Non-equilibrium work relations

arXiv:cond-mat/0511073v1[cond-mat.stat-mech]3Nov2005Non-equilibriumworkrelations.JorgeKurchanPMMH-ESPCI,CNRSUMR7636,10rueVauquelin,75005Paris,FRANCEAbstractThisisabriefreviewofrecentlyderivedrelationsdescribingthebehaviourofsys-temsfarfromequilibrium.TheyincludetheFluctuationTheorem,Jarzynski’sandCrooks’equalities,andanextendedformoftheSecondPrincipleforgeneralsteadystates.Theyareverygeneralandtheirproofsare,inmostcases,disconcertinglysimple.1IntroductionInthelastfewyears,agroupofrelationswerederivedinvolvingthedistributionofworkdoneonasystembynon-conservativeand/ortime-dependentforces[1,2,3](forreviews,see[4]).Itseemstheyhavenotbeennoticedbeforethenineties[5],eventhoughinmostcasestheproofsrequiretechniquesthathavebeenaroundformanydecades.IshalldiscussthemhereinthecontextofsystemsthatarepurelyHamiltonian,or,whenathermostatisneeded,incontactwithastochasticthermalbath[6,7].Inordertostresstheirsimilarity,beforegoingtotheactualproofs,Ishalllistandcommentthem:•‘Transient’FluctuationRelation[1]:startingfromaconfigurationchosenran-domlywiththeBoltzmann-GibbsdistributionattemperatureT=β−1,andapplyingnon-conservativeforcestheworkdoneWisarandom1variabledis-tributedaccordingtoalawP(W),whichclearlydependsupontheprocess.Wehavethentherelation,independentofbothmodelandprocess:lnP(W)−lnP(−W)=βW(1.1)Anequivalent,andsometimesmoredirectformisobtainedbymultiplyingtheexponentialof(1.1)bye−λWandintegrating.For.realλ:heλWiequil=he−(β+λ)Wiequil(1.2)whereh•iequildenotesaverageovertrajectoriesstartingfromthermalequilib-rium.•StationarystateFluctuationRelation[2,8,6,7]:Inthisversion,weconsideradrivenstationarysystem.Wecomputetheworkoveraperiodoftimetstartingfromdifferentinitialconfigurationschosenwiththestationary,(ingeneralnon-Gibbsean)distribution.Aformulalike(1.1)alsoholds,butthistimeonlyforlarget,whenwecanwritethatP(W)convergestoalarge-deviationform:lnP(W)∼tζWt+smallerordersint(1.3)•Jarzynskiequality[3]:Startingfromanequilibriumconfiguration,wevarysomeparameterα(e.g.magneticfield,volume,...)ofthesystemfromα1toαNatanarbitraryrate,andrecordtheworkdoneW.Averagingoverseveralindividualrealisations,wehavethat:he−βWiα1=e−β[F(αN)−F(α1)](1.4)1Evenifthedynamicsaredeterministic,sincetheinitialconditionisitselfrandom.1NotethepuzzlingappearanceoftheequilibriumfreeenergyatthefinalvalueαNoftheparameter,eventhoughthesystemisnotassumedtoequilibrateatthatvalue!Clearly,theJarzynskirelationimpliesthesecondprinciplehβWi≥β[F(αN)−F(α1)](1.5)throughJensen’sinequalitylnhAi≥hlnAi.•ExtendedSecondLawfortransitionsbetweennon-equilibriumstationarystates[9]Consideradrivensystemwithdynamicvariablesxandexternalfieldsα(e.g.shearrate,temperaturegradient,etc)thatadmitsnon-equilibriumsteadystateswithdistributionρSS(x;α)=e−φ(x;α).Wenowtakeastationarystateatα=α1andatarbitraryspeedmakeatransitionfromα1toαNinsuchawaythatstationarityisachievedatαN,butnotnecessarilyatalltheinter-mediatevalues.Repeatingtheexperimentwithmanyinitialconditions,thefollowingequalityholds:e−Rdt∂φ(x;α)∂α˙α=1(1.6)Again,usingJensen’sinequalityweobtain:*Z∂φ(x;α)∂αdα+≥0(1.7)Thecontentofthisinequalityisthatforaprocessα1,α2,...,αNthereisaboundonafunctionalofthetrajectorieswhoseformisgivenentirelybythevaluesαtraversed.Theanalogywiththesecondlawiscompletedbyshowingthattheequalityholdsforaprocessthatcanbeconsideredtobeinthestationarydistributionatalltimes,sincethen:*Z∂φ(x;α)∂αdα+=Zdxdαe−φ(x;α)∂φ(x;α)∂α=Zdx[ρ(x;αN)−ρ(x;α1)]=0(1.8)Theseprocessesplaytheroleofreversibleprocessesfornon-drivensystems.Intheparticularcaseinwhichthestationarystatesforallαareequilibriumstates,wehavethatρSS(x;α)=e−βE(x,α)e−βF(α)φSS(x;α)=β(E(x,α)−Fα)(1.9)and(1.6)and(1.7)reducetoaformoftheJarzynskiequalityandthesecondlaw,respectively.2NOTES:•Thefluctuationrelationsarestatementsabouttheassymetryofthethedistri-butionofworkaroundzero,andnotaroundthemaximum.Theyinvolvenegativeworktailswhichareusuallyveryrare:heatflowingfromcoldtohotreservoirs,fluidsforcingrheometerstomovefaster,etc.Theseeventsbecomeallthemorerareinmacroscopicsystems,sincetheirprobabilityisexponentiallysmallinthesize.Thishasspurredfromthebeginningthesearch(withsomesuccess,see[10])of‘local’versionsofthefluctuationrelationinvolvingtheworkonasubsystem,whosefluctuationsaremoreeasilyobservable.•Similarly,theaverageinJarzynski’sequalityEq.(1.4)isdominatedbytra-jectorieshavingaverylowvalueofWwhichoccurwithextremelylowprobability–exponentiallysmallinthesystemsize.Thisisinstrictanalogywith‘annealed’averagesindisorderedsystems,whicharedominatedbyveryraresampleswithunusuallylowfreeenergy.•ThatthetransientfluctuationrelationandtheJarzynskiequalityareclosedrelativescanalreadybeseenbyconsideringaprocesswhereinitialandfinalexternalfieldscoincideα1=αN.InthatcaseF(αN)−F(α1)=0,andtheJarzynskirelationisaconsequenceofthefluctuationrelation,since:he−βWi=ZdWP(W)e−βW=ZdWP(−W)=1(1.10)OnecanalsodeduceJarzynski’srelationfromtheTransientFluctuationTheo-rembyconsideringthedistributionofworkdonebyaconservativeforceswitchedonatt=0,andusingthefactthattheinitialdistributionisGibbsean.Amoregeneralconnectionhasbeengiv