The McLennan-Zubarev steady state distribution and fluctuation theorems

0.2 MB·

by  Mitsusada M. Sano

The McLennan-Zubarev steady state distribution
and fluctuation theorems
MitsusadaM.Sano
GraduateSchoolofHumanandEnvironmentalStudies,
KyotoUniversity,Yoshida-Nihonmatsu-cho,Sakyo,Kyoto,606-8501,Japan
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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 dis-
tribution. 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 re-
sponse,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