Application range of Jarzynski’s equation for boundary-switc

PHYSICALREVIEWE77,042101 2008

ApplicationrangeofJarzynski’sequationforboundary-switchingprocesses

JaeyoungSung*

DepartmentofChemistry,Chung-AngUniversity,Seoul156-756,Korea

Received1October2007;revisedmanuscriptreceived23February2008;published2April2008 Jarzynski’sequation JE hasbeenknowntorelatefreeenergychangeofasystemtostatisticaldistributionofworkdoneonthesystemforanarbitraryprocess.Inthepresentwork,we rstestablishthevalidityconditionofJEforboundaryswitchingprocesses.ThevalidityconditionofJEisexaminedforanexampleofspontaneousirreversibleprocesses,forwhich,obviously,JEdoesnothold.We ndthatthefreeenergydifferencebetweentwocon gurationalstateswithdifferentphase-spacevolumecannotbecorrectlyestimatedbyJEforanyadiabaticboundaryswitchingprocess.DOI:10.1103/PhysRevE.77.042101

PACSnumber s :05.70.Ln,82.20.Wt,87.10. e

Freeenergyisoneofthecentralconceptsinthermody-namicsandstatisticalthermodynamics,whosequanti cationisofgreatinterestinmanyproblemsofscience.ItiswellestablishedinconventionalthermodynamicsthatfreeenergydifferenceFB FA FBA betweentwoequilibriumstates,AandB,ofasystemisequaltotheworkdoneonthesystemduringtheisothermalreversibletransitionprocessfromstateAtostateB.However,forothertransitionprocesses,therehadnotbeenanyquantitativerelationshipbetweenfreeen-ergydifferenceandworkbeforeJarzynskiproposedhisequationadecadeago.

Jarzynski’sequation JE relatesthedifference FBA =FB FA ofthefreeenergyofstateBfromthatofstateAtostatisticaldistributionPA→B W ofworkWdoneonthesystemduringanarbitrarytransitionprocessfromstateAtostateBby

exp FBA =

exp W PA→B W dW, 1

with beingtheinversetemperature 1 .AsEq. 1 suggeststhatfreeenergydifferencecouldbemeasuredfromtransitionprocessesotherthanthereversibleone,ithasdrawnmuchattention.JEwasrederivedforavarietyofmodelsystems 2–8 ,andveri edexperimentallyforasingleRNAstretch-ingprocess 9,10 .

AnexceptiontothistrendforJEwasCohenandMauzer-all’squestionaboutthecorrectnessofJEforageneralirre-versibleprocess,duringwhichdistributionofasystemdevi-atesfromtheBoltzmanndistributionandtemperatureisnotwellde ned 11,12 .Inresponsetothecriticism,JarzynskipresentedanotherderivationofEq. 1 andkepthisassertionthat,ifaninitialstateAofthesystemisathermalequilib-riumstatewithtemperature 1,Eq. 1 holdsforirreversibleprocessesaswellasreversibleoneseventhoughthetem-peratureofthesystemisnotwellde nedordeviatesfrom 1duringthedynamics 13 .Afterward,whiledif cultyorinef ciencyofitspracticalapplicationhasbeennoted 14–16 ,JEhasseemedtobeacceptedasageneralequationthatholdsforanysystemundergoinganarbitraryprocess 6–8,17–26 .

*Jaeyoung@cau.ac.kr

1539-3755/2008/77 4 /042101 4

However,recently,itwasshownforanexactlysolvablemodelthatpredictionofEq. 1 forfreeenergydifferenceisdependentontheshapeofthetransitionpathwhereasfreeenergydifferencecannotbe 27 .Thereforeitisnowanim-portantissuetode nethevalidityconditionofJE.Inthiswork,we rstestablishthevalidityconditionofJEforanadiabaticboundaryswitchingprocess.Recently,PresseandSilbeydiscussedimportantissuesinpracticalapplicabilityofJEtomacroscopicsystemsforthecasewhereJEisformallycorrect 16 .Incomparison,thequestionweaddresshereisunderwhatconditionJEisformallycorrectforaboundaryswitchingprocess.

InJarzynski’sderivationsofJEforaparameterswitchingprocess 1,13 ,itisimplicitlyassumedthatboundarycondi-tionsimposedonasystemremainintactfromtheparameterswitchingprocess.Nevertheless,de ningtheapplicationrangeofJEforaboundaryswitchingprocessseemsneces-sarybecauseweareofteninterestedinfreeenergydifferencebetweentwoequilibriumstatesde nedbydifferentbound-aryconditions.WhenweidentifythecanonicalequilibriumstateofasystembytemperatureTandotherstateparametersR,thelatterstateparametersRoftenrepresentaboundaryconditionoraconstraintimposedonmicroscopicvariablesofthesystemintheequilibriumstate.Forexample,theca-nonicalequilibriumstateofasystemofgasparticleswhosepositionvectorsrjsatisfytheconstraintrj V,withVbeingavolumecon ningthegassystem,mayberepresentedby T,V .Theequilibriumcon gurationalstateofachainpoly-mercomposedofonlythosemicroscopiccon gurationalstatesofthechainpolymersatisfyingconstraint ri rf =RETEwithriandrfbeingthepositionvectorsoftheinitialandthe nalunitsofthechainpolymermaybeidenti edby T,RETE c.Additionalexamplesforequilibriumstatesiden-ti edbyboundaryconditionsincludeequilibriumcon gura-tionalstatesofamolecularpairidenti edbythepairsepa-ration,equilibriumcon gurationalstatesofabiopolymeridenti edbyitsradiusofgyration,andsoon.Intheseex-amples,boundaryconditionsofsystemsaredependentonvaluesofstateparametersofthesystems.

InthecasewherethesysteminstateAissubjecttoaboundaryconditiondifferentfromthatofthesysteminstateB,thephase-spacedomain eq A accessibletomicroscopicstatesofthesysteminstateAisdifferentfrom eq B ofthesysteminstateB.We ndthat FBApredictedbyEq. 1 iscorrectforanadiabaticboundaryswitchingprocessonlyifthelattertransformsthephase-spacedomain eq A ofthe

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042101-1

BRIEFREPORTSPHYSICALREVIEWE77,042101 2008

initialequilibriumstateAinto eq B ofthe nalequilibriumstateBattheveryendoftheadiabaticswitchingprocess.Thevalidityconditionisdiscussedfortwoadiabaticexpan-sionprocessesofanidealgassystem.TheLiouvilletheoremindicatesthatthevalidityconditioncannotbesatis edforanyadiabaticboundaryswitchingprocesswhenJEisappliedtoestimatethefreeenergydifferencebetweencon gura-tionalstateswithdifferentphase-spacevolumesfromeachother.

Freeenergydifference F10betweentwoequilibriumstates, T,R=R0 and T,R=R1 ,isgivenby

stateofoursystemattimetevolvedfrominitialmicroscopicstate 0forthegivenadiabaticprocess.Theworkdoneonthesystemwithinitialstate 0duringtheadiabaticprocessintimeinterval 0,tS isgivenby

W 0 =H1 tS 0 H0 0 .

4

exp F10 =

eq R1

d exp H1

.

2

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