedby the spatial chargeseparation.

Performingan inverseFourier transformation,we obtain for IR — R’I ~

(~P2)~R,T (~p(R,t) ?ip(R’, t’)~

2 2 2enk. k.D.+,~T.IT~kD 1= cc k~D1+k~D~e (4ITDAr)

312 exp[—(R—R’)I4DAT]. (5.24)

In thecase — R’l ~ A~eqs. (5.17), (5.20),and (5.21) give

= e~n~3/2 exp[—(R—R’)2I4Der]. (5.25)(4ITDe r)

In the equilibrium case(T~= T1), one obtainsinsteadof (5.24) [104]

e~ne 3/2 exp[—(R—R’)2/4DAT], DA= 2DeDi

2(4ITDAT) Dc + D1

Thus, weseethat in thecaseof frequentcollisions theevolutionof thechargedensityfluctuations(and,consequently,the particle density fluctuationsfor free chargedparticles)is determinedby diffusionprocessesand the electromagneticinteraction of the particles results in a renormalizationof thediffusion coefficients. Such a renormalizationmeansthat with increasing IR — R’I unipolar diffusionchangesto ambipolardiffusion. Comparingeqs. (5.24) and (5.25) with the transitionprobabilitiesofBrownian particles[121],we seethat the correlations(5.24) and (5.25) are determinedby randomlywalking chargedparticles.

Let us consider,finally, thespaceand time evolutionof themoleculardensityfluctuations,using thegeneralequation(5.10) and the solutions (5.4), (5.5) obtainedfor the fluctuation propagators.Theresult is

= hlei J (2)~J~ exp{i[k. (R - R’) - wr]) JdPJdP’ [14~eikw(1’,P’)fei(P’) + c.c.]

n~1f (2)~J~ exp{i[k. (R — R’) — T]} [i~’~’(k, w)GJk~+ c.c.], (5.26)

where

366 Yu.L. Klimontovicheta!., Statistical theory of plasma—molecularsystems

GIk~= ~ ~t’ci {i — W(z~1)~_ W(Zei),~[1 — W(Zei)] — ~z~iW(zci)}

=i1dP’ fei(~’)~

— J W~k.V’+~Vci-

The generalanalysis of theseequationsis very difficult. However, in the importantcaseof largecollision frequencies(~‘ei~ w, T ~‘ l/i.~)it is possibleto investigatethe fluctuationspectrumin detail. Inthecaseunderconsideration

i~~(k,w) = w2:i~i

2Dci ir~(k,w) = — ~:~i (i —

(ci) k Sei (ci) (ci) (ci)W

13 (k, w) = —~—— ~ ~ (k, w) = ~23 (1~,w) = ~12 (k, w)ei

(ci) w + 3ik2D~~ (ci) w — ~~k2Dei

~22 (k, o) = ~ ‘ ~31 (k, w) = ~~‘ei

w+~ik2D.(ci) 5 (ci) (ci) 3 ciW32 (k, w) = 3 ~12 (k, w) , ~T33 (k, w) = il/el

Gtk~= (1/izi~~,0,1/ji/ei)

det i~(k, w) (w + jk2Dci)((U — kU + ik2F)(w + kU + ik2F),

where we usedthe following notation:

Dci = S~i/i/ei , = ~ ‘ F = Tci/Ml/ci -

Taking into accountthe explicit form of ir~ (k, w), eq. (5.26) for the fluctuationspectrumreduces

to the following relatively simple result:

= 2flci Re(~~’(k,~GJk~) 2flci Im ( + ik2D~1)(w— kU+ik

2F)(w + kU + ik2F)’

where

~_h14 ~ (C~ ~ A 1— 2 L3 ~~ciM ~ “ fl~jCpM]

= Tci flcj/t’ci, A = fleMC’p1)ci~ Ci,. = 5/2M, C~,= 3/2M.

The latter equation describes the well-known hydrodynamic fluctuation spectrum [18], if thequantities~ ~ and A can be consideredasthe coefficientsof diffusion, dynamicviscosity and heatconductivity.

Yu.L. K!imontovichet al., Statistical theoryof plasma—molecularsystems 367

The fact that thesecoefficientsare relatedto each otherby only one dissipative parameter~ej isdeterminedby the chosenmodel of the collision integral.This is the main defectof the BGK model.The problemmay be removedusing appropriatemodels(see,for instance,ref. [107]).

In the caseof the adiabatic(o kU) and isobaric (w k2F) spectralregionsthe latterequationyields

~kw25~~ei 2+~k4A2 0—k2F<<kU,

4Fk2s~~= (2 — k2U2)2+ 4w2k412~ w kU,

which is in agreementwith well-known results [18,25].The spaceand time evolutionof the fluctuationsis describedby

= nci((4 D

2 exp[—(R ~R’)2I4DeiT]

+ C~,IR —R’HUT D{[(~R — R’~— UT)2]14FT)). (5.27)

It follows from eq. (5.27) that besidesthe diffusion processes, correlations due to sound waveexcitation exist. Obviously, this mechanismcan give correlationsover greater distancesthan thediffusion processes.

Notice that eq. (5.27), as well as the general relation (5.25), is in agreementwith the well-knownOnsagerhypothesis [122]. According to this hypothesis,hydrodynamic fluctuation correlationsaredescribedas the evolution of initial perturbationson the basis of the hydrodynamicequations.

Notice that radiationfriction andcollisional dissipationdo not changethe radiationintensity andtheenergy density of the fluctuation microfields in the transparencydomain. Thesequantitiesare stilldescribedby Kirchhoff’s law (3.43) and by the Rayleigh—Jeansformula (3.44),

1~= TT((w/c)VET(w), w), (5.28)

4ir c2 3/2( \ _______

U~= (~~ ~°~‘ TT((w/c)VET(w),w). (5.29)ire

This is quitenatural,becausein the transparencydomainthecontribution to the radiationintensityandenergydensityis givenby thewhole volumeof thesystem,leadingto energylevelsthat correspondto black-bodyradiation. However, outsidethe transparencydomain it is necessaryto use generalrelationsof the type (3.41),to calculatethe energeticcharacteristicsof thespontaneousfields.

The formal coincidenceof eqs.(3.43), (3.44)with eqs.(5.28), (5.29)doesnot imply identityof theircontents.In the lattercase,in addition to Cherenkovradiationof the fluctuationfields, they takeintoaccountthe radiationgeneratedparticle collisions and the radiationfriction of the bound electrons.

368 Yu.L. Klimontovichet al., Statistical theoryof plasma—molecularsystems

5.1.3. Bremsstrahlungin plasma—molecularsystemsIn section 5.1.2 we have shown the possibility to calculate the spontaneous emission spectra in a

plasma—molecularsystem using the theory of large-scalefluctuations. Separatingthe part of thespectrumrelated to particle collisions, it is possibleto estimatethe energeticcharacteristicsof thebremsstrahlung. This method of calculation may be called kinetic.

There is also anotherapproachto investigatingbremsstrahlungprocesses,which maybe called thefluctuationalapproach.In the frameworkof this approachthe bremsstrahlungfields areinterpretedasthe result of fluctuation microfield scatteringby the fluctuations of currents and chargedensities[46,123, 124]. The efficiency of the bremsstrahlung theory was illustrated in refs. [123,124], includingbremsstrahlung studies in a bounded plasma [125].

The possibility of using the advantagesof incoherentscatteringtheory is an essentialfeatureof thefluctuational description of bremsstrahlung. In particular,it is possibleto describe bremsstrahlung fieldson the basis of a standardsolution of the problemfor electromagneticfield excitation by a givendistribution of external currents. We use this possibility in our further calculations. A consistentdescriptionof the correlationsbetweenparticlesof all kinds in the investigationof collective brems-strahlung processes is an additional advantage of the fluctuational approach.

Accordingto the fluctuationalmodel of bremsstrahlungwe now calculatethe bremsstrahlungfieldson the basisof eqs. (4.3), (4.4) and (4.24). We assumethat the contributionsof slow kinetic processesto the evolutionof smoothedmicroscopicfluctuationscan be neglected.This makesit possibleto omitthe averagedquantitieson the left-handsides of eqs. (4.3) and (4.4). The formal solutionsof theseequationscan now be representedin the form of eqs. (2.25) and (2.26) [126],

~a(X, t) = ~N~°~(X,t) — f dt’ f dX’ Wa(X,X’, T)

x ~Fa(R’~V’, t’). [flafa(X’, t’) + M[5(X’, t’)], (5.30)

= ~ t) — f dt’ Jd~’ Wab(2~, ~‘, T)

x ~ ~1(R’ + a1r’, V’ + a1v’, t’). (~+ a1 kabfab(~”t’) + aNab(~”t’)].

1 (5.31)

However, the latter relationsinclude, in addition to the linear (in the fluctuations) terms, nonlineartermswhich describethe fluctuation interactions.

In first orderof perturbationtheory eqs. (5.30) and(5.31) coincidewith eqs. (4.5) and(4.6) andthesetof eqs. (2.23), (5.30) describesthe linear fluctuationsin plasma—molecularmatter(section5.1.2). Insecondorder in the fluctuations this set of equationsdeterminesthe bremsstrahlungfields underconsiderationand hasthe form

curl ~E~2~(R,t) = — -~ ~B~2~(R,t)

curl ~B~2~(R,t) = -~- ~E~2~(R,t) + ~ ~J(2)Ind(R t) + ~ ~JNL(R t), (5.32)C dt C ua.b,ab C u=a,b,ah


where the inducedcurrents ~J~(R, t) are related to the electromagneticfields 6E~2~(R,t) and~B‘2~(R,t) by linear equationswith the susceptibilitytensors(4.16), (4.25),and thesourcecurrentsforbremsstrahlungfields are of the form

~JNL(R t) = ~ea J dt’ J dX’ JdPVWa(X,X’, T)

x SF~(R’,V’, t’). ~ &N~(X’,t’), a = e, i, (5.33)

~JNbL(R t)= —~ eiJ dt’JdYl’JdPdx(V + a1v)

X ~ ~‘, r) ~i~’~(R’+ a1r’, V’ + a1.v’, t’)~(~7 + a1 i--) ~ t’).

(5.34)

Here, the quantities&E~(R,t), ~BW(R,t), ~N~(X, t) and &N~(~’,t) are the linear fluctuationscalculatedin section5.1.2.

According to eq. (5.32), the bremsstrahlungfields bE~2~(R,t) and bB 2~(R,t) are excited by the

currents6JNL (R, t), which are the products of the nonlinear interactionsof the fluctuationsof the

electromagneticfields and the smoothedphasedensities.The solution of eq. (5.32) for the Fouriercomponentsis in the caseof an infinite medium asfollows [126]:

— ~ A~(k,w) 6J~, (5.35)

where

~JNL~NL ~ NL

U ku ~ ckw+0 cikw ‘

—. 2 ~dk’ f dw’1dP

“ekw — lee J (2ir)3 j 21T J k . V— ~ —

X [(1— k’ V/(U’)t511 + k~V/w’]~ ~~e,k_k,w_w(1~) ~

~ = —~ e1e1 J (2ir)3 J~ JdPdx JdP’ dx’~CIkW(x,P, x’, P’)(V + a

1v)

x exp(ia1.k’ . r’)[(l — k’ . (V’ + a,,v~)/wl)~1+ k(V. + a1.v)Iw’]

x ~ + a~~-~)8NCI,k_k,W_~(x,F’) 8E~W

e,e1 I dk’ Ido. I= ~ oi (2ir)

3 J ~ J dPdxn,,~_w(_iYJn(aik~r)Jm(aik•v/wo)

370 Yu.L. Klimontovichetal., Statistical theoryof plasma—molecularsystems

x exp(ia1k’. r)[(w — k . V + a1,k’ v)~1+ k~(V+ a1v)]

>< V+vmw0/(k.v)+rnw0/(kr)w _k.V_(n+m)wo+iyd

x (~+ a~~-) ~Nej k-k’ ~~(X , F’) ~ (5.36)

Thesegeneralrelationscan be simplified if onerestrictsoneselfto consideringbremsstrahlungin thehigh-frequencyregion and retainsthe termsof first order in the atomic parameterin the equationforthe molecularcurrent. In such a case

~jNL(c) — iee dk’ f dw’ 1i~k1 (k1 kikw — (w + il/c)me j (21T)

3 J 2ir ~ ~ik + \.(~) —

+ ~7 ~jk — ~ ~j] ~ ~ (5.37)

~j~L(el) = 2 eciw J dk’ J ~ [eei~ei,~k4w + ik ~ei~k~w] ~ (5.38)m(w (Uo+17 w) (2ir) iT

Here,

= f dPdx ~ei~k,~w(X, F),

~‘~ei3k4w = ecif dFdx r ~ei3k4w(X, P),

z~k=k—k’, L~w=w—w’, ~J~.j= ~if=c,i,ei

The quantities~ and ~ are the fluctuations of the molecularnumber densityandmolecularpolarization,respectively.The aboverelationscanfurtherbe simplifiedin thepotential fieldapproximation[6E~j = — ik ~4~j, ~ ~j is the fluctuationof the electric potential].The result is

~jNL(c) — e~ f dk’ f ~1_~:.~k’ + k’ k . ~ + i~kk~k’ t~twkw — (w + ivc)mc J (2ir)3 j 2ir \, (i~k)2 ~ (z~k)2‘°

X ~ ~ (5.39)

Finally, if one takesinto account that the basic contribution to bremsstrahlungis given by thewavenumberrange k -~ w/C and that to the correlation functions by the range k’ ~ W/Se~ i.e.,k/k’ -~ Sc/C~ 1, thenit is possibleto expandthe integrandsin k/k’ andretainthe first two terms of theexpansion.In zerothorder in k/k’ one obtains


bJNL(e) = ec I dq I dx (1)e (1)

w (w + ive)mci (2ir)3 J ~q6P—q,1~w64~qx

&JNL(ei) = iecjw I dq I dx (1)Ic) m(2 — w~+ 17e1) J (2ir)3 J ~ q[ecj ~ci,—q,1~w+ ~P~—q,is1 ~thWUI ‘ qx

= w — x, where~ = ik ~ arethefluctuationsof theboundchargedensityand~ arethefluctuationsof the chargedensityin the electronsubsystem.

It follows from the aboverelations that the calculation of the correlationfunctions of the sourcecurrentsfor bremsstrahlung,(~JNL

6JNL )~ (a- = e,ei), consistsof the evaluationof quantitiesof thetype (ABC* D*), whereA, B, C, D areFouriercomponentsof thefluctuationsofthe relevantphysicalquantities (particle density, chargedensity, potential). Neglecting higher-ordercorrelations,thesequantitiescan be calculatedon the basisof the following representation[46]:

(ABC*D*) = (AB)(C*D*) + (AC*)(BD*) + (AD*)(BC*).

The resultsof calculationswith eqs. (5.38) and (5.39) are

(~JNL~JNL)e161T2(_..~_~ I dq f dx 1m~w/i (2ir)

3 J 21T

x { ~(~~(O)2)~(~p(O)2)~~~~[X~(q,i~k,i~w)+ X~(Ak,q, x)][X1(q, i~k,L~w)+ X~’(Ak,q, x)]

+ ~ + (~p~°~2)~)x

1(q,i~k,~w)Y~(q,l~k,~w)} , (5.40)I qx

16ir22 2

(~JNL&JNL)ei— ecjw

kw 2 2 22 ci22m[(w —we) +7 w]

~f dq I dx / 2 2 ci ‘~‘ (6p~0~2)°~q.q.J (2ir)3 J 2 I\eei(~n)~k4u ~ ‘s(q,x)~2 q2ifc,ei

ci ‘ (0)2\ei1 (~P )1~k4~c~P1

+ 2 e(q,x)e(~k,~cü)I2[~(q, i~k,Aw) + X~(~k,q, x)]

x [.~(q, Ak, Aw) + X~(Ak,q, x)]

( ~ (0)2’. ci ‘ (0)2’. cP 1~k4~(c6P1qx+(bp~°~2)~ix)q,

+e(q, x)e(Ak, Aw)12 —~ ~(q, Ak, A&)) . (5.41)

Here,


X~(q,Ak, Aw)= [1 + x~(Ak,Aw) + Xci(~’,Aw)]x~(q,Ak, Aw),

1 / kAkAw k-’qAw\x,(q,Ak, Aw) = —~ (\q, + q 2 + Akq iAk w

Y~(q,Ak, Aw) = [1 + x~(Ak,Aw) + Xci(~’,Aa)]

X [1 + x~(Ak,Aw) + Xei(~’Aw) + xc(q, x)]*x,(q, Ak, Aw)

— xe(q, x)[1 + ~1(Ak,Aw) + Xci(~’Aw) + xe(q,x)]*x1(Ak, q, x), (5.42)

X1(q,Ak, Ao) = [1 + x~(Ak,Aw) + Xc(~’Aw)],

q

Y~(q,Ak, Aw) = (~ [1+ x~(Ak,Aw) + Xe(~,Aw)] Ak~q ~xci(q,x))

X [1 + x~(Ak,Aw) + Xe(~’Aw) + xei(q,x)], Ak= k— q.

These relations take into accountall termsof the expansionin the quantity klq. Keeping only the firsttwo termsof the expansion(which correspondto dipole andquadrupoleradiation in a collision ofparticles), we have

L 1NL ~(&1~~‘j )icw = (~JNL aJNL)if(I) + ~JNL ~JNL\if(Il) (5.43)j Ikw

228ire f dq fdxq1q3 1(~JNL ~JNL)e(I) = m~(w2+ v~) (2~)~i 2ir q

4 e(q,x)e(-q,Aw)~2

x { ~ Ao) + xei(q, Aw) — ~ x) — xei(q,x)~2+ ~ + ~ + x~(—q,Aw) — xci(q,Aw) + xe(q, x)~2},

qx(5.44)

228iTec I dq Idx 1(

6JNL~JNL\e(ll) =j lkw 2 2 2me(w + ~e) ~(2ir)

3 J 2ir e(q,x)e(—q,Aw)~2

1 (0)2’ e (0)2~)e (5.45)q (k

1_4~4 qj)(k1—4~4q)(~p i_Q,~WK.~P /qx’x

(~JNL ~JNL)ei(I) = 8ir2e~

1w2 dq I dx q

1q1m[(w -we) +y w]I(2ir)3J ~ q42 2 22 ci2 2

/ 2 2 ci ~X ~2e~(~n )—q,t~w 2 qx

k(q, x)I if=c,i,ei~ (ap(o)

2\eiIqx i

+ e(q x)e(—q,Aw)~2 x~(q,Aw) + xe(q,Aw) — x1(q, x) — xe(q, x)~

2

(0)2\e +~P(O)2Y~~q4w[(~P Iqx

+Ie(q, x)e(—q,Aw)f2 1 + x~q,Aw) + xe(q, Aw) + xci(q, x)(2), (5.46)

Yu.L. Klimontovichetal., Statisticaltheory ofplasma—molecularsystems 373

/&JNL ~JNL ei(l1) — 8ir2e~~2 f dq ~~ Ii i )~w — m2[(w2 — 2)2 + ~eI2~2] J (2ir)~J 2iT q4

/~ (0)2\ci ,, (0)2\ci , \ I\ P /—q4w\ P Iqx ~‘ k~q ~ k~qx e(q,x)e(—q,Aw)~2‘\ ~ q2 ~ ~— —~~~—q

1

In obtaining the correlation functions (5.45) and (5.47) we neglectthe terms taking into accountcross-overcollisions betweenparticles of different kinds and keep the terms related to electron—electronandatom—atomcollisions. This is so becausethecontributionsof suchtermsto thecorrelationfunctions (&JNL &JNL)cr (a- = e,ei) arenegligibly small, if the susceptibilities,y1(k, w) andXei(k, (U), or~1(k,(U) andXc(k,u) aresmall for thosefrequenciesx and u — x for which theintegrandsin eqs.(5.45)and (5.47) give themain contributions.

It shouldbe rememberedthat eqs. (5.44)—(5.47)are written in thepotentialapproximationof thefluctuation fields. In the case of a pure plasma this approximation is valid, becausethe bremsstrahlungfields are generatedby the particle interactionfor distancesof the order of r -~ e~!T during a timeinterval T — rise, andthereforethecondition for the field to be quasi-stationary,kciw —~ cTir c/se ~is satisfied to a high accuracy.However, this picture may changein the caseof bremsstrahlunggenerationdue to atom collisions. In this casethe correlationfunctions (~JNL ~JNL ) describedbyeqs. (5.43), (5.46) and (5.47) should be supplementedwith the correlation functions taking intoaccounttransversefluctuation fields,

JNL NL ci — 16ir2e~~w2 f dq f dx 1

i 6J~)~w — m2[(w2 — (U~)2+ 7cI2~2] J (2ir)3 J 2ir x2je~(q,x) — q2c2Iw2~2

x ~ — q~q1iq

2) ~if =c,i,ci

(~p(O)2)ci ~

+ “~ ‘ (ôq — q1q1iq

2) ~ (3J~”2)~~), (5.48)s(—q,Aw) ifc,i,ei

where CT (q, x) and (~J~2) are the transversedielectricpermittivity andthe correlationfunction ofthe transversecurrentcomponents[see,eqs. (5.11), (5.13)].

Usingtheaboveeqs.(5.44) and(5.45) it is easyto reproducethe resultsfor thebremsstrahlungin aninfinite plasma.To do this, onemust take t2ei = 0 [Xci(” w) = 0, (&p~°~2)~= 0]. It should be pointedout that in this casethe correlationfunction(5.44) leadsto expressionsfor thebremsstrahlungintensitywhich coincidewith thewell-known resultobtainedin ref. [123].At thesametime,calculationson thebasis of a correlationfunction of the type (5.45) producean intensity of quadrupolebremsstrahlungwhich differs from the resultobtainedin ref. [124],but coincideswith thatgiven in refs. [127,128] onthe basis of other methods.It can be establishedthat this difference is due to the fact that thecalculationsof thebremsstrahlungsourcesin ref. [124]do not takeinto accounttwo termsproportionalto klq. (The differencebetweenthe resultsobtainedin refs. [124]and[127]wasdiscussedin ref. [127],but its naturewas not established.)

374 Yu. L. Klimontovichet a!., Statistical theory ofplasma—molecularsystems

If the distributionsof ions and atoms arein equilibrium, the integrationover x in eqs. (5.44) and(5.46) maybe performedusingthe following approximaterelations:

2 ci (0)2 i 2(~fl )kw = 2lTflciô(W), (6p )kw = 2irn~e~~(w),

~2,,. 2(0)2 ci ~ 1 pm

(~.. )kW—A2[ô(ww0)+~(w+w0)1.-t (00

Simultaneouslyfor a rarefied molecularsubsystemone must take s(k, w~)—~1.Following the aboveprocedureof separatingthe collective and individual contributionsto brems-

strahlung,the region of integrationover q mustbe divided into two subregions,q < kmin andq > kmin,which determinethecontributionsof different spatialregionsof particleinteraction.

The greaterof the two quantitieske and (1)oiS~can be takenas kmin in our further calculations.Let us consider the collisional contribution related to the interactionof electronswith ions and

moleculesat small distances.In this cases(—q, w ±w~)—~ 1, x~(—q,w ± (Un) ‘~-‘0. Becausethe integralsare divergent,they must be regularized,restricting the upper limit of integrationto kmax (seesection3.1.4). The result is

(aJNL ~JNL)e(LP.) = ~ ~ {n~en~ kmaxSe/W)— ~C]~

2 me V I S~k~ax I (w ++ tleieei r ~ L 2 expt,~— 2 2L.m~+1 (w + f3w~)

2kmaxSc

2V2kmaxSe I (w+f3w~)I 2 2 /

+ln I r’ r =e im c - ~5.49w+f3w

0~i 2 j’ c

The first term correspondsto electron—ion collisions and the second one to electron—moleculecollisions.

As regardsthecorrelationfunction (~JNL 6JNL);I(i.P.) it is describedby the relation

(~JNL ~JNL)e~i.P.) = ~ 3~ r~c 2 ei2 2

(w to~) +7 (0 Sc

x {in(~ kmaxS ~iw) + me ~ [ S~k~ 2 exp(_ (w +~)2)2m ~ (+f~o)~)

2kmaxSc

V~ kmaxSc 1 (w + ~3)2 2 2+ in j 2 , rei = eeilmc - (5.50)

w+I3wo CUo

Here, the first term is relatedto the transformationof a longitudinal field into bremsstrahlungdue tothe interactionwith the fluctuationsof the moleculardensityand the other termsdescribethe fieldtransformationby fluctuationsof a boundcharge.

Yu.L. K!imontovichet a!., Statistical theoryof plasma—molecularsystems 375

Assuming that the correlationfunctions (5.49) and (5.50) are additional terms to the correlationfunctions (3.13) and (3.14) calculatedin the collisionlessapproximationand using the fluctuation—dissipation theorem,one can find the relevant correctionsto the imaginaryparts of the dieiectricsusceptibilitytensors (3.12). This makes it possible to estimatethe dissipative constantsrelatedtocollisionalprocesses.The resultsof such calculationsare

2 -2 ci(U . 1(0 (07c . pe ci pm e

&Xij(k,~))TO”ei+ “c,ci)’ ~~11(k,~v) 2 22 2 ci2(U (o w~)+(U7c

wherev~and1’~cjarethe frequenciesof electroncollisionswith ions andboundpairs,respectively,y~

is the broadeningof the molecularline due to collisions with the electrons,

= ~ e~,e~n1ln(V~kmaxsc/Iu~), (5.51)

2 [~ e~e~~n~1m~ (w +“e~ci3 ~me T~

2 ~~±i (O~

I \/~~ k~axS~ ( ( + I3wo)2\ 1X [111 ~ + ~ + (w + ~3~)2 exp~— 2k~ax5~)]~ (5.52)

ci ~ [~ e~e~n~f= 3 1’~rn T~2~,ln(V~kmaxseiw)

1 mc ‘ç’ ( + 130)2 I k~maxS~ (_ (o + /3(U)2 \ V’~kmaxSc+ — Li 2 I 2 exp~ 2 2

1+in - (5.53)~ m ~ (Uo L (w + l3w~)

2kmax5e °~+ P~O

Comparingeqs. (5.51) and (5.52) with (3.63a) and(3.63b) one seesthat the resultsof the collisionfrequencycalculationsobtainedin the framework of the bremsstrahlungtheory coincide with thosegiven by thekinetic approach.The only differenceis in the frequencydependentfactorsappearingin

‘1c,ei’ but in the casew 4 ui~this differenceis negligible.Let us also considerthe contributionsof collectiveprocessesto bremsstrahlung.In theequilibrium

case such effects could be neglected. However, if thesystemunderconsiderationis not in equilibrium,thecontributionof the rangeq < kmjn is comparableto the quantities(5.49) and (5.50). In fact, in therangewhere collective fluctuationsexist, thecorrelationfunctionsof electronbremsstrahlungsourcescan be written as

(~JNL ~JNL);(I)(coll) = ~ ~ 1 Jdq~4’(-~_T~(—q,w)Z~n~5(Res(—q, to))

e.n.m q2s2 1(5.54)

ec m ~ ~


24(6JNL ~jNL\c(JI)(colI) = rcc I dx dq (k, —4 ~__qq1)(k1 —4 ~ q.)

2 22J I/ /kw

8lraec XI.1Oi.i

x Te(—q, o—x)T~(q,x)6(Re e(—q, —x))~(Re e(q,x)), (5.55)

where Tc (k, w) is the partial effective temperatureof collective fluctuationsinducedby the electronsources,

(0)2 cTe(k,w) = 2ira ç~p )kWik Im e(k, w).

The molecularcorrelationfunctionsare

24 2(~JNL~JNL\ci(l)(colI) = ~ ‘~ei Jdq ~(-~- T~(—q,o)~(Ree(—q, w))

22 ci2 2j /kw ir ((U2(U0) +7 (U q (U

22q Sc ~ Tj—q, + ~(U0)8(Rer(—q, (U +

m w

(5.56)

24 2(~,JNL~JNL \ ci(II)(coll) = rcic 0)

i I Ikw 8ir3e~

1( — ~)+ J —v— J dq(k~—2 ~ qj)(kj —2 ~ q.)2 22 ci2 2

x T~(—q,Aw)T~~(q,x)6(Re e(—q,Ao))~(Rer(q, x)). (5.57)

Here,

Tci(k, oi)= 2ir (6p~°~2)~,,ik2Im e(k,o),

and the quantity TL(k, (U) is given by eqs. (3.28). Let us recall that eq. (5.55) does not take intoaccountatom—atomcollisions.

Finally, the additional term to the molecularcorrelationfunctionsrelatedto transversefluctuationshasthe form [125]

242(~JNL ~jNL\ci(T) = rcic ~ ~2ci Jdq (~— q

1q11q2)2 22 ci2 2

/ /kw ir[(w — (Un) + y (U ]22 2x {o~’T~(—q,w)~(Re[s~(—q,a) — q c 1w ])}

+ ______ dx q T (q, x)T~~(—q,Aw)1ci~i ‘ 2

4irn. xAoi cix ~(Ree(—q,x))ô(Re[e~(—q,Aw)— q2c2/&2]), (5.58)

whereTT(k, w) is given by eqs. (3.28).

Yu.L. Klimontovich et al., Statistical theoryofplasma—molecularsystems 377

Sincein the caseof a nonequilibriumsystemtheeffectivetemperaturesmaybemuchhigherthanthetemperatureofthesystem,it is obviousthat in sucha casethecontributionof thecollectivefluctuationsto bremsstrahlungcan be anomalouslylarge.Let us illustrate this by the exampleof the correlationfunction for the electroncurrent in the caseof a plasma—molecularsystemwith a two-temperatureelectrondistribution,

fe(1’) = (2irTcme)~312exp(—F2I2mcTc)/3 + (2iTmcTE)312exp(—P2/2meTe)(1 — /3),

1—/34/3, TE~”Te~

In this case

(~JNL~J~4L):(I)(coll) = 9~e(~ 1) (~)3(~) j=1,2 F(wi)e~ne(Z~n16(o.— w~)

+ 3 (q)2 ~ n~ 6(u + f3w~— ~ (5.59)

wherethe quantities(U~andF(o) aregiven by eqs. (3.17).The quadrupolepart of the electroncorrelationfunction is describedby

(~JNL~JNL)e(lI)(coII) = (6~~— k~k1Ik

2)((~J~)2)~+ (k1k1/k

2)((~J~)2)~ (5.60)

242 2 3 2((~JNL)2)e = 256 (~) ~ ~ ~(u.— 2o.~),

= ~J~L)2): , (q0Ik~)

2=[31n(TE/Tc)—21n(1_/3)]1 . (5.61)

In the equilibrium caseq0 is equalto kc and TE = Te, and therefore,the collective contribution is

relatedto the collisional one [seeeqs. (5.49), (5.50)] as in A, — ln(kmaxikD). In the nonequilibriumcase,however, TE ~‘ Te This meansthat the contributionof eqs. (5.59)—(5.61) should be takenintoaccounton an equalfooting with (5.49).

As regardsthemolecularterm, it follows from eq. (5.46) that

r2c48ira2 ~2e2nn / ~

/ ~

7NL ~ 1NL \ ci(I)(coII) — ci pe e e ci ( q0 ‘~\ j UJ / kw — 2 2 2 ei2 2 1 1

95c[(~ü 0)~) +7 to] “~c” T~

F(o1)[ô(w — u~)+ ~ (~) ~ ~ 6(a + /3o~— wj)]. (5.62)

The contribution of the transversefields to the correlation function of molecularbremsstrahlungsourcesis as follows:


2 4NL NL ei(T) 4 reiC ~‘ei(~J, &Jj )icw = 2 2 2 ci2 2 ~Re ~T(~)

3(0) ~ +7 (U

_____ 1+ ~ 2 2 F(w1)—~Tei((~w—U)i~iC)\/ReET((U—(Ui)~coi)[ResT(w—(Ui)]

3~2)6ii.11.2 0)0)pm

(5.63)

Using the fluctuation—dissipation theorem in the equilibrium caseone can calculatethe correctiontothe imaginarypart of the susceptibilitytensorand find the collectivecontribution to the dissipativeconstant.The correlationfunction (5.63) leads to the following additive correction for the molecularpolarizationdissipativeconstant:

~ei(coll) = ~ e~w~(yRe ET() + ~ 2 F(o~)T~ [VRe ET(w — w~)]3). (5.64)3 mc J1.2 0) 0)pm JflC

One can see that the first term coincideswith the transversepart of the radiation friction coefficient(3.57).

Thus, one is led to the conclusionthat the generationof electromagneticradiation due to thescatteringof transversewavesby fluctuationsof themolecularnumberdensityis an inverseprocesswithrespectto the dampingof the electromagneticfield due to radiationfriction.

Using the correlationfunctionsof the bremsstrahlungsourcesit is easyto calculate the intensityofthe bremsstrahlung.It can be doneon the basis of eqs. (3.45).

5.2. Half-space

5.2.1. Fluctuationpropagators,dielectric responsefluctuationsand the correlation functionsof sourcesIf a specularly reflecting boundary is present in the system, the fluctuation propagatorsshould be

calculatedon the basis of eqs. (4.12), (4.15), (4.18) and (4.19) with the boundaryconditions

Wif(X, X’, r)~~0= W~(X,X’, T)~zo , w~(X,~‘, T)~~1= — w~(X~,~‘, T)~0

(5.65)p(d)(x ~‘, T)~~1= p(d)(xt ~‘, T)~0.

As regardsthe solution of eqs. (4.12) and (4.18), it can be found on the basisof comparativelysimplecalculations,if one transformsthese equationsto integral equationswith a difference kernel. Theprocedurefor such calculationsin the caseof eq. (4.15) is given in ref. [129].

Finally, the fluctuationpropagatorsfor a semi-infinite system,asbefore,can be representedin termsof linear combinationsof the relevantpropagatorsfor an infinite medium,

w~(X,X’, T) f dk J ~ exp{i[k1 . (R1 — R~)— wr]}

(2ir) 2ir

x {14’ifkw(1’~F’) exp[ik~(Z— Z’)] + Wifkw(P, P’t) exp[ik~(Z+ Z’)]} , (5.66)


whereWQ,kO(P,P’) is given by eqs. (5.2), (5.3) or (5.4). The latter relation describesthe principle ofprobability summationfor different alternativetrajectories.

The solution of eqs. (4.19) is morecomplicateddue to the z-dependenceof the coefficient 77k’. Inorder to simplify the problemas much as possible,in our furthercalculationswe restrict ourselvestodescribingthe particularcaseswhenthe influence of the boundaryon the kinetic coefficientscan beneglected.This may be donefor electromagneticprocesseswhosespatialscalesin Z aremuchgreaterthan the characteristicspatial scaleover which the boundaryinfluences the coefficient 7. [seeeqs.(3.109)]. In thesecasesthe probability summationprinciple is still valid, leading to

w~(X,~‘, T) J dk~J 4~exp{i[k1 . (R1 — R~)— (UT])(2ir) 2ir

x {w~j(F,x’, F’) exp[ik~(Z— Z’)] + w~j(F,x~f,F’~)exp[ik~(Z+ Z’)]) -

(5.67)

If one knows thesolutions (5.66) and (5.67), it is easy to calculate the dielectricresponsefunctionsand thecorrelationfunctionof the fluctuationsources,using thegeneralrelations(4.16),(4.17),(4.25)and (4.26). In particular, one can showthat the dielectric susceptibilitytensorsare still describedbyeqs. (3.73) and (3.74) with G~(k,(U) andF~(k,(ii) replacedby

G~(k, ) = — 4~nif JdFJdF’ V((co — k1 .V1) + Vk1. ‘))1~rk(p, F’), a- = e, i,

= 4ir~~iflci JdFf dF’dx’ w~(F,x’,F’) 9fci(X’,F) a-ei,

4irie2n I I / Of (F’) Of (F’)\ -

F~(k,w) = — J dPJ dF’ (\V ,~, — V~ )WifkW(F~P’), a- = e, i,(U

=0, oei,

where WifkW(P, F’) and w~j(F,x’, F’) aregiven by eqs. (5.3) and (5.7).Thegeneralrelationsfor thecorrelationfunctionsof the fluctuationsourcesarenot changedeither.

Theyare describedas beforeby eqs. (3.75) and (3.83); however, oneshould now usethequantities(~j~O)~j~O))~ given by eq. (5.9).

Becausethegeneralstructureof the responsetensorshasthesameform asin thecaseof collisionlessplasma—molecularmatter,it is obvious that thegeneralsolution of theproblemof field excitation byany given currentsis not changedsignificantly. The only differencebetweenthe relevantsolutionsisthat in the general equations(3.78) and (3.79), instead of the permittivity tensor e,

1(k, (U) for acollisionlesssystem,dielectrictensorswith x7,(k, w) given by eq. (5.8),which takeinto accountparticlecollisions, shouldbeused.In turn,this showsthat the generalformulaefor the microscopicfluctuationspectrain a semi-infinite plasma—molecularsystem,eqs. (3.84)—(3.95),can be used to describethespectraof large-scalefluctuations if we also replacethe correlation functions of the collisionlessfluctuationsourcesby the relevantcorrelationfunctions(5.9) or


= e~niff dFf dF’ WifkW(P,P’)fif(P’) + c.c., a- = e, i,

- 2 1 (k~T)2fei(X,P)

=_lecincijdFdx(kV)22+ici(kV)~ a-e1,

(5.68)

in addition to the replacementof the dielectric susceptibilitytensorsmentionedabove.Obviously,the formal coincidenceof the generalrelationsfor collisionlesssystemsandsystemswith

collisional processestakeninto account doesnot mean that their physical contentsare identical anddoes not leadto the same final results. In particular, it is quite natural that particle collisions andradiation friction processeslead to additional dissipationof surfaceexcitationsjust as in the caseofvolume excitations. In fact, for surfaceexcitationswith a spectrumdescribedby eq. (3.82), the wavedamping rate .~ in the caseof a collisional systemshould be replacedby

F =— 2 (22)2k~ 2(1 + 8)0.1k (w~— ~2)2 + (U~0)~mI(1+ ~‘)

+ ci2 + (8)1/2 ~kiSc ) -

~k1 0~1~c(0

4j— ‘~~) ~ ~kj e(k, wk).\,~

5.2.2. Electromagneticfluctuationsand spontaneous emissionfrom a plasma—molecularhalf-spaceAs mentionedabove,particlecollisions andradiationfriction processesdo not influencethe general

solution of the field excitation problem and the generalstructureof the sourcecorrelations,andtherefore, the general form of eqs. (3.84)—(3.93), which describe the electromagneticfluctuationspectrain a semi-infinite plasma—molecularsystem,is not changed.It is easyto verify this statementusing the solution of the problem of electromagneticfield excitation, eqs. (3.78), (3.79) [with anaccuracyup to the replacementof ~q(k,w) by that calculatedon the basis of eq. (5.8)] andtaking thestatisticalaverageof the relevantbilinear combinationon the basis of eqs. (3.83) and (5.9).

Let us consider, as an example, large-scalefluctuations of the electron charge density in thequasi-staticapproximation.The spectrumof thesefluctuationsis describedby

(~Pk~Pk;Yk1W= (~P

2)~ 21T[~(k~— k~)+ o(k5 + k~)j+ (aPk ~Pk~Yk’~ + PkPk;)~~ , (5.69)

where K~P2Ykw is given by eq. (5.17) and

cVS c , c*

K~Pk~8Pk;)k1U.J (k~,k~)+J(k~,k~),

jc k k’ — 4~k1 4(k’, w) 1— L(k1, (U) k

2Ie(k, (U~2 e*(kl, (U)

x {[1 + x~(k,w) + Xci(k, (U)](~P~°~2)~+ x~(k,~ + (~p(O)2)~,]}, (5.70)

—2 2 * , (0)2 if

/ ~cS 4e k1 Xe(k,(U)X~(k,(U) ~ dk~/‘.~3p )kw

\ Pi~ Pk~/k1w — ITIL(k±,~12 s(k, (U)E*(k~,(U) ~ k4 Ie(1~,w)12 -

Yu.L. Klimontovich etal., Statistical theory ofplasma—molecularsystems 381

Here, the quantities(~p(0)2) ~ andx~(k, w) aredescribedby eqs. (5.68) and (5.12), respectively.

In the equilibrium caseeqs. (5.69) and (5.70) yield

= [ö(k~ — k~)+ o(k~+ k~)]Im([1 + x1(k,(U) + Xei(k, w)] x~(k,w)/r(k, ))(&Pk

0Pk~lk1~

+ 2Tk~Im(~e(k,W)Xe(k’,w)Ir(k, w)E(k’, )L(k1,w)), (5.71)irw

where thex~(k,w) aregiven by eq. (5.19).In the spectralrange W4i’e~ o0 kiseii)e; kzSe<<Z/e (T~l/Ve, Ri~R.~i>>Se/Pe,Z>>Se/Pe) the

quantities~ w) and (bp~0~2)~~should be takenin the form (5.20) and (5.21). This leads to the

following spectrum[93,103]:

k2k~DeDi / k~D1 T. k~De

(~Pk~~Pk~)k1~ = 2n~e~Iik

2DAI2 ~Di+~De)2 + T~(~D1+~D~)2)

2 k~ k~DeDiX 2ir[ô(k~ — k~)+ 5(k~+ k~)]+ 8ineee~ + k~D~

( ~2I ______

[(W_jk2Di)_(TiITe)(W_jk2De) DI k~+k~ (Z’4I~’)

w+ik2D ‘2 1Al I k2D o—ik’2DX[ x

~k~D1+ k~De w — ik’

2DA (Z’ ~‘ £~1 SelVe)

lo (Z~’) 1k~De w+ik2D I I

________ _______ (5.72)— k~D

1+ k~De~ + ik2DA X + ik’2D~)— (Ti! Te)(W+ ~k’2De) (Z’ ~ £~‘,Se/Ve) I

lw+ik’D,j jOn the other hand, in the range IR±— Rh ~k~’,Z, Z’ ~ k~’, the fluctuation spectrumis

determined,as before,by the spectrumof fluctuation sourcesand thereforein sucha caseonly thevolume part of thecorrelationswill be present.However,evenin thevolume part thereis an additionalterm,

k2D(8Pk &Pk)k~W= 2neee2 + .kDI2 2ir[S(k~— k~)+ ~5(k~+ k~)].

The space—timedistributionof the fluctuationsrelatedto thespectrum(5.72) hasthe following formfor IR±—Rhl>>k~1,Z~’k~1[93,103]:

382 Yu.L.Klimontovich eta!., Statistical theoryof plasma—molecularsystems

~~p(R, t) ~p(R’, t’)~ = e~ne k~Di+(T,ITe)k~De 13/2 exp[—(R1 — RI)2I4DATI

x {exp[—(Z — Z’)214DAT1 + exp[—(Z + Z’)2I4DAT]}

+ e~nek~k~(Di— Dc) 8~ 1 1k~(k~D

1+ k~De)V’~t~ (4i~DAr)312k~R

1— RI!3

x 4~DAT$3~2[A(~)+ 2~A’(~)][1-(1- TilTe) ~2~2D

~ (i —2 ~ D,_De) exp[—(Z2I4DAT+ kDz)]], z’~~1,

X kD(Z + Z~)~3I2(~[1- ~((Z + Z’)/V4~DAT)][(3 - 4~)A(~)+ 8~(1-

x V4~DAT exp[(Z + Z’)2!4DAT} - A(~)- 2~A(~))

x ~ [~D1 + (Tj!Te)k~De]exp[—(Z + Z’)

2/4DAT], Z’ ~ k~,SeIPe,

I e e (573)

A(/3)=10(13)exp(—f3), A’(f3)=dA(f3)Idf3, f3=(R~—RI)

2I8DAT.

In the case R1 —Rh!~k~’;Z, Z’~kj’; T~lIPe~one obtains

(~p(R,t) ~p(R’,t’)~= e~ne 3/2 exp[—(R~— Rh)2I4DeTI

(4ltDeT)

x {exp[—(Z — Z’)2I4DeT] + exp[—(Z + Z’)2I4DeTI} . (5.74)

Comparing(5.25)and (5.24)with (5.74) and(5.73),we seethat thepresenceof a boundarychangesthe spaceand time distribution of the large-scalefluctuationsfor differentcorrelationdistances.It isseenthat when R

1 — RI ~‘ k~ the influenceof a boundaryleadsnot only to an additional termdescribingparticle reflection from the boundary,but also to essentiallylong-rangecorrelations.Itfollows from (5.73) that the correlationsin the directionsparallel to the boundarydecreaseaccordingnot to an exponential,but to an inverse powerlaw.

In the equilibrium case, for a hydrogen-like plasma subsystem [k~= k,~= k2DI2, DeID

1(mi/me)

112~ 1] eq. (5.73) reducesto


(bp(R, t) ~p(R’,t’)y = ~e~ne 3/2 exp[—(R1 — RI)2/4DAT]

(4ITDAr)

{exp[—(Z + Zl)Z/4DAT] + exp[—(Z — Zl)2/4DAT]}

+ e~n~(iIr0) 1

2k~IR±— RI~3(4ITDAT)312

2DATkD exp[—(Z2/4DAT) — kDZ], Z’ 4x

(Z + Z’) exp[—(Z + Z’)2/4DAT], Z’ ~ k~,~

It canbeassumedthat the correctionsproportionalto the inversepowerof IR — RI I arerelatedtothe evolution of static long-rangecorrelations,

~ exp[-~D(Z+Z’)].

The presenceof a boundaryis also important for moleculardensity fluctuations. In this casethespectrumof fluctuationsis as follows:

~ = 1~ei[1T~j’ (k, w)GfkW + c.c.] 2i~[6(k~ — k~)+ t5(k5 + k~)].

In the gas-dynamiclimit this gives the following space—timedistribution:

(Bn(R, t) &n(R’, t’)yt = flei[(1 — ~)(

4~D~~T)3~[exp( RJ4D~1T)+exp(—R~/4Dejr)]

+ 3/2 ( -D exp[—(R_ — UT) I4FT]

‘—s (4i~Fr) ‘

R —UT 2+ R exp[—(R+ — UT) /4FT])j~

+ (5.75)

R~= [(R1 — RI)

2 + (Z ±Z’)2]”2.

If one knows the responsefunctions and the correlation function of the sources,it is possible tocalculatethe radiationintensity for a spontaneousfield emittedfrom a plasma—molecularsystemintothe outermedium. The final resultof suchcalculationscanbe representedin thesameform asin thecaseof microscopicfluctuations,namely,

1~= ~I~(1 — R~)+ ~I~~(1 — R~), I~= (w2e~I41T3c2)T~,, (5.76)

where ~ and ~ can be found on the basis of eqs. (3.100) and (3.101) with E.1(k,(1.)) and

(ôj~°~~j~O)) ~ calculatedin the collision approximation.The analysis of such relationsin the generalcaseis a very difficult problem. It requiresa lot of


numericalcalculations.A detailedinvestigationof radiationspectrafor the spontaneousemissionof asemi-infiniteplasmais given in refs. [102,130]. In the presentpaperwerestrictourselvesto an analysisin the “cold” plasmaapproximationwith emphasison investigatingthe role of the molecularsubsystemand the featuresof the spectrumrelatedto the presenceof a boundary.

We startfrom the casewhenthe Dopplerbroadeningof the molecularline is negligible (~ye~~ kSei).

In sucha case

2/ (U (0j pe pm

e~1(k,w)= 8~1e(w)=3~\l— I + — 2 2 ei~ ‘se) Ct) —wo+1)’ Ce)

and the impedancesand reflectivity coefficientscan be found in an explicit form,

~ rp = —(k~1/k~)~‘/r(w) , r~ r~= —k2/k~1, r~= ~ = 0,

k51= [(w21c2)E(w)— k~]”2= (w/c)[e(w) — ~‘ sin2i9i”2

= [(w2/c2).~—k~]”2= (w!c)1/~cosi~

— s~)cosi~—V~r)—~sin2t~2 — ~ 2

“ s(w)cos i~+ \1~”.Je(w)— ~ sin2~ ‘ ~ cos t~+ \/e(w) — ~ sin~

Considerthe practically importantcase Xei(w)j 4 1. It follows from the latterrelationsthat for smallangles [Re~(w) — ~sin~>0]

1 — R = ~ cos i~n(oi) Re e(w) (~+ ReX~~(~)[2~sin2t~’— ReE(w)][cos ~ — \/~n(o.)]

~ [Re e(w) cos i~+ V~n(w’jf \ Re e(o) 2[cos 0 + V~n(oi)]n2(w)

— Im2e(w) ~[2~ sin21~— Re e(w)] ~

4n(a)[Re ~(w)cos i~+ V~n(w)] /

1— R = 4V~~i~n(w) (~— Re X~1@°)u’(w) V’~c0si~— Im

2e(w)~cos2i~[n(w)+V~cosi~]2 \ 2n2(w) n(i)+\/~cosi~ 4n2(o)[n(o.)+V~’cosi~]

(5.78)

I -•2 2 2 2 •e,n(~.)= VRe s(w) — e sin 0, Xei(’~)= ~WpmI(W — + i’y w).

On the other hand for large angles and comparativelysmall frequences[~sin21~> Re s(u)] oneobtains

1— R — 2V~’cosi~Re s(w)Im e(w) 2~’sin2i~—Re~(w)— Re2e(w)cos2~+ ~[~sin2t~— Re E(w)] [~‘sin2i~— Re ()]1/2

1 — R — 2V~cosi~Im e(w)S — [~sin2t~— Re ~‘ ~j3/2

Yu.L. Klimontovich etal., Statisticaltheory ofplasma—molecularsystems 385

If thereis no molecularsubsystem,the radiationintensitycan be characterizedby asharptransitionfrom small levels[in the range(U < ~pe1(1 — ~‘ sin2~)1/2] to the black-bodyradiationlevel in a mediumwith dielectric constant~‘, near the cut-off frequencyfor a transverseelectromagneticfield w,~=

(Upe![1 — ~~sin2~]U2.This behaviour of the spectral distribution is explained by the fact that forfrequencies(U <ó~U~electromagneticwaves cannot propagatein the medium and therefore, thecontributionto the radiationis given only by athin near-surfacelayer with a thicknessof the orderofthe penetrationdepthfor electromagneticwaves.On the contrary,for (U> w,~the contributionto theradiationis given by a large volume,which leadsto a radiation intensity closeto that of black-bodyradiation.

Thepresenceof a molecularsubsystemleadsto additional featuresin the intensityspectrumnearo~.As follows from eqs. (5.78), (5.79), the molecularcontribution results in a strong line on thebackgroundof plasma radiation (for ~ < °~U~) or in a resonanceminimum (for wo> W~~~).Thelinewidth maydiffer from its naturalvalue.

If thereis no plasma(i.e., thereis no cut-off for the electromagneticwaves),for 1 — ~‘ sin2~>0 theintensityspectrumshould reproducethe spectrumof black-bodyradiation(exceptfor smalldeviationsin the vicinity of w~)andfor 1 — ~‘ sin2~<0 representa broadenedmolecularline. In the latter case,however, the absenceof radiation far from wo is related to the total internal reflection of waves(generatedin the volume of the system) from the boundary,but not to the condition for wavepropagationas it is in the caseof a plasmasystem.

The qualitative conclusionswhich follow from eqs. (5.78), (5.79) are confirmed by numericalcalculations.Below we give typical examplesof radiationspectrafor different valuesof the quantity

Wpeh0pm, which characterizesthe ratio of free and boundelectronnumberdensities.The figures below show the dependenciesof the reduced intensities I~’~IIOW(here, ‘Ou, =

W2ET/87T3c2)on the dimensionlessfrequencyw/w~andthe angle~ betweenthe externalnormal to theboundaryand the chosendirection of radiation. Calculationswere performedon the basis of eqs.(5.76), (5.77) for 0~pm’~0o= 5 x 10~,v~Iw~= 10~.The solid curves relate to the spectrum of p-polarizedwaves, andthe dashedcurvesto the spectrumof radiation of s-polarizedwaves.

As mentionedabove,the radiationspectrumof purely molecularmatterwith ~= 1 (1 — ~ sin2i~>0)is characterizedby anintensityof the orderof the black-bodyradiationintensity,exceptfor thevicinityof the resonancefrequency(fig. 5. la).

i!/I~~ _________

~Z~a~ ~/z\~\:~\~ 0.8 I~/Io~

b

‘\.1

/ Dli

3

/ a.Z

(&-&.b)JO~)~ ~ (Ia)-W0)1O~I4,

-~G -29 9 420 ~4O —49 -20 0 +20 +~

Fig. 5.1 Radiation intensity spectrumof semi-infinitemolecularmatter. 1, 0 = 0; 2, 0 = irI6; 3, 0 = ir/4; 4, 0 = ir/3; (a) i= 1, (b) ~= 10.

386 Yu.L.Klimontovich et a!., Statistical theory ofplasma—molecularsystems

L IW/IO& ——-—------

a ~• ‘tt) C~3

0.6 0.~

0.11II I

0.2 3:I 0.6 I

08 1 LII ~‘g 2.~2 2~6 ~

~

0.8 / It—. 0.4

I

0.4 / / 0.2

0.2 ~/~o ____________________________

0.6 i~ L4 18 2~2 26 3. o’4 d.~ t’2 i.6 2

Fig. 5.2. The radiationintensity spectrumof semi-infiniteplasma—molecularmatterasa function of frequency,angleand relativeconcentrationoffree and boundparticles. 1, 0 = 0; 2, 0 irIó; 3, 0 = srI4; (a) w~Iw

0= 0.5, (b) a~~~Iw0= 1, (c) w~Iw0= 1.5.

If thereis aplasmasubsystem(fig. 5.2), the typical featuresof the radiationof asemi-infiniteplasmaappear,namely, a sharpincreaseof the intensitynearthe cut-off frequency.

Thepresenceof moleculesleadsto an intensityminimumfor w~>w~U~(justas in the caseof a purelymolecularsystem)andto a molecularline nearw~whenw~< w~(figs. 5.2b, c). However,dueto theinteraction with the plasma subsystem,additional broadeningof the line and a deformationof thelineshapebecomepossible (fig. 5.3).

The effectsdescribedabovedependstronglyon the quantity ~,becauseit determinesthe possibilityfor spontaneousfields to penetrateinto the externalmedium.The effect of ~ on the radiationspectraissignificant evenin the caseof purely molecularmatter (figs. 5.la and 5.lb). This is explainedby thereducedangularrangefor which penetrationof fields into the externalmediumis possible(~<

8max

arcsiniiV~’j. Thus for ~> 8max the radiation intensityis different from zero only nearthe resonancefrequency(curves 2, 3 in fig. 5.lb). At the same time the radiation intensity in the direction of thenormal to the surfaceis close to that for black-bodyradiation(curve 1 in fig. 5.lb).

Yu.L. Klimontovichef a!., Statistical theory ofplasma—molecularsystems 387

a

I ,(~~-&0)LCY’~o ,

-40 -20 a +20 ~40 -40 -20 0 +20 #‘iU

Fig. 5.3. Structureof the resonanceintensity line for o~/w0= 1 (other parametervaluesarethe sameas in fig. 5.1).

5.3. Layer

5.3.1. Fluctuation propagators, dielectric response and source current correlation functions for aplasma—molecular layer

In the caseof two plane-parallelspecularly reflecting boundaries(Z = 0, Z = L) the boundaryconditionsfor the fluctuationpropagatorsare as follows:

Wa(R,P, X’, r)1Z0L = 14’a(1~,pt, x,, T)Iz~oL

w~(R,P, f”, T)1z.OL = —w~(R,pt ~ T)IzoL , (5.80)

(d)ç q~,, \ — — (d)1 ~

P~~ T, Z=O,L — Pz \ ‘ ‘ ‘ T) Z=0,L

Justas in the caseof a half-spaceonecan find the solutionsof eqs. (4.12), (4.15) and (4.18) for thefluctuation propagatorstogetherwith the initial conditions (2.28), (4.20) and boundaryconditions(5.80) by transformingtheseequationsto integral equationswith differencekernelsfor the Fouriercomponentsof the propagators.

The final result is

fdkldw .

W~(X,X’, T) = J (2ir)3 J ~ exp{i[k

1 . (R.L — R1) — WT]}

x ~ {1~ck~(P,P’) exp[ik5(Z — Z’ + 2nL)]

+ J~iç~~(p,pit) exp[ik~(Z+ Z’ + 2nL)]} , (5.81)

where W~(P, P’) is the Fouriercomponentof the relevantcorrelationpropagatorin the caseof aninfinite medium,whichis describedby eqs. (5.2), (5.3)or (5.4).As is seenfrom acomparisonof (5.81)

388 Yu.L. Klirnontovich eta!., Statistical theory ofplasma—molecularsystems

with (3.123),the relationbetweenthepropagatorsfor a plasma—molecularlayerandan infinite mediumin the caseof large-scalefluctuationsis the sameas for the caseof collisionlesssystems.

The vectorpropagatorw~(X,a”, T) is also describedby a relation of the type (5.81),

(d) _______ (Uw (X~~~’~T)_—j 3 j ~—exp{i[k1.(R1 —R1)—wrj}(2ir) LIT

x ~ {w~](P,x’, P’)exp[ik,(Z — Z’ + 2nL)]

+ w~(P,x’t, P’~)exp[ik~(Z+ Z’ + 2nL)1} , (5.82)

where w~j(P,x’,P’)is given by eq. (5.7).Since the structureof the propagatorsreproducesthe structureof the transitionprobabilities for

collisionlesssystems,it is obviousthat in order to take into accountthe contributionof collisions, it isenoughto substituteinto the general relations(3.127)—(3.139)the dielectricpermittivity tensorsandsourcecorrelationfunctionscalculatedon the basis of eqs. (5.8), (5.9) and (5.19).

5.3.2. Large-scalefluctuationsandthe spectrumofspontaneousemissionfrom a plasma—molecularlayerCalculationsof the fluctuationspectraand the radiationintensitymaybe performedin the sameway

as in the caseof a collisionlesslayer.As mentionedabove,the generalequations(3.127)—(3.139)arestill valid with an accuracyup to the replacementof r

11(k, w) and K&1~°~~ by the relevantquantitiesfound with the collisions takeninto account.

Becausethe presenceof a secondboundarymakesall the calculationsmuch more complicated,inthe presentsectionwe restrict ourselvesto an analysisof the spectral,as well as the spaceand timedistributions in the limit S~/P~~ — RI! ~ k1i~,r ~ 1 ~ In this case

e 2 4LDek~~~

6Pk )k, = eene . 2 2 (6mn+ ~m.-n)-= CS W+lkmDe

ci . 3mei~ ~k~

5)k~~1o= ~1flei (w + ~k~Dei)[(W + ik~F)2 — k~U2I+ c.c.)2L(6mn+ 6). (5.83)

The discretenessof the wavenumberand the possibility for sound waves to propagateresult in aresonanceseriesin the spectrumof (an(R) ~n(R’)) just as Tonks—Dattnerresonancesappearin thecaseof a plasmalayer. In fact, when R

1 — RI! — Ur ~‘

(~nkCm~k~)k~ = 2~n~w6[k~— (w2/U2 — m2IT2/L2)J ~2L(6mn + 6m,), (5.84)

= ~-~-~—°~ —~-- ~ cos(mITZ/L)cos(mITZ’IL)Sei 2L ,n=-N

x J0(1[w

2IU2 — m2IT2/L2 R1 — Rh), N = [WL!UIT], (5.85)

i.e., for ~ = mITUIL we havecorrelationmaximain the frequencyspectrum.


Wealso give the space—time distributions of the density fluctuations. They are

(~p(R,t)&p(R’, ti))e e~n~ ~ [exp(—R~/4DeT)+exp(—R~+/4DeT)], (5.86)

(4ITDeT) ~R~+= [(R1 — RI)2+ (Z ±Z’ + 2nL)2]”2,

(~n(R,t) ~n(R’,ti)yi = nej ~ [(1 — C~/C~)(4ITDe

1T)

x [exp(—R~./4D51r)+ exp(—R~+/4D51r)]

+ ~ 1 312(R~_— UT exp[—(R5 — Ur)2!4Fr]C~,(4ITFT) R~_

+ ~UT exp[—(R5+ — UT)2/4FT])]. (5.87)

As follows from the aboveresults,whenL ~ ~‘ s~/v~it is enough to keep in eq. (5.86) terms withn =0 and the term with n = —1 that dependson (Z + Z’). This leads to the following result for asemi-infinitesystemZ >0, or Z < L [131]

(~p(R,t) 6p(R’, t’)~ = e~ne3/2 exp[—(R1 — RI)2/4Der]{exp[—(Z — Z’)2/4DeT]

(4ITDeT)

+ exp[—(Z + Z’)2I4DeT] + exp[—(Z + Z’ — 2L)2I4DeT]}. (5.88)

If Z and Z~4 L, the latter term in (5.88) is small and (5.88) coincides with (5.74). On the other hand,if Z and Z’ are of the order of L (i.e., we considerthe region nearthe boundaryZ = L), then weshould omit the second term in (5.58)and we obtain an analogue of eq. (5.74) for the half-space Z < L.

Let us consider the opposite caseof a thin layer. In order to do this, we use the followingrepresentationsfor the correlationfunctions,which are equivalentto (5.86) and (5.87):

(~p(R,t)&p(R’, t’)~ = ~!~! 4ITDeT exp[—(R~~~RI)2/4DeT]

x ~ exp[—(m2IT2IL2)D5r] cos(mirZ/L) cos(mITZ’/L),

(&n(R, t) 3n(R’, t’)~’ = ~ ((1 — C~/C~)4ITDe1T exp[—(R1 — RI)2/4DeiT]

x ~ = exp[— (m2ir2IL2)Der] cos(mirZ/L)cos(mirZ’IL)

+ ~ ~ exp[— (m2ir2/L2)De1T]cos(mITZ/L)cos(mirZYL)

x J ~ k±J0(k±IR±—RII)exp(—k~D~1T)cos\/k~+ (mlr/L)2Ur).


Obviously, in the caseof a thin layer,L 4 one shouldtake only the term with n = 0. Theresult is

~p(R, t) ap(R’, t’)~ eene 4ITDeT exp[—(R1 — RI)2/4D~rJ,

i.e., correlationsin the directionsalong the layer are \/4ITDeT!L times (\/4iiDeT/L ~‘ 1) larger thanthose in an infinite medium. Becausethesecorrelationsare relatedto particle diffusion, one mayassumethat diffusion along the two plane-parallelboundariesis more intense than in the caseof asystemwithout boundariesor a semi-infinite system.

A similar pictureis observedin the caseof molecules.In the caseof a thin layer(L ~ \b~r) oneobtains

(6n(R, t) ~n(R’,t’)~’ = ~ — C~,IC~)exp[—(R1 — RI)

2I4DeiTJ

+ ~— f ~ k1J0(k1~R,— RI!) exp(—k~Fr)cosk~Ur).

The intensityof the spontaneousradiationis alsoradically changed,though it is still describedby amodified Kirchhoff law, 1~= ~ + ~

Let us recall that in the case under considerationthe effective temperaturesand absorptioncoefficientsshould be calculatedwith radiation friction andparticle collisions takeninto account.

Now we considerthe main featuresof the radiationspectraof a plasma—molecularlayerrelatedtothe presenceof moleculesandboundaries,restrictingourselvesto the “cold” plasmaapproximation.Inthe caseof a uniformly broadenedline the absorptioncoefficientshavethe form

F = — 1 r(w) cos t~— iV’~\/r(w)— ~sin2~cot((wLI2c)\/r(w) — ~sin2O)2

2 r(w) cost~+ i\/~\/e(w)— ~sin2~cot((wL/2c)\/r(w) — ~sin20)

— 1 e(w)cos~ + i\/~\/r(w) — ~ sin2O tan((wL/2c)~r()— E sin20) 2

2 e(w)cos i~ — i\/~’\/e(w)— ~sin2i~tan((wL/2c)\/r(w) — ~sin20)

— — 1 \/~)cos19—i\/r(w)— ~sin

2~cot((wLI2c)\/r(w)— ~sin21~)22 \1~(~cosi~ +i\/~(w)— ~sin2t~cot((wL/2c)\/r(w) — ~sin2i~)

— 1 \/~jcos i~+i\/~(~.~)— ~sin2i~tan((wLI2c)\/r(w) — ~sin21~)22 ~/~(~cos i~ — i\/r(w) — ~‘sin2~tan((wL/2c)\/e(w)— ~sin2~)

In the caseof a thick layer [Im(wL/2c) [r(w) — ~ sin2i9j112~ 1] eqs. (5.89) reduce to the relevantquantitiesfor a semi-infinite system. In the oppositecase[Im(wL / 2c) [r(w) — ~ sin2i~]”24 11, insteadof (5.89) we can usethe following approximaterelations:


~4cosi9wL Imr(w)Re2s(w) _____ ~sin2’i~ I2c1/~ k(o)!2 ‘ 2c ~(w)I2 cos~ m

+ _ 2[Re e(w)—~‘sin2i~}2wL -cV~cosi~ Ime(w), F~~0.

Thus it is seen that in the given approximation the radiation intensity is proportional to Im X5~(°4in the

rangew -= con, i.e., thereis a resonancedueto the radiationof molecules.However,the linewidth mayexceedits naturalvaluebecauseof the dissipationin the plasmasubsystem.

On theotherhand,a resonancerelatedto plasmaoscillationsis observedin the p-polarizedradiationspectrum at ~ ~ 0 [F Im Xe(o~)/ I s(w)1

21. The absenceof such a resonancein the s-polarizedspectrum(and also at t~= 0) can be explainedby the fact that plasmaoscillationsgive acontributiononly to the z-componentof the fluctuation field.

For somevaluesof the parametersof the systemanda thicknessof the orderof the wavelengthofthe radiation the following conditionmaybe satisfied:

Re~e(w)—~sin219wL/2c---~mIT,m=1,2,...,

which leadsto an effectiveincreaseof wave dissipation(or acontributionto spontaneousradiation)andnew resonancesdueto multiple reflectionsof wavesfrom the boundaries.

As mentionedabove,when Im e(w) .oLIc2 Re2s(w) 8~’1 the intensity spectraare close to that forblack-bodyradiation.At the sametime,whenIm e(w) COLIC Re2e(w) 4 1, the intensity is comparativelysmall. Since, in the range —~ o~the imaginary part of s(co) is resonant, this implies that forIm Xei(~o)(U

0L/4C> 1, but Im X~(~o+ i~o)w0LI4c <1, the effectivewidth 2z~oof the intense(of theorderof ‘Ow) radiationrangeof the spectrumis much largerthan 7d1,

= ~ \/i~_.~ (5.90)o~ 2co~ w~ C

Theseconclusionsarein good agreementwith the resultof numericalcalculations,which weremadefor the same parameter values of the system as in the case of a half-space.

In the caseof a thin layer (fig. 5.4) there is a resonanceline at w ~— w~and a relatively broadmaximumnearthe plasmafrequency(for t~~ 0) in the spectrum of the radiation. As the width of theresonanceneara0 in this caseis larger than the natural line width, however,when the thicknessincreasesthe width of the resonancealsoincreases(fig. 5.5). Such an increaseis most effectivewhen

~ 0 (fig. 5.6a).The presenceof a molecularresonanceis a characteristicfeature of radiation spectraalso for a

thicker layer (fig. 5.6). In such casesthe width of the resonanceis much larger than the naturallinewidth even for a purely molecular layer. However, when COo>~pe’ ~ > arcsin(1/~),the linebroadening is not observed in practice. This can be explained by the fact that when 0 > arcsin(1/~)contributions to the radiation are given only by a subsurfacelayer.

A further growth of the layer thickness leads to the appearance of new geometricresonances(figs.5 .6b, c), which are determined by the condition


0.5 / a 025 c

0.4

/0.30.2.

0.2

0.1 2

J ul/4~0 0:2 04 0.6 0.8 d. £2

.7. r

b0.1

0:5k. - ~

Fig. 5.4. The radiationintensity spectrumas afunction of frequencyand relativeparticle densitiesin thecaseof a thin plasma—molecularlayer. 1,O = ir/6; 2. 0 = sr/4; (a) W~~/W= 0.5, (b) w~/w

0= 1. (c) w~lw0= 1.5; ~= 1, oiL/c 0.lsr.

Re\/E(w) — E sin2~wLI2c —

or(0pe 22 21/2

~resm / . 2 [1+ m IT (C/WpeL) I ‘ ~0resm~‘

V 1 — r sln

Theseresonancesarerelatedto the excitationandinterferenceof transverseelectromagneticwavesinthe systemunderconsideration.

Finally, for a large thicknessof the layer, we observea further growth of the molecularresonancewidth (figs. 5.7, 5.8). In this case the resonancelooks like the spectrum of the correspondingsemi-infinitesystemwith the linewidth describedby eq. (5.90).

Let us notice, finally, the existenceof resonancesgiven by the dispersionof the molecularsubsystem(fig. 5.9),

COpm 2 2 —2 22 21/2

~LW

0

Theseresonancescan be observedwhen (vejlwo)(wo/topm)2 < 1.

Yu.L. Klimontovichet at., Statistical theory ofplasma—molecularsystems 393

-40 -20 0 +20 +40I(~

3/Zoe~J102

(O-6)~tA~1~-4o -~o +20 +~~o

Fig. 5.5. Structureof themolecularresonanceline in thecaseof a thin layer. 1, II = 0; 2, 0 = sr/6; 3, II = ir/4; (a) w~,Iw,= 1, (b) w~,Iw0= 1.5;

~1, w~L/c0.1sr.

As was mentionedearlier the aboveresultsare relatedto the caseof a “cold” plasmasubsystem.Taking into accountthe thermal motion of the particles,it is possible to investigatethe additionalfeaturesof the radiation spectraassociatedwith the possibility of the existenceof new collectiveexcitations[56,132].

Here,we shouldfirst of all drawattentionto a resonanceseriesin thelow-frequencyrange(w <

of thep-polarizedspectrumin thecaseof a nonisothermalplasma,Te ~ T~(fig. 5.10).Theseresonancesare relatedto the excitationand interferenceof low-frequencyion-soundwavesnearthe frequencies

2

t0resm 1 +k~Ik~[1 + 3(k~+ k~)Ik~],k~= k~+ (mrr/L)2.

They are observedfor ve/wpe<102. In the oppositecasevelwpe~ 10_2, the main contribution to theradiationintensityis givenby particlecollisions and,therefore,ion-soundresonancesdisappear.Thereis also an additionalcondition on the layer thickness,keL~ 70, becausewhenkeL>70 the distancesbetweenthe resonancesbecomeof the orderof their width and,therefore,we seeonebroadmaximum.

The influenceof thermaleffects on the radiationspectrain thefrequencyrangew~1< Co < is also

dependenton the plasma parametersand layer thickness.For example, when topeLIc<0.1ir andVs/Cope ~ 102 this influenceis negligible,but with a decreasingcollision frequencythe role of thermal

394 Yu.L. Klimontovich eta!., Statistical theory of plasma—molecularsystems

~ .~ z a

0.2 3

0.4 r~/I~~

~ (~1u)~~0.8,‘.sir O.’~ 8.6 0.8 1. 12w / l0~

0.7 b

Q6~\

0.4 ~MIIH~~iII! rI’’

0.3 II U I’ II

:: ~ - ) ~, ~ III 02 - -

..--

06 1 1:2 /.4 16 1:8 i. iC2 1’4 £6 £8 20

Fig. 5.6. The radiationintensity spectrumof a plasma—molecularlayer as a function of the relative particle density. w)L/c= Sir; 1. 0 = 0; 2,O = ir/6; 3, 0 = sr/4; the otherparameterscoincide with thosefor fig. 5.4.

effects becomesimportant.This is explainedby the increasedp-polarized intensityin the anomalousskin-effectdomain due to the excitation of longitudinal strongly dissipativewaves.

As the thicknessof the layer increases,the contributionof thermaleffects to the radiationspectrabecomessignificant. For instance,when ~~peJ~C = O.4ir the influence of the thermal motion of theelectronsis observedevenfor a collision frequencyPc/COpe~ 10~.A further increaseof L leadsto adominantcontributionof thermaleffects.

The w-dependenceof the radiation intensities and the comparativecontribution of the thermaleffects in the range(~pj< ~ < ~

0peare illustrated by figs. 5.11 and5.12. Notice that, beginningfrom athicknessWpeL/C = 2IT, the radiation spectrafor p- and s-polarizedwaves in the given rangedo notdependon the layer thickness.This may be explained by the effective screening of the field fordistanceslessthan L — 2ITCIWpe and, therefore,thereis no differencebetweena semi-infinite systemand a layer.

We shouldnotethe differencebetweenthe spectraof p- ands-polarizedwavesfor i~ � 0 in the rangewhere the influenceof thermal effects is dominant. If in the s-polarizedspectrumthere is only themaximum at w ~ O~2C0pe~then the spectrumof p-polarizedwavesis characterizedby two maxima at


1i517

-- a

0.8

0.? . ~-40 -20 0 ~20 •4QTP.~IT

0.8 1w f.Low

-40 -20 0 420 ~4’0

Fig. 5.7. Radiation spectrumof a thick layer. 1, 11 = 0; 2, 11 = ~r/6;3, 11 = irI4; (a) i= 1, (b) i= 10; w0LIc = lO

2ir.

~ sO.2w~and O.2w~5< to <O~

8tüpe~This is due to an additional mechanismof p-polarizedwavegenerationrelatedto the transformationof longitudinalcollectiveexcitationsat the boundary.A similarsituationwas observedin the theoryof the anomalousskin-effect,which predictsdifferentabsorptioncoefficientsfor the casesof normal andoblique incidenceof waves[133].

Describingthe radiationspectrain the high-frequencyrange(to ~ w~~)it shouldbenotedthatin sucha casethe s-polarizedspectrumis not influenced by the thermalmotion. This shows that transversewave generationdue to particle collisions is the dominantmechanismof s-polarizedradiation.

Onthe contrary,in the caseof a p-polarizedwave the influenceof thermaleffectsmaybe importantevenwhen ~ — 10_i (fig. 5.13). A decreasingcollision frequencyleadsto an increasingeffectof thethermalmotion andqualitativechangesin the spectrum,andthe only maximumobservedin the caseofa “cold” plasma (fig. 5.4a, b) splits into a resonanceseries (fig. 5.14) (Tonks—Dattnerresonances[134]). As is seen in the caseof a thin layer (k

5L =1OIT), the role of collisions for v~/to~~i03 is

negligible.The resonancefrequenciesare given by

1 + (k~/k~,)ReW (Coresm’~m5e)= 0, (5.91)

396 Yu.L.Klimontovich et a!., Statistical theory ofplasma—molecularsystems

~ b

a

0.6 ~‘~“ ~

0/ o:a I L2 /4/0

Fig. 5.8. The radiationspectrumof a thick layeras afunction of therelativeparticledensities. w0L/c= lO

2ir; theotherparametersarethesameasin fig. 5.6.

underthecondition Re WI ~ Im WI. If (Oresmil~m5e> 1, this conditionis satisfiedandeq. (5.91)yields

m = ~~~e[1+ 3(s~/c2)sin2i~+ (mIT/keL)21

This equationgives the following distancesbetweenthe resonances:

~res = ~res m+1 ~resm = ~(m + 1)(IT/keL)241.

Notice that in the caseof low-frequencyresonancesone obtains~Wres/Wpj = IT/keL, i.e., the relativedistancebetweenthe resonancesfor keL ~ 1 is greaterthan for high-frequencyresonances.

If (COresm/kmSe)2< 10, the condition Re WI ~ IIm W~is not satisfied.This showsthat the maximumnumberof athermalresonanceis equalto the integerpart of thequantitykeL/\/I~iIT. Thiscorrespondsto the resultsof numericalcalculations(fig. 5.14).

Calculationsalso showthat the distancesbetweenthe resonancesbecomesmallerwith increasingLand in thecaseof comparativelylarge dissipation only a broad maximum appears.A more detailed


~ ~1.0

05

0.6

0.6

-0:2 O~4 o:~ o.~ 10 (2

0.4

0.9

0.51Z2

0.7

______________________________________ 0:2 i~~ ~8 ~o ~.a3 -Z -~ 0 1 2 3 Ct)

0 Fig. 5.10. Low-frequency spectrumof p-polarized radiation in theFig. 5.9. Geometric splitting of a molecular resonance in the radiation case of a nonisothermal plasma layer. w~,LIc= Our, c/se= 60, 0 =

spectrum of a molecular layer (0 = 0, i= 10, w,,,,1w0=5 x 10’). ir/’1, v~Ioj~,= iO~,~= 1, T~IT= 100.

1~/L.1O2 a r~IL.to2 a

a

1.28

4.

a, a~ 0.3 °~‘ 4

02 ~4 a~ Q6

1’

b

8~b

_04

at 02 03 04 05— ~2 04 OS ~d

Fig. 5.11. The intensity of s-polarized radiation as a function of the Fig. 5.12. Same functions as in fig. 5.11 for p-polarized radiation.layerthickness in the anomalousskin-effect domain. v,Iw,,, = i03,0 = ir/4, 1, w~,L/c= OuT, 2, w~

5LIc= 0.2i~,3, w,,Llc= O.4~r,4,w,,~LIc= 2ir, (a) cis,= ~, (b) cis,= 60.


p(41 SkI

to 1.2 1.4’ 1.0 LI 1.2 i.~ o-/t~pe

Fig. 5.13. High-frequencyspectrumof p-polarizedradiation for large Fig. 5.14. Thermalsplitting of a plasmaresonancein thespectrumofcollision frequencies.r,Iw~= 10 ‘, w~~L/c= Our; 1. c/sC =60; 2, p-polarized radiation. ~ = l0~, w~L/c= Our, 0 = ir/8: 1,c/se= 100; 3, c/se= x; the uppercurvescorrespondto 0 = ir/4, and c/se= ~ 2, c/sC = 100: 3. c/s.= 60.the lower ones to 0 = ir/8.

discussionof the numericalanalysisof radiation spectrain the caseof alayer is given in refs. [56,1321.The spontaneousemission spectra for systemswith nonplanarboundaries are described in refs.[135,136].

6. Concluding remarks

The readersof this papermay convincethemselvesthat the statisticaltheory of boundedplasma—molecularsystemsis rathercomplicatedevenin the caseof an atomic descriptionin the frameworkofthe classicalmodel of atomicoscillators.On the otherhand, such a theory remainsvalid and maybeeffectively used for studyingmanyphenomena.

Of course,it is also necessaryto developa quantumstatisticaltheory of boundedplasma—molecularsystems.However, thereare many problemsin the way of such studies.We can only hope that thepresentpaperwill stimulateinvestigationsin this field. This will be very usefulfor the solutionof manypracticalproblems.

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Statistical theory of plasma-molecular systems