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The McLennan-Zubarev steady state distribution and fluctuation theorems MitsusadaM.Sano GraduateSchoolofHumanandEnvironmentalStudies, KyotoUniversity,Yoshida-Nihonmatsu-cho,Sakyo,Kyoto,606-8501,Japan 7 1 0 2 n a Abstract J 6 TheMcLennan-Zubarevsteadystatedistributionisstudiedintheconnectionwithfluctuationtheorems.Wederivethe 1 McLennan-Zubarevsteadystatedistributionfromthenonequilibriumdetailedbalancerelation.Then,consideringthe cumulantfunctionor cumulantfunctional,two fluctuationtheoremsforentropyand forcurrentsare proved. Using ] h thefluctuationtheoremforcurrents,thecurrentisexpandedintermsofthermodynamicforces.Inthelowestorderof c thethermodynamicforce,wefindthatthetransportcoefficientsatisfiestheOnsager’sreciprocalrelation. Inthenext e order,wederivedthecorrectiontermtotheGreen-Kuboformula. m - Keywords: Nonequilibriumsteadystate,McLennan-Zubarevsteadystatedistribution,Fluctuationtheorems t PACS:05.40.-a,05.60.-k,05.90.+m,05.20.-y a t s . t a 1. Introduction m - Systemsoutequilibriumare filledin thenature, forinstance,physical, chemical,andbiologicalphenomena. In d anisolatedsystem,anonequilibriumstatespontaneouslyrelaxestothermodynamicalequilibrium.Ontheotherhand, n a nonequilibrium steady state (NESS) is maintained or sustained by flows of energy and matter from the outside o c of a given system. Macroscopically, the fluxes and the entropy production are constant in the NESS. However, [ microscopically speaking, the fluxes and the entropy production fluctuate around macroscopic steady values. To elucidate the nature of these fluctuations is an issue of nonequilibrium statistical physics in last two decades, for 1 v instance,inrefs.[1,2,3,4,5,6,7,8,9,10,11,12,13]. 7 Theentropyproductionis themostinterestingquantityinthe NESS, since itmeasuresirreversibilityofa given 8 system.Thefluctuationoftheentropyproductionhasasymmetry,thatis,intheNESS,thelogarithmoftheprobability 4 ratiooftheentropyproductionisrelatedtotheentropyproductionitself. 4 0 1 P (σ) . lim ln τ =σ, (1) 1 τ→∞τ Pτ(−σ) 0 7 where σ is the entropy production and P (σ) is the probability that the system exhibits the entropy production σ τ 1 in time interval τ. This relation is called the fluctuation theorem, which was first found in numerical data of the : v simulations of the Nose´-Hooverthermostat systems[1]. After that, a carefulexaminationusing dynamicalsystems i theoryandergodictheory,haselucidatedthephenomenaobserved[2,3]. Asaresult,itisshownthatthefluctuation X oftheentropyproductionisgovernedbytimereversalsymmetryofthesystem. Theprobabilityratioineq.(1)isa r a ratio of probabilitiesof forwardpath and its time reversedpath. Thefluctuationtheoremis the consequenceof the timereversalsymmetryinthesteadystate. Asanotheraspect, thefluctuationtheoremaffectstransportphenomena. Inref.[4],itwasformulatedasanextensionofOnsager’sreciprocityrelation. Furthermore,thefluctuationtheorem is not restricted to dynamical systems, and was also confirmed for the Langevin system[5], for general stochastic systems[6],andforthemasterequation[10,13]. Inref.[6],forstochasticsystems,thefluctuationtheoremisrewritten intermsofthe cumulantgeneratingfunctionµ(λ), i.e., µ(λ) = µ(1−λ). We alsocallthissymmetrythefluctuation theoremfortheentropyproductionorLebowitz-Spohnsymmetry.Usingthenonequilibriumdetailedbalancerelation, microscopicderivationsofthefluctuationtheoremwere attemptedfora stochasticsystem[7]andfora Hamiltonian PreprintsubmittedtoPhysicaA January18,2017 system[8]. Fordynamicalsystems, theproofofthe fluctuationtheoremis still notin a satisfactoryformexceptthe caseoftheGaussianthermostatsystemsorNose´-Hooversystems. HereweemphasizethatwetreattheHamiltonian dynamicalsystems,notstochasticsystems.WeovercomedeficiencyofmathematicalrigorinJarzynski’streatment[8]. Asweshallsee,thedeficiencyisduetothefactthathedoesnotknowthesteadystatedistribution. Inaddition,our formulationiscloselyrelatedtotheworkbyLebowitzandSpohn[6]. Theydonotderivethefluctuationtheoremin thesection7, entitled”Stochasticandthermostattingheatreservoirs”,of [6]. Ourresultsmaybewhattheytriedto attempt. Independentfromthefluctuationtheorem,thederivationofthesteadystatedistributioninaformlikeacanonical ensemblewas oneof themesin nonequilibriumstatistical physics. Itwasderivedby McLennan[14, 15, 16] andby Zubarev[17, 18]. Nowtheir steadystate distributionis calledthe McLennan-Zubarevsteadystate distribution. The McLennan-Zubarevsteadystatedistributionwasveryformallyderived. Forstochasticsystems(themasterequation andtheLangevinequation),themeaningoftheMcLennan-Zubarevsteadystatedistributionisinvestigatedbyseveral authors[19,20, 21]. Intheseworks, thesteadystate distributionnearequilibriumiswrittenina similarformto the McLennan-Zubarevsteadystatedistribution. Thus,theMcLennan-Zubarevsteadystatedistributionisconsideredas an approximationof the steady state distribution,tentatively. However,in recentyears, in the considerationsof the steady state thermodynamics[22, 23, 24, 25], the McLennan-Zubarevsteady state distribution was derived through anotherroute, i.e., the nonequilibriumdetailedbalancerelation, which is closelyrelated to the fluctuationtheorem. In addition, the McLennan-Zubarevsteady state distribution was also derived for quantum systems[26, 27]. These observationsstronglystimulateustoreconsidertheMcLennan-Zubarevsteadystatedistributioninaconnectionwith thefluctuationtheorem. TheaimofthispaperistoderivetheMcLennan-Zubarevsteadystatedistributionfromthenonequilibriumdetailed balancerelation, which is a key relationto derivethe fluctuationtheorem, and, in addition, to provethe fluctuation theoremsforthe entropyproductionandforthe currents,by usingthederivedMcLennan-Zubarevsteadystate distribution. Thus, we establish an exactconnectionbetween the McLennan-Zubarevsteady state distribution and the fluctuationtheorems.WeconfirmthattheMcLennan-Zubarevsteadystatedistributionisexact,asfarasweadmitthe nonequilibriumdetailed balance relation. For stochastic Markoviandynamics, the nonequilibriumdetailed balance relationisjustified[28]. Unfortunately,however,wecannotjustifythenonequilibriumdetailedbalancerelationinour Hamiltonianformulation.Thisrelationisanassumptionmoreorless. Webelievethattheassumptionofthisrelation holds. As shown later, the nonequilibriumdetailed balance relation is strongly related not only to the McLennan- Zubarevsteadystatedistribution,butalsotothefluctuationtheorems. Thenonequilibriumdetailedbalancerelation, eq.(53),impliesallweneed,i.e.,McLennan-Zubarevsteadystatedistributionandtwofluctuationtheorems. Asdone for the master equation[11, 12], we obtain correction terms to the Green-Kubo formula, that is, the non-linear response,byusingthecumulantgeneratingfunctionalforthecurrents. Itissurethatthederivedcorrectionsareuseful forinvestigationsonthetransportproperty. Thispaperisorganizedasfollows.In§2,someknownresultsontheMcLennan-Zubarevsteadystatedistribution are summarized. In § 3, after a preparationof the setting and notations, the McLennan-Zubarevsteady state distri- butionisderivedfromthenonequilibriumdetailedbalancerelation. In§4, usinga similardiscussionto thatin the previoussection,thefluctuationtheoremfortheentropyproductionisproved. In§5,thefluctuationtheoremforthe currentsis proved. In § 6, the nonequilibriumdetailed balance relation is argued. In § 7, we investigate the mean current. The mean current is expressed in a power series of thermodynamicforces. We derive the linear response (i.e.,theGreen-Kuboformula)andthenon-linearresponseinthelowestorder.Asaby-product,weobtainnon-trivial relationsofthecurrentcorrelationfunctions.Thesenon-trivialrelationsareakindofrepresentationsoftime-reversal symmetry.In§8,wesummarizetheresultsofthispaper. 2. TheMcLennan-Zubarevsteadystatedistribution Inthissection,wesummarizeasettingupfordefiningtheMcLennan-Zubarevsteadystatedistribution.Thebasic facts on the McLennan-Zubarev steady state distribution are in [18] in details. The derivation of the McLennan- Zubarevsteadystatedistribution,whichisnotthederivationin[18],willbeshowninthenextsection. First,wesummarizeknownresultsofnonequilibriumthermodynamics.Weconsideranisotropicone-component 2 fluid.Theentropydensityssatisfiesthefollowingentropybalanceequation[29,30]. ∂s +div(j )=σ[s], (2) ∂t s wherej istheentropycurrent s Q µ∗j j = su+ − . (3) s T T HereT isthetempe[6]. rature. uisthelocalfluidvelocity. Qistheheatcurrent. jisthediffusioncurrent. Thefirst termoftherighthandsideofeq.(3)istheentropycurrentofthefluid. Thesecondtermistheentropycurrentofthe energyflow.Thethirdtermistheentropycurrentofthediffusion.µisthechemicalpotential. Weset u2 µ∗ =µ− . (4) 2 Theenergycurrentj isgivenby ε j =Q−u:σ′. (5) ε σ′istheviscousstresstensor. ThestresstensorTisrelatedtotheviscousstresstensorasfollows. T =−Pδ +σ′. (6) ij ij ij Pisthehydrostaticpressure.Theproductofvectorandtensoru:σ′is (u:σ′) = u σ′. (7) i j ji Xj Theentropyproductionrateisgivenby 1 u µ∗ σ[s]=j ·∇ +(−σ′):∇ − +j·∇ − . (8) ε T! (cid:18) T(cid:19) T ! Thenotationofthetensorproductis u ∂ u (−σ′):∇ − = (−σ′) − i . (9) (cid:18) T(cid:19) Xi,j ij ∂xj (cid:18) T(cid:19) From the expression of σ[s], we see that the entropy production rate is (the irreversible part of the current of the conservedquantity)×(thegradientoftheintensiveparameter). Eachtermisrelatedtotheenergyconservationlaw, themomentumconservationlaw,andtheparticlenumberconservationlaw. Thisfactisimportantwhenweconsider theentropyproductionrateofagivensystemnotonlyinamacroscopicdescription,butalsomicroscopicdescription. Now we turn to microscopic description of a one-component fluid system. Consider a system with N point particleswithmassm. TheinteractionpotentialisgivenbyV(|q −q |). TheHamiltonianis i j N p2 N H = i + V(|q −q |). (10) i j 2m Xn=1 Xi<j Theequationsofmotionis dq ∂H dp ∂H i = , i =− . (11) dt ∂p dt ∂q i i Inthispaper,weusethefollowingnotationforthevariablesofpositionandmomentum.Γ=(q ,q ,...,q ,p ,p ,...,p ). 1 2 N 1 2 N TheHamiltonianH isafunctionof{q}N and{p}N . TheHamiltonianmaywrittenasH = H({q},{p}). Symmetry i i=1 i i=1 i i whichthesystempossessesisimportant.ThisHamiltonianhasasymmetry. H({q},{p})= H({q},−{p}). (12) i i i i 3 Time-reversalsymmetryoforbitscanbewrittenas q (t)=q(−t), p (t)=−p(−t). (13) i,R i i,R i FortheorbitΓ(t),thetime-reversedorbitiswrittenasΓ (t). R Thissystemhasthreeconservationlaws. (1)Particlenumberconservationlaw: ∂n(Γ,x) +div(j(Γ,x))=0. (14) ∂t n(Γ,x)istheparticledensity.j(Γ,x)isthecurrentdensityofparticles. (2)Energyconservationlaw: ∂H(Γ,x) +div(J (Γ,x))=0. (15) ∂t H H(Γ,x)istheenergydensity. J (Γ,x)istheenergycurrentdensity.(3)Momentumconservationlaw: H ∂p(Γ,x) +div(T(Γ,x))=0. (16) ∂t T(Γ,x) is the momentum tensor. Here the functions appeared in the above conservation laws are the density of particles, N n(Γ,x)= δ(q −x), (17) i Xi=1 themomentumdensity, N p(Γ,x)= pδ(q −x), (18) i i Xi=1 andthecurrentofparticles, N p j(Γ,x;t)= iδ(q −x). (19) i m Xi=1 Inordertocaptureheat,wedefinetheenergydensityisgivenby N p2 1 H(Γ,x) =  i + V(|qi−qj|)δ(qi−x). (20) Xi=1 2m 2Xj,i  Forheatcurrent,theenergycurrentisdefinedas   N p2 1 p 1 J (Γ,x;t) = i + V(|q −q |) iδ(q −x)+ ((p +p )·F )(q −q )δ(q −x). (21) H 2m 2 i j  m i 4m i j ij i j i Xi  Xj,i  Xj,i ThemomentumtensorT(Γ,x;t)is N 1 1 T (Γ,x;t) = (q −q ) (F ) δ(q −x)+ (p) (p) δ(q −x), (22) βα i j β ij α i i β i α i 2 m Xi,j Xi=1 Heretheforceis ∂ F =− V(|q −q |). (23) ij ∂q i j i F (q −q )istheforce,whichthe j-thparticleactstothei-thparticle ij i j 4 As a result, the above three conservation law implies three phenomena, i.e., thermal conduction, momentum diffusion,anddiffusion.Herewedefinethreecurrentscorrespondingtothethermalconduction,momentumdiffusion, anddiffusion. j (Γ,x;t) = J (Γ,x;t) (24) 0 H j (Γ,x;t) = T(Γ,x;t) (25) 1 j (Γ,x;t) = j(Γ,x;t). (26) 2 Nextwedefinethreequantities. P (Γ,x) = H(Γ,x) (27) 0 P (Γ,x) = p(Γ,x) (28) 1 P (Γ,x) = n(Γ,x). (29) 2 Hereµ∗isthechemicalpotential,whichisdefinedas mu2 µ∗ =µ− , (30) 2 whereuisthevelocityfield. Ifthevelocityfieldexists,thechemicalpotentialwouldbesubtractedbytheamountof thevelocityfield. β(x)istheinversetemperaturedividedbytheBoltzmannconstant. 1 β(x)= , (31) k T(x) B HereT(x)isthetemperaturefield. Wealsodefinethefollowingquantities. F (x) = β(x) (32) 0 F (x) = −β(x)u(x) (33) 1 F (x) = −β(x)µ∗(x). (34) 2 ThethermodynamicforcesX =∇F aredefinedas i i X (x) = ∇β(x) (35) 0 X (x) = −∇(β(x)u) (36) 1 X (x) = −∇(β(x)µ∗). (37) 2 Forconvenience,wedefineavector,whichcombinesthreethermodynamicforces. X (x) 0 X(x)= X1(x) . (38)  X2(x)  Innonequilibriumthermodynamics,thelocalentropyproductionrateσ[s]isdefinedaseq.(8).Nowwedefinethe microscopiclocalentropyproductionrateσ[Γ,x]forourparticlesystem. σ[Γ,x] = j (Γ,x;t)·X (x) m m Xm = J (Γ,x;t)·∇β(x)+T(Γ,x;t):∇(−β(x)u)+j(Γ,x;t)·∇(−β(x)µ∗). (39) H Preciselyspeaking,thisquantityisthelocalentropyproductionratedividedbytheBoltzmannconstantk . Equation B (39)isafirstkeyassumptioninthispaper. Otherassumptionsarethenonequilibriumdetailedbalancerelation,and thetime-reversalsymmetry,whichwillbeexplainedlater. Thesethreeassumptionsarethekeyrelationsinthispaper. 5 Thelocalequilibriumstateischaracterizedbyalocalequilibriumdistribution.Thatis ρ (Γ)=Z−1exp − dx F (x)P (Γ,x) , (40) ℓ 0  Z m m   Xm  where Z = dΓexp − dx F (x)P (Γ,x) . (41) 0 Z  Z m m  ℓstandsforlocalequilibrium.ThisdistributiondoesnoXtmdescribeasteadystate.Toobtainthesteadystatedistribution, we should treat the deviation from the local equilibrium. The deviation includes the time-integral of the entropy productionrate. Here we “simply” write down the McLennan-Zubarevsteady state distribution which appeared in the book of Zubarev[18].Bytheabovesetting,theMcLennan-Zubarevsteadystatedistributionisgivenasfollows[14,15,16,17, 18]. 0 ρ (Γ) = Z−1exp − dxF (x)P (Γ,x)+ dx dtj (Γ,x;t)·X (x) . (42) “ss”meansthesstseadystate. HereΓisΓXm=Z(q ,q ,m...,qm,p ,p ,.X.m.,Zp ). ZZi−s∞thenmormalizationmconstant. 1 2 N 1 2 N 0 Z = dΓ exp − dxF (x)P (Γ,x)+ dx dtj (Γ,x;t)·X (x) . (43) Z  Xm Z m m Xm Z Z−∞ m m  j (Γ,x;t) is theenergycurrentdensity(m = 0), the momentumcurrentdensity (m = 1), andthe currentdensityof m the particle number (m = 2). It might be that an explanation on the notations is needed here. The coordinates of Γ and x is the coordinateof the orbitsand of the density, respectively. j (Γ,x;t) is the density of the current. The m McLennan-Zubarevsteadystatedistributionisnearthelocalequilibriumstate,eq.(40). Thedeviationfromthelocal equilibriumis 0 dx dtj (Γ,x;t)·X (x). (44) m m Xm Z Z−∞ This term of eq. (42) expresses the total amount of the entropy production for a given orbit. ρ (Γ) has a physical ss meaningthatatinfinitepast(t = −∞),thesystemisinalocalequilibrium. Thenafter,uptothepresent(t = 0),the systemproducesthewholeoftheentropyproduction.Thisamountofthetotalentropyproductionisintheexponential function. 3. DerivationoftheMcLennan-Zubarevsteadystatedistribution Inthissection,wederivetheMcLennan-Zubarevsteadystatedistributionfromthenonequilibriumdetailedbal- ancerelation.AssumptionsareneededtoderivetheMcLennan-Zubarevsteadystatedistribution:(a)theentropypro- ductionrate,eq.(39),(b)thetime-reversibility,eq.(57),and(c)thenonequilibriumdetailedbalancerelation,eq.(59). ThenonequilibriumdetailedbalancerelationisneededtoderivetheMcLennan-Zubarevsteadystatedistributionand thefluctuationtheorems.Thetimereversibilityisessentialtothefluctuationtheorems.Beforestartingthederivation, weprepareasettingandnotationsforlateruse. 3.1. Preparation Inthissubsection,wepresentthenotationsandthetime-reversalsymmetry,whichareimportantforourformula- tion. HerewedonotfollowMcLennan’sderivation,butwillpresentanotherderivationbyusingthenonequilibrium detailedbalancerelation.TheMcLennan’sderivationispresentedinAppendixAforcomparison. Considerasystemwhichconsistsofparticles. Thetotalsystemhasthesystemandtwobaths. TheHamiltonian ofthetotalsystemis H = H +H +U. (45) tot S B 6 Bath 1 Bath 2 System 1 1 2 2 ( ) ( ) 1 2 ( ) ( ) 1 2 Figure1:Theconfigurationofthesystemandthebaths:Thesystemissandwichedbetweenthebaths1and2,whosetemperaturesandchemical potentialsaredifferent(T1 , T2,µ1 , µ2). Hamiltonians HS,Hi(B),andHi(int)(i = 1,2)describethesystem,thei-thbath,andtheinteraction betweenthesystemandthei-thbath,respectively H istheHamiltonianofthesystem. H istheHamiltonianofthebaths. U istheinteractionbetweenthesystemand S B thebaths.TheconfigurationofthewholesystemisdepictedinFig.1. Thebathiisinequilibriumandhastemperature T (i = 1,2), and has chemical potential µ (i = 1,2). The system (the bath i) has N (N) particles, respectively, i i s i N = N +N +N . ThetotalsystemhasN particles. All N particlesareidentical. Imaginethatthesystemisunder s 1 2 theheatconductionandtheparticleflowwiththisnonequilibriumboundarycondition.WedefinetheHamiltonianof thetotalsystemasfollows. H = H (Γ ), (46) S S s 2 H = H(B)(Γ(i)), (47) B i b Xi=1 2 U = H(int)(Γ(i),Γ ). (48) i b s Xi=1 Γ is the coordinates and momenta of the system. Γ(i) is the coordinates and momenta of the bath i. H(B) is the s b i Hamiltonianofthebathi. H(int) representstheinteractionbetweenthesystemandthebathi. Thestateofthesystem i andthebathsisdeterminedbythecoordinatesandthemomenta. Γ =(q ,···,q ,p ,···,p ), (49) s 1,s Ns,s 1,s Ns,s Γ(i) =(q ,···,q ,p ,···,p ). (50) b 1,i Ni,i 1,i Ni,i TheHamiltoniansaredefinedas Ns |p |2 H (Γ )= j,s +V (q ,···,q ), (51) S s 2m s 1,s Ns,s Xj=1 Ni |p |2 H(B)(Γ(i))= j,b +V(q ,···,q ), (52) i b 2m i 1,i Ni,i Xj=1 H(int)(Γ(i),Γ )=U (q ,···,q ,q ,···,q ). (53) i b s i,s 1,i Ni,i 1,s Ns,s WewritedΓ = Ns dq dp anddΓ(i) = Ni dq dp . Itisclearthatthesystemisdrivenoutofequilibriumwith s i=1 i,s i,s b j=1 j,i j,i thenonequilibriuQmboundaryconditionsT1Q,T2andµ1 ,µ2.Inthesystem,theheatconductionandtheparticleflow areachieved. 7 The time-reversaloperationis essentialto the considerationof thefluctuationtheorem. Flippingthesign of the momentaisabasicoperation.Wedefinethe∗-operationby Γ∗ =(q ,q ,...,q ,−p ,−p ,...,−p ). (54) s 1,s 2,s Ns,s 1,s 2,s Ns,s The“whole”systemconsideredinthispaperisaHamiltoniandynamicalsystem.Butthesystemisacertaindynamical systemwithdissipationbytheexistenceofthebathsorreservoirs.TheHamiltoniandoesnotcontainthemagneticfield andhasthesymmetryofeq.(12).WedenotethewholephasespacebyS. SisS=Ss×Ss×Sb(1)×Sb(1)×Sb(2)×Sb(2) = c m c m c m S ×S ,whereSs(Sb(i))isthecoordinatespaceofthesystem(thebathi),respectively.andSs (Sb(i))isthemomentum s b c c m m spaceofthesystem(thebathi),respectively.WedenotetheflowofthiswholesystembyΦ. Ifweusethenotationof Γ(t)=(Γ ,Γ(1),Γ(2))forthestate,theflowactsas s b b Γ(t+τ)=ΦτΓ(t). (55) Wecanwritethetime-evolutionofthedistributionusingthePerron-Frobeniusoperator. ρ(γ;t+τ)= dΓδ(ΦτΓ−γ)ρ(Γ;t). (56) Z We call the delta function in the above equation the integralkernel of the Perron-Frobeniusoperator. This type of deltafunctionswillbefrequentlyusedlater. Inthispaper,wemayfixτinΦτ. Then,aftersomemanipulations,we shalltakethelimitτ→ ∞. Thus,itisconvenienttodefinethemapT asT = Φτ. Bythemapwhichthewholemap T isrestrictedtothesystem,thephasespacevolumeelementofthesystemcontracts. ThemapT satisfiesthefollowingtimereversalsymmetry. (T(Γ))∗ =T−1(Γ∗), (57) foranyΓ∈S.ThissymmetrywillbeusedlaterandisthesecondkeyrelationtoderivetheMcLennan-Zubarevsteady statedistributionandtoprovethefluctuationtheorems. 3.2. Derivation We denotetheentropyproductioncontributionfromaspecificpathν → γ, byΣ[ν → γ],whereν,γ ∈ S . This s quantityisgivenby 0 Σ[ν→γ]= dx dtj (Γ ,x;t)·X (x), (58) m s m Xm Z Z−τ where γ = (T(Γ)) with Γ = ν. The integral of x is taken over the system. Hereafter without specifying, dx s s represents the integral over the system, not over the whole system. The third key relation in our derivation isRthe nonequilibrium detailed balance relation. For various derivations of the fluctuation theorem, the nonequilibrium detailedbalancerelationwasused[7]. Thenonequilibriumdetailedbalancerelationforourcaseisgivenby P[ν→γ]=eΣ[ν→γ]P[γ∗ →ν∗], (59) whereP[ν → γ]isthekernelofthePerron-Frobeniusoperatorwhichcastsfromthestateνtothestateγ. Sincewe consideradynamicalsystem,thefunctionP[ν→γ]isgivenbyadeltafunction. P[ν→γ]=δ((T(Γ)) −γ). (60) s Equation (59) is our assumption, since we cannot derive eq. (59) from the equations of motion by first principle. However,thenonequilibriumdetailedbalancerelationgivesustheconnectionbetweenentropyproductionrateand thestabilityofthesystem[2,3]. Thiscanbeeasilyconfirmedbyrewritingδ-functionandintegratingeq.(59)with respecttoν=Γ . Weget s ∂(T(Γ)) −1 s =eΣ[ν→γ]. (61) (cid:12)(cid:12) ∂Γs (cid:12)(cid:12) (cid:12) (cid:12) (cid:12)(cid:12) (cid:12)(cid:12)8 Thus, the entropyproductionis directlyrelated to the stability of a givensystem. Here we set ν = Γ , Γ′ = T(Γ), s γ=(Γ′) ,andΓ′′ =T−1(Γ). s Letusconsidertimeevolutionofagivenstate. Weassumethattheinitialstateisνandtheinitialdistributionisin acanonicalstate,i.e.,alocalequilibriumstate. ThiscanberealizedbydefiningthedistributionP (Γ ) ν s P (Γ )=ρ (Γ ), (62) ν s ℓ s where ρ (Γ)=exp − dx{β(x)H(Γ,x)−β(x)u(x)·p(Γ,x)−β(x)n(Γ,x)µ∗(x)} . (63) ℓ " Z # In P (Γ ), ν representsthe initialstate. After time evolutionduringtime intervalτ, the system will be foundin the ν s stateγ∈S withtheprobability s ρ (γ)= dΓ P (Γ )δ((T(Γ)) −γ). (64) ν s ν s s Z LetusstartthederivationoftheMcLennan-Zubarevsteadystatedistribution. Usingthenonequilibriumdetailed balancerelation,eq.(59),weget ρ (γ) = dΓ P (Γ )P[ν→γ] ν s ν s Z = dΓ P (Γ )eΣ[ν→γ]P[γ∗ →ν∗] s ν s Z = dΓ P (Γ )eΣ[ν→γ]δ((T(T(Γ)∗)) −ν∗). (65) s ν s s Z HerewenotethatT(Γ)∗ =(T−1(Γ∗)). Thus,weobtain ρ (γ) = dΓ ρ (Γ )eΣ[ν→γ]δ(Γ∗−ν∗) ν s ℓ s s Z = dΓ ρ (Γ )eΣ[ν→γ]δ(Γ −ν) s ℓ s s Z = ρ (ν)eΣ[ν→γ]. (66) ℓ Wenotethattakingthelimitτ→∞,thesteadystateisobtainedas ρ (γ)= lim Z−1ρ (γ). (67) ss τ ν τ→∞ whereZ isthenormalizationconstant,whichisgivenby τ 0 Z = dΓ exp − dxF (x)P (Γ ,x)+ dx dtj (Γ ,x;t)·X (x) . (68) τ Z s  Xm Z m m s Xm Z Z−τ m s m  Insertingtheexpressionforρ (γ)intoEq.(67),wefind ν ρ (γ) = lim Z−1ρ (ν)eΣ[ν→γ]. (69) ss τ ℓ τ→∞ Thus, from the nonequilibrium detailed balance relation, eq. (59), we have derived the McLennan-Zubarevsteady statedistribution,eq.(42). 4. Fluctuationtheoremfortheentropyproduction Inthissection,weprovethefluctuationtheoremfortheentropyproductionasasymmetryrelationofacumulant generatingfunction.Nowdefinethecumulantgeneratingfunctionµ(λ). 1 τ µ(λ)= lim − ln exp −λ dx dtj (Γ ,x;t)·X (x) . (70) τ→∞ τ *  Xm Z9 Z0 m s m +ss Here h···i is the averageover the McLennan-Zubarevsteady state distribution. Note that the McLennan-Zubarev ss steadystatedistributionincludesthenormalizationconstantZ. Weshallprovethatthiscumulantgeneratingfunction hasthefollowingsymmetry,whichLebowitzandSpohndiscoveredforstochasticsystems[6], µ(λ)=µ(1−λ). (71) WecallthisrelationtheLebowitz-Spohnsymmetry. First,wedefinetheprobabilitythattheentropyproductionisX alongapathν → γduringthetimeintervalτby pF(X;γ). τ pF(X;γ)= dΓ P (Γ )δ((T(Γ)) −γ)δ(X−Σ[Γ →(T(Γ)) ]). (72) τ s ν s s s s Z Nextconsidertheprobabilitythattheentropyproductionis X duringthetimeintervalτ. Wedenotethisprobability by pF(X). pF(X)isgivenbyintegratingeq.(72)withrespecttoγ. Thus,wehave τ τ pF(X) = dγ pF(X;γ) τ τ Z = dγ dΓ ρ (Γ )δ((T(Γ)) −γ)δ(X−Σ[Γ →(T(Γ)) ]) s ℓ s s s s Z Z = dΓ ρ (Γ )δ(X−Σ[Γ →(T(Γ)) ]) s ℓ s s s Z = dΓs ρℓ(Γs)eΣ[Γs→(T(Γ))s]e−Σ[Γs→(T(Γ))s]δ(X−Σ[Γs →(T(Γ))s]). (73) Z InsertingTT−1intheaboveequation,weget pFτ(X) = dΓs ρℓ(Γs)TT−1eΣ[Γs→(T(Γ))s]e−Σ[Γs→(T(Γ))s]δ(X−Σ[Γs →(T(Γ))s]) Z = dΓs ρℓ((T−1(Γ))s)eΣ[(T−1(Γ))s→Γs]e−Σ[(T−1(Γ))s→Γs]δ(X+Σ[(T−1(Γ))s →Γs]). (74) Z Here we have used T−1(Σ[Γ → (T(Γ)) ]) = −Σ[(T−1(Γ)) → Γ ]. In the last line, we have used the result of the s s s s AppendixB(i.e.,anti-linearity). dΓF(Γ)TG(Γ)= dΓG(Γ)F(T−1(Γ)). (75) Z Z Takinglim dX e−λX fromthelefthandside,weget τ→∞ R lim dX e−λXpF(X) = lim dX e−λX dΓ ρ ((T−1(Γ)) ) τ s ℓ s τ→∞Z τ→∞Z Z ×eΣ[(T−1(Γ))s→Γs]e−Σ[(T−1(Γ))s→Γs]δ(X+Σ[(T−1(Γ))s →Γs]) = lim dΓs ρℓ((T−1(Γ))s)eΣ[(T−1(Γ))s→Γs]e−Σ[(T−1(Γ))s→Γs]eλΣ[(T−1(Γ))s→Γs] τ→∞Z = lim Z dΓs ρss(Γs)e−(1−λ)Σ[(T−1(Γ))s→Γs]. (76) τ→∞ Z Inthelastline,wehaveusedeq.(69). Thus,weobtain 0 lim dX e−λXpF(X) = lim Z exp −(1−λ) dx dtj (Γ ,x;t)·X (x) τ→∞Z τ τ→∞ *  Xm Z Z−ττ m s m +ss = lim Z exp −(1−λ) dx dtj (Γ ,x;t)·X (x) . (77) τ→∞ *  10 Xm Z Z0 m s m +ss