Fluctuating forces in plasmadynamics

Volume71A, number4 PHYSICSLETTERS 14 May 1979

FLUCTUATING FORCES IN PLASMADYNAMICS

W. ROZMUS a andLA. TURSKI b, 1aInstituteof NuclearResearch,00-681 Warsaw,Polandbinstitutesof GeophysicsandTheoreticalPhysics,WarsawUniversity, 00-681 Warsaw,Poland

Received29 December1978

Using theconceptof theVlasov—Landau—Langevinequationwe discussthestructureoffluctuating forcesa la LandauandLifshitz in plasmadynamicalequationsfor weakly coupledonecomponentplasma.Explicit expressionsfor stresstensorandheatcurrentvectorcomponentscorrelationfunctionsarederiveddisplayingnon-markovianbehaviour.

The origin andstructureof fluctuating forcesadded from the Chapman—Enskogin the derivationof theto thehydrodynamicalequationsby LandauandLif- plasmadynamicalequationsfrom the kinetic equation.shitz [1] attractnow a greatdealof attention[2,31. Thispointwas fully discussedby Baus [7]. FromhisTheseforcesplay an importantrole in variouspractical analysis(cf. also ref. [8]) it follows that correctplasma-applicationsfor examplein hydrodynamicalinstabil- dynamicalequationsfor the OCPare non-localin timeitiesandturbulencephenomena[4,5]. i.e. haveexplicitly time dependenttransportcoefficients

Themicroscopicderivationof the LandauandLif- [8]. In ref. [8] we havederivedthesenon-localequa-shitzexpressionsfor the correlationfunctionsof the tions,startingfrom the linearizedVlasov—Landaukineticfluctuatingforceswaspresentedfor the first timeby equation,usingthe projectionoperatorstechniqueBixon andZwanzig[6]. Theyhaveusedthe socalled combinedwith the Grad’s 13 momentsexpansion.TheBoltzmann—Langevinequationandconventional Grad’stechniqueis beingusedhere to studythepredic-Chapman—Enskogprocedure. tionsof the Vlasov—Landau—Langevinequationwritten

In this notewe follow the line of argumentsfrom below,wherethe Fourier transformwith respecttoref. [61 in studyingthe natureof fluctuatingforces~ spacevariableswas performed,plasmadynamicalequationsdescribingthe long-wave- - -

- a~h(k,u,t)+ik.uh(kvt)+V(h)—c(h)lengthevolutionof a onecomponentweakly coupled kplasmaOCP.To thebestof our knowledgethis hasnot = F(k, ~, t~- (1)

beendiscussedin the literature.The essentialdifferencebetweenthelong-wavelength In theaboveh(k,u, t) is the dimensionlessdeviationof

behaviourof the OCP and thatof neutralsystemsstems the exactone.particledistribution functionfrom itsfrom thefact that the OCP supportsthe fastmodeof equilibrium value~fB (lB beingthe Boltzmanndistribu-excitations(plasmamode)down to thezero-wavevec- tion andn themeanparticledensity)~k andC standtor. This rendersapplicationsof conventionalkinetic for linearizedVlasovandLandaucollision operators,theoryconcepts— like the local equilibriumdistribution respectively.Notice that in contrastto the neutralpar-— invalid and forcesusto adopta proceduredifferent tide casethe operator~‘k= Vk C containssingular

k-dependencein its Vlasov term.This leadsto the exis-1 tenceof thefast modeat k = 0.

Researchsupportedin partby the Polish Ministry of ScienceHigherEducationandTechnologyGrantNo. MR.1.7 andby The F(k, u, t) on the RHSof eq.(1) representsthethegrantfrom NationalBureauof Standardsmadeavailable fluctuatingsourceaddedhere in accordwith ideasofvia theMaria Sklodowska-CurieFoundation. ref. [6]. Since thewhole operatoron the LHS of eq.(1)

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is local in time, then recallingbasicpropertiesof the We adoptheretheGrad’s 13 momentstechnique.InMon formulationvia which eq.(1) canpresumablybe practicethis meansthat we haveto insertunit opera-justifiedoneconcludesthat the randomsourceFhasto torsspannedby the Hermitepolynomialsin thevelocitybe deltacorrelatedin time, i.e. space,betweenall the operatorsoccurringin eq.(4)

andthenrestrictthenumberof allowedpolynomials(F(k1,U1, t1) F(k2,~2’ r2)) to thefirst 13. To completecalculationswe also need

= E(k1,u1 k2,u2) 6(t1 — t2), (2) eigenfunctionsandeigenvaluesof theL-operator.Itsufficesto calculatethemby perturbationandwith

whereC..) denotesaveragingoverthe initial equilibrium accuracyup to thesquareof the wavevectork. Theensemble, resultingexpressionsfor correlationfunctionsbetween

The apriori unknowncoefficientE canbeinferred variouscomponentsof the stresstensorP,1 andfrom the requirementthat thevelocity-spacecorrelation betweencomponentsof theheatcurrentvector arefunction written belowdisplayingclearlynon-markovian

character:C’(k1,u1,t1k2,o2’ t2) = (h(k1,u1,t1)h(k2,u2,t2))be timetranslationinvariant.E is givenentirelyin terms <~i1(”i,ti)Pim(k2, t2)) = kBTh(kl+ k2)of equilibriumcorrelation functions.Assumingfor thelattera lowest order approximationwithrespectto the X(&iit

5jm +6im6jl_(2I3)~5ijt5lm)plasma-parameterwe obtain x {O(t

1 — r2)fl(t1 — t2)+ o~(t2— t1) ~(t2 — t1)} , (5)

E(k uk u)1’ 1 2’ 2 (q1(k1,t1)qJ(k2,t2))k~T26(k

1+k2)61J

= —2C(ul)(nfB(u2))’ 6(u1 — u2) 6(k1+ k2). (3)x {8(t1—t2)K(t1—t2)+0(t2—r1)K(t2—r1)} . (6)

Havingthis, onecanwrite down the formalexpressionfor the correlationfunction C. Fourier transformingin Here8(t) is thestep functionand~(t) andic(t) aretimetimewe have dependentshearviscosityandthermalconductivity,

respectively.Both thesequantitieswereevaluatedin2irC(k u1,w1k2,u2,w.,)

1 ref. [8] andare expressedin termsof matrix elements= 6(k1 +k2)6(~o1+ w2)(—iw1 +L1)~ of the operatorC(u) in the nonhydrodynamicalpart

i of the Gradbasis.Explicitly we haveX(iw2 +L~)— E(k1u1k2,u2). (4)

In theaboveLa = ik Da — ~k and * denotescomplex ~(t) = nkB T exp(—C1t),conjugation. ,c(t) = nkBT

2(5/2m)exp(—C2t), (7)

It is possiblenow to checkthat thedynamicalstructurefactorS(k,w) obtainedfrom eq.(4) after wherethe velocity spaceintegrationfulfills two first mo- ~~/1 1

- C1 —(IOir ‘ ) w,,~ln~g) C2=(2/3)C1,mentsrelationsfor OCPas it should.

The fluctuatingforcesin theplasmadynamical and is the plasmafrequencyandg is the plasmaequationsare recognizedasusual asthe divergences parameter(g ~ 1).of fluctuatingpartsof thestresstensor andthe Eqs. (5) and(6) are our main result.Theytell usheatcurrentvector~ Thesefluctuating partsare thateventhougheq.(1) was clearlymarkovianthelinear functionalsof the distributionh(k,u, t) thus fluctuatingforcesin plasmadynamicalequationsaretheir correspondingcorrelationfunctionsare expressed non-markovian.This is in accordwith thefact thatentirelyin termsof Cfrom eq.(4). Thishowever plasmadynamicalequationsareby itself non-localinrequiresexplicit evaluationof the inverseoperators time andthat simpleminded“localization” of theseoccurringin eq.(4). In the processof doingso, due equationsis inconsistent[7,8]. Forneutralparticlesto thereasonsoutlinedin theintroductorypart,we onecanevaluatecorrelationfunctionslike eqs.(5)haveto departfrom the argumentationof ref. [6]. and(6) also usingthe Grad’s 13 momentsmethod.

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Now the localizationof thetransportcoefficientsis in hydrodynamicsandthepossibility of regardingthepossibleresultingin replacementof the curly brackets non-hydrodynamicalpart of the initial value for thein eqs.(5), (6) by 26(t1—t2) timestransportcoefficient, linearizedkinetic equationas randomquantity.WehopeTheseareoriginal LandauandLifshitz formulae(cf. to returnto theseproblemsin a separatepublication.also ref. [3]).

It is possibleto understandeqs.(5), (6) on thebasis Referencesof theMon formulation.Indeed,we know from thisgeneraltheorythat the correlationfunction of the [11L.D. LandauandE.M. Lifshitz, Soy.Phys.JETP 32

generalizedLangevinforcesoccurring in the Mon equa- (1957)618.tionhaveto be equal(modulostaticcorrelationfac- [2] K.T. MashiyamaandH. Mori, J. Stat.Phys.18 (1978)385.

tors)to the memoryfunctionappearingin thesame [3] C.P.Enzand L.A. Turski, submittedfor publication.[4] For a reviewcf. H. Haken,Synergetics(Springer,Berlin,equation.Adding the fluctuatingforcesto theplasma- 1977).

dynamicalequationsfrom ref. [8] it is easyto seethat [5] S.A. Orszag,in: Fluid dynamics,ed.R. Balian(Gordon

this is really thecase,i.e. the correlationfunctionseqs. and Breach,New York, 1973),LesHouchesproceedings.

(5), (6) determinethememoryfunctionsin ref. [8]. [6] M. Bixon and R.W.Zwanzig,Phys.Rev. 187 (1969)267.

We haveconcludedthatonecanmakemuchmore [7] M. Baus,Physica79A (1975)377.[8] W. Rozmus,submittedfor publicationin J. PlasmaPhys.

generalstatementsconcerningthenatureof fluctuating (1979).

forcesin hydrodynamicalequations,memoryfunctions

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