Chapter Microscopic Reversibility The Principle of Microscopic Reversibility was formulated - PDF document

Chapter2 MicroscopicReversibility The PrincipleofMicroscopicReversibility wasformulatedbyRichardTolman[ 14 ] whostatedthat,atequilibrium,“anymolecularprocessandthereverseofthatpro- cesswillbetakingplaceontheaverageatthesamerate”.Applyingthisconcept tomacroscopicsystemsatlocalequilibriumleadstotheruleof detailedbalances (Sect. 2.2 uxes,totheformulationofthecelebratedreciprocityrelations(Sect. 2.3 )derived byLarsOnsagerin1931,andtheuctuation-dissipationtheorem,(Sect. 2.4 )proved byHerbertCallenandTheodoreWeltonin1951.Inthischapter,thisvastsubject matteristreatedwithacriticalattitude,stressingallthehypothesesandtheirlimi- tations. 2.1ProbabilityDistributions Dene: € ( x ,t) thattherandomvariable x (t) hasacertainvalue x attime t . 1 € Thejointprobability ( x 2 ,t 2 ; x 1 ,t 1 ) thattherandomvariable x (t) hasacertain value x 2 = x (t 2 ) attime t 2 and,also,thatithasanothervalue x 1 = x (t 1 attime t 1 . € Theconditionalprobability ( x 2 ,t 2 | x 1 ,t 1 ) thatarandomvariable x hasacertain value x 2 = x 2 ) attime t 2 ,providedthatatanother(i.e.previous)time t 1 ithasa value x 1 = x (t 1 ) . Bydenition,when t 2 �t 1 , x 2 ,t 2 ; x 1 ,t 1 ) = ( x 2 ,t 2 | x 1 ,t 1 )( x 1 ,t 1 (2.1) 1 Hereandinthefollowing,weusethesamenotation, x ,toindicateboththerandomvariableand thevaluethatitcanassume.Wheneverthismightbeconfusing,differentsymbolswillbeused. R.Mauri, Non-EquilibriumThermodynamicsinMultiphaseFlows , SoftandBiologicalMatter,DOI 10.1007/978-94-007-5461-4_2 , ©SpringerScience+BusinessMediaDordrecht2013 13 142MicroscopicReversibilityInastationaryprocessallprobabilitydistributionsareinvariantunderatime.Thereforeforstationaryprocessesthethreedistributionfunc-tionssimplifyasfollows.,t)independentof.Inthefollowing,whenthereisnoambiguity,thetime0willbeIftheprocessisalsohomogeneous,thenallprobabilitydistributionsareinvari-antunderaspacetranslation.Thereforeforstationaryhomogeneousprocessesthethreedistributionfunctionssimplifyasfollows.independentof,),).Notethatforstationaryandhomogeneousprocessesthejointprobabilityandtheconditionalprobabilityareequaltoeachother;infact,theratiobetweenthemisgivenbythesimpleprobability,whichinthiscaseisaconstant.Now,letusconsiderastationaryprocess.Itsprobabilitydistributionfunctionsarenormalizedasfollows:Consequently,showingthat,since0,wehaveBasedonthesedenitions,foranyfunctionswecandenetheaverages,)()g()g()(.Obviously,whiletherstaverageisaconstant,thesecondisafunctionof 2.2MicroscopicReversibility15Wecanalsodenetheconditionalaverageofanyfunctionsoftherandomvariable(t)asthemeanvalueofattime,assumingthat,i.e.,.e.,x,|x0,0]dx.(2.7)ThisconditionalaveragedependsonandonNow,substituting()into(),weobtainthefollowingequality,)g()g()()(thatis:)g(2.2MicroscopicReversibilityForaclassical-bodysystemwithconservativeforces,microscopicreversibilityisaconsequenceoftheinvarianceoftheequationsofmotionundertimereversalandsimplymeansthatforeverymicroscopicmotionreversingallparticlevelocitiesalsoyieldasolution.Moreprecisely,theequationsofmotionofanparticlesystemareinvariantunderthetransformationarethepositionsandthevelocitiesoftheThisleadstotheso-calledprincipleofdetailedbalance,statingthatinastationarysituationeachpossibletransitionbalanceswiththetimereversed,sothat,(),()(),()AswesawinSect.,thissameconditioncanbeappliedwhenwedealwiththermodynamic,coarsegrainedvariablesatlocalequilibrium,i.e.whenfluctfluctisthetypicaluctuationtime.Infact,forsuchveryshorttimes,forwardmotionisindistinguishablefrombackwardmotion,astheybothareindistinguish-ablefromuctuations.First,letusconsidervariables(t)thatareinvariantundertimereversal,e.g.theyareevenfunctionsoftheparticlevelocities.Inthiscase,another,perhapsmoreintuitive,waytowritetheprincipleofdetailedbalanceistoassumethatthecondi-tionalmeanvaluesofavariableattimesareequaltoeachother,which 162MicroscopicReversibilityor,equivalently,Multiplyingthislastequationby,itcanberewrittenas(x(x(x,),wherewehaveappliedthestationaritycondition.Asexpected,thisequationisiden-ticalto(),withNow,denethecorrelationfunctionforastationaryprocessas:()()x0.Applying(),weseethatfrommicroscopicreversibilityweobtain:()),thatis,()x()Fromthisexpression,applyingEq.(),weseethatanotherformulationofmicro-scopicreversibilityis:Now,considerthegeneralcasewhereisanarbitraryvariablewhich,undertimereversal,transformsintothereversedvariableaccordingtotherule,1,whenthevariableisevenundertimereversaland1,whenitisodd.Atthispoint,Eq.()canbegeneralizedas:Inthefollowing,wewilldenotebythosevariableshaving1,i.e.thoseremaininginvariantundertimereversal,andbythosevariableshavingi.e.thosechangingsignundertimereversal,(e.g.velocityorangularmomentum). Notethat,since(x(,then,implyingthatalloddvariableshavezerostationarymean.Inmostoftheliterature,-and-variablesaregenerallyreferredtoas-and-variables. 2.3Onsager’sReciprocityRelations2.3Onsager’sReciprocityRelationsAssumethefollowinglinearphenomenologicalrelations:(i.e.neglectinguctua-wherethedotdenotestimederivative,arereferredtoasthermodynamicuxes,arethegeneralizedforcesdenedin().Thatmeansthatthisequationholdswhenweapplyittoitsconditionalaverages,ThecoefcientsaregenerallyreferredtoasOnsager’s,orphenomenological,coefcients.NowtakethetimederivativeofEq.(),consideringthatiscon-Consideringthat,weobtain:Thesearethecelebratedreciprocityrelations,derivedbyLarsOnsager[]inInthepresenceofamagneticeldorwhenthesystemrotateswithangularve-,theoperationoftimereversalimplies,besidesthetransformation(thereversalofaswell.Therefore,theOnsagerreciprocityrelationsbe-Inthefollowing,wewillassumethat;however,weshouldkeepinmindthatinthepresenceofmagneticeldsoroverallrotations,theOnsagerrelationscanbeappliedonlywhenarereversed.AcleverwaytoexpresstheOnsagercoefcientscanbeobtainedbymulti-plyingEq.()byandaveraging:Nowtake0andapplyEq.()toobtain:=Šsym 182MicroscopicReversibilitywherethesuperscript“sym”indicatesthesymmetricpartofatensor,i.e.sym .Thisisoneofthemanyformsoftheuctuation-dissipationtheorem,whichstatesthatthelinearresponseofagivensystemtoanexternalperturbationisexpressedintermsofuctuationpropertiesofthesysteminthermalequilibrium.AlthoughitwasformulatedbyNyquistin1928todeterminethevoltageuctuationsinelectricalimpedances[],theuctuation-dissipationtheoremwasrstproveninitsgeneralformbyCallenandWelton[]in1951.In(isthevelocityoftherandomvariableasitrelaxestoequilib-rium.Therefore,consideringthattendstoforlongtimes,weseethat(t)dtandthereforetheuctuation-dissipationtheoremcanalsobeformu-latedthroughthefollowingGreen-Kuborelation:(t)symshowingthattheOnsagercoefcientscanbeexpressedasthetimeintegralofthecorrelationfunctionbetweenthevelocitiesoftherandomvariablesattwodifferentNow,considertheoppositeprocess,wheretherandomvariableevolvesoutofitsequilibriumposition.Therefore,applyingagainEq.(),butwithnegativetimes,weobtain: 2d symshowingthattheOnsagercoefcientscanbeexpressedasthetemporalgrowthofthemeansquaredisplacementsofthesystemvariablesfromtheirequilibriumvalues.Theseresultsareeasilyextendedtothecasewherewehavebothvariables,i.e.evenandoddvariablesundertimereversal.Inthiscase,thephe-nomenologicalequations()canbegeneralizedas:(xx)(xy)(yx)(yy)  S x;Yi=1  S arethethermodynamicforcesassociatedwiththe-variables,respectively.Withthehelpofthesequantities,thereciprocityrelations()weregeneralized TheGreen-Kuborelationisalsocalledtheuctuation-dissipationtheoremofthesecondkind.See[ 2.3Onsager’sReciprocityRelationsbyCasimirin1945as[(xx)(xx)(xy)(yx)(yy)(yy)Substituting(),()and()intothegeneralizedform(1.26)oftheen-tropyproductionterm, kdS weobtain: kd (xx)(yy)ThisshowsthatneithertheantisymmetricpartsoftheOnsagercoefcients(xx)(yy),northecouplingtermsbetween-variables,(xy)(yx),giveanycontributiontotheentropyproductionrate.Itshouldbestressedthat,whenweapplytheOnsager-Casimirreciprocityrela-tions,wemustmakesurethatthevariables(andthereforetheirtimederivatives,oruxes,aswell)areindependentfromeachother,andsimilarlyforthethermody-namicforces.Comment2.1Inthecourseofderivingthereciprocityrelations,wehaveassumedthatthesameequations()governboththemacroscopicevolutionofthesystemandtherelaxationofitsspontaneousdeviationsfromequilibrium.ThisconditionisoftenreferredtoasOnsager’spostulateandisthebasisoftheLangevinequation(seeChap.Theuctuation-dissipationtheorem,Eqs.()and(),canbeseenasanaturalconsequenceofthispostulate.Comment2.2Thesimplestwaytoseethemeaningoftheuctuation-dissipationtheoremistoconsiderthefreediffusionofBrownianparticles(seeSect.).First,considerahomogeneoussystem,followasingleparticleasitmovesrandomlyanddeneacoefcientofself-diffusionas(onehalfof)thetimederiva-tiveofitsmeansquaredisplacement.Then,takethesystemoutofequilibrium,anddenethegradientdiffusivityastheratiobetweenthematerialuxresultingfromanimposedconcentrationgradientandtheconcentrationgradientitself.AsshownbyEinsteininhisPh.D.thesisonBrownianmotion[],whentheproblemislin-ear(i.e.whenparticle-particleinteractionsareneglected),thesetwodiffusivitiesare Asshownin[],whenuxesandforcesarenotindependent,butstilllinearlyrelatedtooneanother,thereisacertainarbitrarinessinthechoiceoftheindependentvariables,sothatattheendthephenomenologicalcoefcientscanbechosentosatisfytheOnsagerrelations.Onsagerstatedthat“theaverageregressionofuctuationswillobeythesamelawsasthecorre-spondingmacroscopicirreversibleprocess”.Seediscussionsin[ThisprocessissometimescalledKnudseneffusion 202MicroscopicReversibilityequaltoeachother,thusestablishingperhapsthesimplestexampleofuctuation-dissipationtheorem.Althoughwetakethisresultforgranted,itisfarfromobvious,asitstatestheequalitybetweentwoverydifferentquantities:ononehand,theuc-tuationsofasystemwhenitismacroscopicallyatequilibrium;ontheotherhand,itsdissipativepropertiesasitapproachesequilibrium.2.4Fluctuation-DissipationTheoremAswesawintheprevioussection,theßuctuation-dissipationtheorem(FDT)con-nectsthelinearresponserelaxationofasystemtoitsstatisticaluctuationpropertiesatequilibriumanditreliesonOnsager’spostulatethattheresponseofasysteminthermodynamicequilibriumtoasmallappliedforceisthesameasitsresponsetoaspontaneousuctuation.First,letusderivetheFDTinaverysimpleandintuitiveway,followingtheoriginalformulationbyCallenandGreene[].Assumethataconstantthermo-dynamicforce/kTisappliedtothesystemforaninnitetime0andthenitissuddenlyturnedoffat0.Therefore,at0thesystemwillhaveanon-zeropositionofstableequilibrium,,suchthatminmin istheminimumworkthattheconstantforce,,hastoexerttodisplacethesystemtopositionNow,intheabsenceofanyexternalforce,i.e.when0,themeanvalueofthe-variablerelaxesintimefollowingEq.(),with,(t)(t)(t)expisaconstantphenomenologicalrelaxationcoefcient.Therefore,substituting()into()weobtain:(t)(t) (t)expisatimedependentrelaxationcoefcient.Ontheotherhand,thefunctionisrelatedtothecorrelationfunctionatequilib-.Infact,fromthedenition(),substituting()weseethat:(t)expexp 2.4Fluctuation-DissipationTheoremComparingthelasttwoequations,weconclude:(t)(t).Thisrelationrepresentstheuctuation-dissipationtheorem.Notethat,when0,therelation()isidenticallysatised,since,whileTheuctuation-dissipationtheoremcanalsobedeterminedassumingageneral,time-dependentdrivingforce,(t).Inthiscase,duetothelinearityoftheprocess,wecanwrite:(t) (t)isthegeneralizedsusceptibility,with(t)0for0.Denotingby(\n)(\n)(\n),theFouriertransforms()of(t)(t)(t),respec-tively,wehave:(\n) (\n)(\n).Ingeneral,(\n)isacomplexfunction,with(r)(i),wherethesuperscripts(r)(i)indicatetherealandimaginarypart.Since(t)isreal,wehave:(\n),wheretheasteriskindicatescomplexconjugate,showingthat(r)isanevenfunc-tion,while(i)isanoddfunction,i.e.,(r)(r)(\n)(i)(i)(\n).Analogousrelationsexistsregardingthecorrelationfunction(t).Infact,consideringthemicroscopicreversibility()andtherealitycondition,weob-(\n)(\n)\n),i.e.theFouriertransformofthecorrelationfunctionisarealandsymmetricmatrix.AsshowninAppendix,usingthecausalityprinciple,i.e.imposingthat(t)0for0,weseethatthegeneralizedsusceptibilityissubjectedtotheKramers-Kronigrelation(),sothat(t)canberelatedto(t)as[cf.Eq.(C.30(\n) (\n). Thesameresultcanbeobtainedassumingthattheconstantthermodynamicforceissuddenlyturnedonat0,sothatforlongtimesthesystemwillhaveanon-zeropositionofstableequi-.Inthatcase,redeningtherandomvariable,wendagainEq.(Thisisasomewhatsimpliedanalysis.Formoredetails,see[ 2MicroscopicReversibilitySubstitutingthisresultintoEq.()andconsidering(),weseethattheuctuation-dissipationtheoremcanbewritteninthefollowingequivalentform:(r)(\n) (i)(\n)(s)wherethesuperscripts(s)denotesthesymmetricpartofthetensor.Notethat,since(\n) 2=1 (i)(\n) d\n,usingthedispersionequation()with0,weobtaintheobviousrelation,(r)wherewehaveusedthefactthatisanevenfunction.Thisresultcanbeeasilyextendedtocrosscorrelationfunctionsbetween-and-typevariables,consideringthat(\n)isanimaginaryandantisymmetricma-trix,i.e.,(\n)=Š(\n)=Š\n).Attheend,theuctuation-dissipationrelationbecomes,(i)(\n) (r)(\n)(a)wherethesuperscripts(a)denotestheantisymmetricpartofthetensor.Tobetterunderstandthemeaningoftheuctuation-dissipationrelation,considerthesinglevariablecase, Now,Fouriertransformingthisequation,weobtain()with,(\n) g(Mi\n) i\nLi\nL Ontheotherhand,thecorrelationfunction()gives:(t) geŠ,(2.55) Here,whentheappliedforceisconstant,theequilibriumstatewillmovefrom0toF/(gkT) 2.4Fluctuation-DissipationTheorem23whoseFouriertransformyields:( ) g(M thusshowingthattheFDT()isidenticallysatised.Identicalresultsareobtainedinthemulti-variablecase,wherewehave: istheOnsagerphenomenologicalcoefcient,whileNotethatthesymmetryofisadirectconsequenceoftheOnsagerreciprocitySometimes,itisconvenienttoconsidertheuctuationsofasbeingcausedbyarandomctitiousforce,sothattheinstantaneousvalueof(notitsmeanvalue,whichisidenticallyzero)islinearlyrelatedtothroughthesamegeneralizedthatgovernstherelaxationofthesystemfarfromequilibrium,(t) Inthiscase,consideringthat( ),wehave:( )(kT)( )( )( )then,weobtain:( )(kT) (i)( ) (kT)Therefore,whenthegeneralizedsusceptibilitycanbeexpressedasEq.(),we( )(kT).Infact,inthiscaseEq.()becomestheLangevinequation(seenextchapter), ThisisclearlyequivalenttotheOnsagerregressionhypothesis.Notethathereandinthefollow-denotestheuctuation( 242MicroscopicReversibility istheuctuatingux,satisfyingthefollowingrelation:(t)(t),(t)istheDiracdelta.Thisshowsthatthereisnocorrelationbetweentheparticlepositionandtherandomforce(seeProblem).Infact,itisthislackofcorrelationthatisatthefoundationoftheOnsagerregressionhypothesis,thereforejustifyingtheLangevinequation,asdiscussedinthenextchapter.2.5ProblemsProblem2.1ConsiderasmallparticleofarbitraryshapemovinginanotherwisequiescentNewtonianuid.Increepingowconditions,determinethesymmetryrelationssatisedbytheresistancematrixconnectingvelocityandangularvelocitywiththeforceandthetorquethatareappliedtotheparticle.Problem2.2Consideradriven1Doscillatorofmassatfrequency,withdampingforce,withdenotingthedisplacementfromitsequilibriumposition,0.Determinethespectrumoftherandomforce.References1.Callen,H.B.,Greene,R.F.:Phys.Rev.,702(1952)2.Callen,H.B.,Greene,R.F.:Phys.Rev.,1387(1952)3.Callen,H.B.,Welton,T.A.:Phys.Rev.,34(1951)4.Casimir,H.B.G.:Rev.Mod.Phys.,343(1945)5.deGroot,S.R.,Mazur,P.:Non-EquilibriumThermodynamics.Dover,NewYork(1962),Chap.VIII.46.Einstein,A.:Ann.Phys.(Berlin),549(1905)7.Green,M.S.:J.Chem.Phys.,398(1954)8.Kubo,R.:J.Phys.Soc.Jpn.,570(1957)9.Marconi,U.M.B.,etal.:Phys.Rep.,111(2008)10.Meixner,J.:Rheol.Acta,465(1973)11.Nyquist,H.:Phys.Rev.,110(1928)12.Onsager,L.:Phys.Rev.,405(1931)13.Onsager,L.:Phys.Rev.,2265(1931)14.Tolman,R.C.:ThePrinciplesofStatisticalMechanics,p.163.Dover,NewYork(1938),Chap.5015.vanKampen,N.G.:StochasticProcessesinPhysicsandChemistry.North-Holland,Amster-dam(1981),Chap.VIII.8