78C.JarzynskiSeminairePoincarewhereWdenotestheworkperformedbytheagen - PDF document

78C.JarzynskiSeminairePoincarewhereWdenotestheworkperformedbytheagentthatmanipulatesthepiston.Thisinequalityisnotmandatedbytheunderlyingdynamics:therecertainlyexistmicro-scopicallyviableN-particletrajectoriesforwhichW0.However,theprobabilitytoobservesuchtrajectoriesbecomesfantasticallysmallforlargeN.Bycontrast,fora\gas"ofonlyafewparticles,wewouldnotbesurprisedtoobserve{onceinararewhile,perhaps{anegativevalueofwork,thoughwestillexpectEq.1toholdonaverage:hWi&#x]TJ/;༖ ;.9;Ւ ;&#xTf 2;.52; 0 ;&#xTd [;0:(2)Theangularbracketshereandbelowdenoteanaverageovermanyrepetitionsofthishypotheticalprocess,withthetinysampleofgasre-equilibratedpriortoeachrepetition.Thisexamplesuggeststhefollowingperspective:asweapplythetoolsofther-modynamicstoeversmallersystems,thesecondlawbecomesincreasinglyblurred.InequalitiessuchasEq.1remaintrueonaverage,butstatistical uctuationsaroundtheaveragebecomeevermoreimportantasfewerdegreesoffreedomcomeintoplay.Thispicture,whilenotwrong,isincomplete.Itencouragesustodismissthe uctuationsinWasuninterestingnoisethatmerelyre ectspoorstatistics(smallN).Asitturnsout,these uctuationsthemselvessatisfyratherstrong,interestingandusefullaws.Forexample,Eq.2canbereplacedbytheequality,he�W=kBTi=1;(3)whereTisthetemperatureatwhichthegasisinitiallyequilibrated,andkBisBoltzmann'sconstant.Ifweadditionallyassumethatthepistonismanipulatedinatime-symmetricmanner,e.g.pushedinataconstantspeedandthenpulledoutatthesamespeed,thenthestatisticaldistributionofworkvalues(W)satisesthesymmetryrelation(+W) (�W)=eW=kBT:(4)Thevalidityoftheseresultsdependsneitheronthenumberofmoleculesinthegas,nor(surprisingly!)ontherateatwhichtheprocessisperformed.Ihaveusedthegasandpistonoutofconvenienceandfamiliarity,butthepredictionsillustratedherebyEqs.3and4{andexpressedmoregenerallybyEqs.15and30below{arenotspecictothisparticularexample.Theyapplytoanysystemthatisdrivenawayfromequilibriumbythevariationofmechanicalparameters,underrelativelystandardassumptionsregardingtheinitialequilibriumstateandthemicroscopicdynamics.Moreover,theybelongtoalargercollectionofrecentlyderivedtheoreticalpredictions,whichpertainto uctuationsofwork,[4,5,6,7,8,9]entropyproduction,[10,11,12,13,14,15,16,17,18]andotherquantities[19,20]insystemsfarfromthermalequilibrium.Whilethesepredictionsgobyvariousnames,bothdescriptiveandeponymous,theterm uctuationtheoremshascometoserveasausefullabelencompassingtheentirecollectionofresults.Thereisbynowalargebodyofliteratureon uctuationtheorems,includingreviewsandpedagogicaltreatments.[21,22,23,24,25,26,3,27,28,29,30,31,32,33,34,35,36,37,38]Inmyviewthesearenotresultsthatonemightnaturallyhaveobtained,bystar-tingwithasolidunderstandingofmacroscopicthermodynamicsandextrapolatingdowntosmallsystemsize.Rather,theyrevealgenuinelynew,nanoscalefeaturesofthesecondlaw.Myaiminthisreviewistoelaborateonthisassertion.Focusing 80C.JarzynskiSeminairePoincareterminalstates:W
FFB;T�FA;T(5)HereF;TdenotestheHelmholtzfreeenergyofthestate[;T].Whentheparameterisvariedslowlyenoughthatthesystemremainsinequilibriumwiththereservoiratalltimes,thentheprocessisreversibleandisothermal,andW=
F.Eq.5istheessentialstatementofthesecondlawofthermodynamicsthatwillapplyinSections3-6ofthisreview.Ofcourse,notallthermodynamicprocessesfallwithinthisparadigm,norisEq.5thebroadestformulationoftheClausiusin-equality.However,sincecompletegeneralitycanimpedeclarity,Iwillfocusontheclassofprocessesdescribedabove.Mostoftheresultspresentedinthefollowingsectionsapplyalsotomoregeneralthermodynamicprocesses{suchasthoseinvol-vingmultiplethermalreservoirsornonequilibriuminitialstates{asIwillbrie ymentioninSection7.Threecommentsarenowinorder,beforemovingdowntothenanoscale.(1)Asthesystemisdrivenawayfromequilibrium,itstemperaturemaychangeorbecomeill-dened.ThevariableT,however,willalwaysdenotetheinitialtem-peratureofthesystemandthermalreservoir.(2)Noexternalworkisperformedonthesystemduringthere-equilibrationstage,t,asisheldxed.Inthissensethere-equilibrationstageissomewhatsuper uous:Eq.5remainsvalidiftheprocessisconsideredtoendatt={evenifthesystemhasnotyetre-equilibratedwiththereservoir!{providedwealwaysdene
Ftobeafreeenergydierencebetweentheequilibriumstates[A;T]and[B;T].(3)Whileingeneralitispresumedthatthesystemremainsinthermalcontactwiththereservoirfor0t,theresultsdiscussedinthisreviewarealsovalidifthesystemisisolatedfromthereservoirduringthisinterval.2.2MicroscopicdenitionsofworkandfreeenergyNowletus\scaledown"thisparadigmtosmallsystems,withaneyetowardincorporatingstatistical uctuations.Consideraframeworkinwhichthesystemofinterestandthethermalreservoirarerepresentedasalargecollectionofmicrosco-pic,classicaldegreesoffreedom.Theworkparameterisanadditionalcoordinatedescribingthepositionororientationofabody{orsomeothermechanicalvariablesuchasthelocationofalasertrapinasingle-moleculemanipulationexperiment[27]{thatinteractswiththesystemofinterest,butiscontrolledbyanexternalagent.ThisframeworkisillustratedwithatoymodelinFig.1.Herethesystemofinterestconsistsofthethreeparticlesrepresentedasopencircles,whosecoordinateszigivedistancesfromthexedwall.Theworkparameteristhefourthparticle,depictedasashadedcircleatadistancefromthewall.Letthevectorxdenoteamicroscopicstateofthesystemofinterest,thatisthecongurationsandmomentaofitsmicroscopicdegreesoffreedom;andletysimilarlydenoteamicrostateofthethermalreservoir.TheHamiltonianforthiscollectionofclassicalvariablesisassumedtotaketheformH(x;y;)=H(x;)+Henv(y)+Hint(x;y)(6)whereH(x;)representstheenergyofthesystemofinterest{includingitsinterac-tionwiththeworkparameter{Henv(y)istheenergyofthethermalenvironment, Vol.XVLeTemps,2010IrreversibilityandtheSecondLawofThermodynamicsattheNanoscale83systemevolvinginthefullphasespace(x;y)underatime-dependentHamiltonianH(x;y;(t)).TheresultsdiscussedinSections3-6canallbeobtainedwithinthisframework.Alternatively,wecantreatthereservoirimplicitly,bywritingdowneectiveequationsofmotionforjustthesystemvariables,x.ExamplesincludeLan-gevindynamics,theMetropolisalgorithm,Nose-Hooverdynamicsanditsvariants,theAndersenthermostat,anddeterministicequationsbasedonGauss'sprincipleofleastconstraint.[23,34]AswiththeHamiltonianapproach,theresultsdiscus-sedbelowcanbederivedforeachofthesemodeldynamics.Thissuggeststhattheresultsthemselvesareratherrobust:theydonotdependsensitivelyonhowthemicroscopicdynamicsaremodeled.Sincetheaimofthisreviewistodescribewhatthesecondlawofthermodyna-mics\lookslike"inthepresenceof uctuations,full-blownderivationsof uctuationtheoremsandworkrelationswillnotbeprovided.However,inSections3and4,inadditiontodescribingvariousworkrelationsandtheirconnectionstothesecondlaw,IwillsketchhowseveralofthemcanbederivedforthetoysystemshowninFig.1,inthephysicalcontextmentionedbythenalcommentinSection2.1:thesystemisthermallyisolatedduringtheinterval0t.Theaimhereistoconveysomeideaofthetheoreticalfoundationsoftheseresults,withoutexploringthetechnicaldetailsthataccompanyanexplicittreatmentofthereservoir.[50]3Equilibriuminformationfromnonequilibrium uctuationsThermodynamicsaccustomsustotheideathatirreversibleprocessesaredescri-bedbyinequalities,suchasW
F.Oneofthesurprisesofrecentyearsisthatifwepayattentionto uctuations,thensuchrelationshipscanberecastasequalities.Inparticular,thenonequilibriumworkrelation[6,7]statesthathe�W=kBTi=e�
F=kBT;(15)where(asabove)Tistheinitialtemperatureofthesystemandthermalreservoir,andangularbracketsdenoteanensembleaverageoverrealizationsoftheprocess.Thisresulthasbeenderivedinvariousways,usinganassortmentofequationsofmotiontomodelthemicroscopicdynamics[6,7,8,17,18,9,51,52,50,53,54,55,56,57,58,59,60,61,62,63],andhasbeenconrmedexperimentally.[64,65,66,67]InthefollowingparagraphIwillsketchhowitcanbeobtainedforthetoymodelofFig.1.Imaginethatafterpreparingthesysteminequilibriumat=Awedisconnectitfromthethermalreservoir.Thenfromt=0tot=thethree-particlesystemofinterestevolvesundertheHamiltonianH(x;(t))(Eq.7)aswedisplacethefourthparticlefrom=AtoBusingaprotocol(t).Arealizationofthisprocessisdescribedbyatrajectoryxtx(t)obeyingHamilton'sequations.CombiningEq.9withtheidentitydH=dt=@H=@t(seeRef.[68],section8-2),wegetW=H(x;B)�H(x0;A).WethenevaluatetheleftsideofEq.15byaveragingoverinitialconditions,usingEq.10:he�W=kBTi=Zdx0peqA;T(x0)e�W=kBT=1 ZA;TZdx@x @x0�1e�H(x;B)=kBT=ZB;T ZA;T:(16) Vol.XVLeTemps,2010IrreversibilityandtheSecondLawofThermodynamicsattheNanoscale85asimmediatelyfromEq.15:[31]P[W
F�]Z
F��1dW(W)Z
F��1dW(W)e(
F��W)=kBTe(
F�)=kBTZ+1�1dW(W)e�W=kBT=e�=kBT(20)Here,Pistheprobabilitytoobserveavalueofworkthatfallsbelow
F�,whereisanarbitrarypositivevaluewithunitsofenergy.Eq.20tellsusthatthelefttailofthedistribution(W)becomesexponentiallysuppressedinthethermodynamicallyforbiddenregionW
F,abitliketheevanescentpieceofaquantum-mechanicalwavefunctioninaclassicallyforbiddenregion.ThuswehavenohopetoobserveavalueofworkthatfallsmuchmorethanafewkBTbelow
F.Thisisgratifyinglyconsistentwitheverydayexperience,whichteachesusnotonlythatthesecondlawissatisedonaverage,inthesenseofEq.19,butthatitisneverviolatedonamacroscopicscale.Forsucientlyslowvariationoftheworkparameter,thecentrallimittheoremsuggeststhat(W)isapproximatelyGaussian.InthiscaseEq.15implies[6]
F=hWi�2W 2kBT(21)where2Wisthevarianceoftheworkdistribution.Thisistheresultthatoneexpectsfromlinearresponsetheory.[73,74,75,76]BecauseEq15unequivocallyimpliesthathWi
F,itmightatrstglanceappearthatthisrepresentsamicroscopic,rst-principlesderivationofthesecondlaw,andthusclariesthemicroscopicoriginsofirreversibility.Thisisnotthecase,however.InallderivationsofEq.15andrelatedworkrelations(e.g.Eqs.25,30,31),thearrowoftimeiseectivelyinsertedbyhand.Specically,aquitespecialstatisticalstate(theBoltzmann-Gibbsdistribution,peq)isassumedtodescribethesystemataparticularinstantintime(t=0),andattentionisthenfocusedonthesystem'sevolutionatlatertimesonly(t�0).Ifinsteadtheevolutionofthesystemleadinguptotheequilibriumstateatt=0hadbeenconsidered,thenalltheinequalitiesassociatedwiththesecondlawwouldhavebeenobtained,butwiththeirsignsreversed.Thisemphasizestheimportanceofboundaryconditions(intime),andtouchesonthedeepconnectionbetweenirreversibilityandcausality[77,78,79].Gibbsalreadyrecognizedthatifoneacceptsaninitialequilibriumstategivenbypeq/e�H=kBT,thenvariousstatementsofthesecondlawfollowfrompropertiesofHamiltoniandynamics(seeChapterXIIIofRef.[80]).Similarresultscanbeobtainediftheinitialequilibriumstateisrepresentedbyanydistributionthatisadecreasingfunctionofenergy[81].Interestingly,however,foramicrocanonicalinitialdistribution,inequalitiesrelatedtothesecondlawofthermodynamicscanbeviolated,atleastforsystemswithonedegreeoffreedom[82,83].Letusnowreturntothepictureofourensembleasaswarmoftrajectories, Vol.XVLeTemps,2010IrreversibilityandtheSecondLawofThermodynamicsattheNanoscale87ratherthancomplexphases.Inthequantum-mechanicalcase,thesumoverpathsproducesasolutiontotheSchrodingerequation,whileherewegettheconstruc-tionofanequilibriumdistributionfromnonequilibriumtrajectories.HummerandSzabo[9]haveusedEq.25toderiveamethodofconstructinganequilibriumpoten-tialofmeanforce(afreeenergyprolealongareactioncoordinatethatdiersfromtheworkparameter)fromnonequilibriumdata.ThismethodhasbeenconrmedexperimentallybyBerkovichetal.[85]4MacroscopichysteresisandmicroscopicsymmetryThesecondlawofthermodynamicsismanifestednotonlybyinequalitiessuchasW
F,butalsobythetime-asymmetryinherenttoirreversibleprocesses.Hysteresisloopsneatlydepictthisasymmetry.Asanexample,imaginethatwera-pidlystretchanordinaryrubberband,thenafterasucientpausewecontractit,returningtotheinitialstate.ForthisprocesswegetaclassichysteresisloopbyplottingthetensionTversusthelengthLoftherubberband(Fig.2).Hysteresisconveystheideathatthestateoftherubberbandfollowsonepathduringthestret-chingstage,butreturnsalongadierentpathduringcontraction.Quantitatively,thesecondlawimpliesthattheenclosedareaisnon-negative,HTdL0.Similarconsiderationsapplytotheanalogousstretchingandcontractionofsinglemolecules[86],onlynowstatistical uctuationsbecomeimportant:theran-domjigglingsofthemoleculedierfromonerepetitionoftheprocesstothenext.Intheprevioussectionwesawthatwhen uctuationsaretakenintoaccount,therela-tionshipbetweenworkandfreeenergycanbeexpressedasanequalityratherthantheusualinequality.Thecentralmessageofthepresentsectionhasasimilarring:withanappropriateaccountingof uctuations,thetwobranchesofanirreversiblethermodynamiccycle(e.g.thestretchingandcontractionofthesinglemolecule)aredescribedbyunexpectedsymmetryrelations(Eqs.30,31)ratherthanexclusivelybyinherentasymmetry(Eqs.28,35).Todeveloptheseresults,itisusefultoimaginetwodistinctprocesses,designatedtheforwardandthereverseprocess.[8]TheforwardprocessistheonedenedinSec.2,inwhichtheworkparameterisvariedfromAtoBusingaprotocolF(t)(thesubscriptFhasbeenattachedasalabel).Duringthereverseprocess,isvariedfromBtoAusingthetime-reversedprotocol,R(t)=F(�t):(27)Atthestartofeachprocess,thesystemispreparedintheappropriateequilibriumstate,correspondingto=AorB,attemperatureT.Ifweperformthetwoprocessesinsequence,theforwardfollowedbythereverse,allowingthesystemtoequilibratewiththereservoirattheendofeachprocess,thenwehaveathermody-namiccyclethatexhibitshysteresis.TheClausiusinequalityappliesseparatelytoeachstage:�hWiR
FhWiF(28)where
Fisdenedasbefore(Eq.5)andthenotationnowspeciesseparateaveragesoverthetwoprocesses.Ofcourse,Eq.28impliesthattheaverageworkovertheentirecycleisnon-negative,hWiF+hWiR0:(29) Vol.XVLeTemps,2010IrreversibilityandtheSecondLawofThermodynamicsattheNanoscale91processes.WecanthenusetherelativeentropyD[PFjPR]toassignavaluetotheextenttowhichthesystem'sevolutionduringoneprocessdiersfromthatduringtheother.FromEq.31itfollowsthat[79]D[PFjPR]=WdissF kBT(38)whereWdissFhWiF�
F(39)istheaverageamountofworkthatisdissipatedduringtheforwardprocess.(Simi-larly,D[PRjPF]=WdissR=kBT.)Whiledistributionsintrajectoryspaceareabstractanddiculttovisualize,aresultsimilartoEq.38canbeplacedwithinthemorefamiliarsettingofphasespace.LetfF(x;t)denotethetime-dependentphasespacedensitydescribingtheevolutionofthesystemduringtheforwardprocess(Eq.22),anddenefR(x;t)analogouslyforthereverseprocess.ThenthedensitiesfF(x;t1)andfR(x;�t1)aresnapshotsofthestatisticalstateofthesystemduringthetwoprocesses,bothtakenatthemomenttheworkparameterachievesthevalue1F(t1)=R(�t1).TheinequalityfF(x;t1)6=fR(x;�t1)(40)thenexpressestheideathatthestatisticalstateofthesystemisdierentwhentheworkparameterpassesthroughthevalue1duringtheforwardprocess,thanwhenitreturnsthroughthesamevalueduringthereverseprocess.(Thereversalofmomentainxisrelatedtotheconjugatepairingoftrajectories,Eq.32.)Evaluatingtherelativeentropybetweenthesedistributions,Kawai,ParrondoandVandenBroeck[93]foundthatD[fFjfR]WdissF kBT;(41)wheretheargumentsofDarethedistributionsappearinginEq.40,foranychoiceof1.Thisbecomesanequalityifthesystemisisolatedfromthethermalenvironmentastheworkparameterisvariedduringeachprocess.AswithEq.38,weseethataninformation-theoreticmeasureofthedierencebetweentwodistributions,fFandfR,isrelatedtoaphysicalmeasureofdissipation,WdissF=kBT.Eqs.38and41arecloselyrelated.Thephase-spacedistributionfF=fF(x;t1)istheprojectionofthetrajectory-spacedistributionPF[ F]ontoasingle\timeslice",t=t1,andsimilarlyforfR.Sincetherelativeentropybetweentwodistributionsdecreaseswhentheyareprojectedontoasmallersetofvariables[91,93]{inthiscase,fromtrajectoryspacetophasespace{wehaveD[fFjfR]D[PFjPR]=WdissF kBT:(42)Intheabovediscussion,relativeentropyhasbeenusedtoquantifythedierencebetweentheforwardandreverseprocesses(hysteresis).Itcanequallywellbeusedtomeasurehowfarasystemisremovedfromequilibriumatagiveninstantintime,leadingagaintoalinkbetweenrelativeentropyanddissipatedwork(Eq.43below).FortheprocessintroducedinSec.2,letftf(x;t)denotethestatisticalstateofthesystemattimet,andletpeqtpeq(t);T(x)betheequilibriumstatecorresponding Vol.XVLeTemps,2010IrreversibilityandtheSecondLawofThermodynamicsattheNanoscale93Todiscussthispoint,itisconvenienttoconsiderahypotheticalguessinggame[79].ImaginethatIshowyouamovieinwhichyouobserveasystemundergoingathermodynamicprocessasisvariedfromAtoB.Yourtaskistoguesswhe-therthismoviedepictstheeventsintheorderinwhichtheyactuallyoccurred,orwhetherIhavelmedthereverseprocess(varyingfromBtoA)andamnow(deviously)showingyouthemovieofthatprocess,runbackward.InthespiritofaGedankenexperiment,assumethatthemoviegivesyoufullmicroscopicinformationaboutthesystem{youcantrackthemotionofeveryatom{andthatyouknowtheHamiltonianfunctionH(x;)andthevalue
F=FB;T�FA;T.Assumemoreoverthatinchoosingwhichprocesstoperform,I ippedafaircoin:heads=F,tails=R.Wecanformalizethistaskasanexerciseinstatisticalinference.[95]LetL(Fj )denotethelikelihoodthatthemovieisbeingshowninthecorrectdirection(i.e.thecoinlandedonheadsandtheforwardprocesswasperformed),giventhemicroscopictrajectory thatyouobserveinthemovie.Similarly,letL(Rj )denotethelikelihoodthatthereverseprocesswasinfactperformedandthemovieisnowbeingrunbackward.Sincethesearetheonlypossibilities,thelikelihoodsassociatedwiththetwohypotheses(F,R)sumtounity:L(Fj )+L(Rj )=1(46)NowletWdenotetheworkperformedonthesystem,forthetrajectorydepictedinthemovie.IfW�
F,thenthersthypothesis(F)isinagreementwiththeClausiusinequality,whilethesecondhypothesis(R)isnot;ifW
F,itistheotherwayaround.Thereforeforamacroscopicsystemthetaskiseasy,asthesignofW�
Fdeterminesthedirectionoftime'sarrow.Formally,L(Fj )=(W�
F)(47)where(
)istheunitstepfunction.ForamicroscopicsystemwemustallowforthepossibilitythatEq.5mightbeviolatednowandagain.Bayes'Theoremthenprovidestherighttoolforanalyzingthelikelihood:L(Fj )=P( jF)P(F) P( ):(48)HereP(F)isthepriorprobabilitythatIcarriedouttheforwardprocess,whichissimply1=2sinceI ippedafaircointomakemychoice,andP( jF)istheprobabilitytogeneratethetrajectory whenperformingtheforwardprocess;inthenotationofSec.4,thisisPF[ ].Finally,P( )is(eectively)anormalizationconstant.WritingtheanalogousformulaforL(Rj ),thencombiningthesewiththenormalizationconditionEq.46andinvokingEq.31,weget[97,98,31]L(Fj )=1 1+e�(W�
F)=kBT:(49)Thisresultquantiesyourabilitytodeterminethearrowoftimefromthetrajectorydepictedinthemovie.Theexpressionontherightisasmoothedstepfunction.IftheWsurpasses
FbymanyunitsofkBT,thenL(Fj )1andyoucansaywithhighcondencethatthemovieisbeingshowninthecorrectdirection,whileintheoppositecaseyoucanbeequallycondentthatthemovieisbeingrunbackward. Vol.XVLeTemps,2010IrreversibilityandtheSecondLawofThermodynamicsattheNanoscale95equilibrium.[99]Esposito,LindenbergandVandenBroeckhaverecentlyshownthatinthissituationthevalueof
Stotisequaltothestatisticalcorrelationthatdevelopsbetweenthesystemandthereservoirs,asmeasuredintermsofrelativeentropy.[100]TheHamiltonianframeworkisofteninconvenientforstudyingnonequilibriumsteadystates.Amongthemanytoolsthathavebeenintroducedforthetheoreticalanalysisandnumericalsimulationofsuchstates,Gaussianthermostats{thetermreferstoamethodofmodelingnonequilibriumsystemsbasedonGauss'sprincipleofleastconstraint[101]{haveplayedaprominentroleinrecentdevelopmentsinnonequilibriumthermodynamics.Theterm uctuationtheoremwasoriginallyappliedtoapropertyofentropyproduction,observedinnumericalinvestigationsofasheared uidsimulatedusingaGaussianthermostat[10,11,12,13].Since uctuationtheoremsforentropyproductionhavebeenreviewedelsewhere[21,22,24,29,30,32,33,35,36],IwilllimitmyselftoabriefsummaryofhowtheseresultsconnecttothoseofSections3-6.Thetransient uctuationtheoremofEvansandSearles[11]appliestoasystemthatevolvesfromaninitialstateofequilibriumtoanonequilibriumsteadystate.Lettingp(
s)denotetheprobabilitydistributionoftheentropyproduceduptoatime�0,itstatesthatp(+
s) p(�
s)=e
s=kB:(51)ThisisclearlysimilartoEq.30,exceptthatitpertainstoasinglethermodynamicprocess,ratherthanapairofprocesses(FandR).Eq.51impliesanintegrated uctuationtheorem,he�
s=kBi=1;(52)thatisentirelyanalogoustoEq.15,andfromthisweinturngetanaloguesofEqs.19and20:h
si0;P[
s�]e�=kB(53)Nowconsiderasystemthatisinanonequilibriumsteadystatefromthedistantpasttothedistantfuture,suchasa uidunderconstantshear[10],andlet
s=denotetheentropyproductionrate,time-averagedoverasingle,randomlysampledintervalofduration.Thesteady-state uctuationtheoremofGallavottiandCohen[12,13]assertsthattheprobabilitydistributionp()satiseslim!11 lnp(+) p(�)= kB:(54)Theintegratedformofthisresultis[21]lim!11 ln e�=kB=0;(55)wherethebracketsdenoteanaverageoverintervalsofduration,inthesteadystate.Formalmanipulationsthengiveushi0;lim!11 lnP[�]�;(56)whereP[�]istheprobabilitytoobserveatime-averagedentropyproductionratelessthan�,duringanintervalofduration.TheresemblancebetweenEqs.54 Vol.XVLeTemps,2010IrreversibilityandtheSecondLawofThermodynamicsattheNanoscale97Itremainstobeseenwhethertheunderstandingoffar-from-equilibrium uc-tuationsthathasbeengainedinrecentyearswillleadtotheformulationofaunied\thermodynamicsofsmallsystems",thatis,atheoreticalframeworkbasedonafewpropositions,comparabletoclassicalthermodynamics.Someprogress,however,hasbeenmadeinthisdirection.Forstochasticdynamics,Seifertandcolleagues[54,115,116,117,32]{buildingonearlierworkbySekimoto[118,37]{havedevelopedaformalisminwhichmicro-scopicanaloguesofallrelevantmacroscopicquantitiesarepreciselydened.Manyoftheresultsdiscussedinthisreviewfollownaturallywithinthisframework,andthishashelpedtoclarifytherelationsamongtheseresults.[32]EvansandSearles[22]havechampionedtheviewthat uctuationtheoremsaremostnaturallyunderstoodintermsofadissipationfunction, ,whosepropertiesare(byconstruction)inde-pendentofthedynamicsusedtomodelthesystemofinterest.Morerecently,GeandQian[119]haveproposedaunifyingframeworkforstochasticprocesses,inwhichboththeinformationentropy�RplnpandtherelativeentropyRpln(p=q)playkeyroles.References[32]and[119]makeaconnectiontoearliereortsbyOonoandPaniconi[120]todevelopa\steady-statethermodynamics"organizedaroundnone-quilibriumsteadystates.Whiletheoriginalgoalwasaphenomenologicaltheory,thederivationbyHatanoandSasaof uctuationtheoremsfortransitionsbetweensteadystates[19,121]hasencouragedamicroscopicapproachtothisproblem.[122,123]IntheabsenceofauniversalstatisticaldescriptionofsteadystatesanalogoustotheBoltzmann-Gibbsformula(Eq.10)thishasproventobehighlychallenging.Thisreviewhasfocusedexclusivelyonclassical uctuationtheoremsandworkrelations,butthequantumcaseisalsoofconsiderableinterest.Whilequantumversionsoftheseresultshavebeenstudiedforsometime[124,125,126,127],thepasttwotothreeyearshaveseenasurgeofinterestinthistopic[128,129,130,131,132,133,134,135,136,137,138,139,140,141,142].Quantummechanicsofcourseinvolvesprofoundissuesofinterpretation.Itcanbehopedthatintheprocessoftryingtospecifythequantum-mechanicaldenitionofwork[132],ordealingwithopenquantumsystems[131,137,138,139,140,141,142],oranalyzingexactlysolvablemodels[130,133,135,136],orproposingandultimatelyperformingexperimentstotestfar-from-equilibriumpredictions[134],importantinsightswillbegained.Applicationsofnonequilibriumworkrelationstothedetectionofquantumentanglement[143]andtocombinatorialoptimizationusingquantumannealing[144]haveveryrecentlybeenproposed.Finally,therehasbeenarekindledinterestinrecentyearsinthethermodyna-micsofinformation-processingsystemsandcloselyrelatedtopicssuchastheappa-rentparadoxofMaxwell'sdemon[145].Makinguseoftherelationsdescribedinthisreview,anumberofauthorshaveinvestigatedhownonequilibrium uctuationsandthesecondlawareaectedinsituationsinvolvinginformationprocessing,suchasoccurinthecontextofmemoryerasureandfeedbackcontrol.[146,147,148,149,150]AcknowledgmentsIgratefullyacknowledgenancialsupportfromtheNationalScienceFoundation(USA),underDMR-0906601.ThisarticlewillappearinAnnu.Rev.Cond.Mat.Phys. 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