Universal Ideal Behavior and Macroscopic Work Relation of Linear Irreversible Stochastic Thermodynamics
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a r Universal Ideal Behavior and Macroscopic WorkRelation of Linear Irreversible StochasticThermodynamics
Yi-An Ma ∗ and Hong Qian † Department of Applied MathematicsUniversity of Washington, SeattleWA 98195-3925, U.S.A.October 2, 2018
Abstract
We revisit the Ornstein-Uhlenbeck (OU) process as the fundamental mathemat-ical description of linear irreversible phenomena, with fluctuations, near an equilib-rium. By identifying the underlying circulating dynamics in a stationary process asthe natural generalization of classical conservative mechanics, a bridge between a fam-ily of OU processes with equilibrium fluctuations and thermodynamics is establishedthrough the celebrated Helmholtz theorem. The Helmholtz theorem provides an emer-gent macroscopic “equation of state” of the entire system, which exhibits a universalideal thermodynamic behavior. Fluctuating macroscopic quantities are studied fromthe stochastic thermodynamic point of view and a non-equilibrium work relation isobtained in the macroscopic picture, which may facilitate experimental study and ap-plication of the equalities due to Jarzynski, Crooks, and Hatano and Sasa.
Contents ∗ Email: [email protected] † Email: [email protected] Free energy functions and functional 7 A ( E, α ) . . . . . . . . . . . . . . . 73.2 Dynamic free energy functional Ψ[ f α ( x , t )] . . . . . . . . . . . . . . . . . 83.3 Universal equation of state of OU process . . . . . . . . . . . . . . . . . . 9 A Derivation of work equalities 19
A.1 The Jarzynski equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19A.2 Crooks’ approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Gaussian fluctuation theory is one of the most successful branches of equilibrium statisticalmechanics [1, 2]. Since the work of Onsager and Machlup [3, 4], the Ornstein-Uhlenbeckprocess (OUP) has become the stochastic, mathematical description of dynamic, linearirreversible phenomena [5]. It has been extensively discussed in the literature in the past[6, 7, 8, 9]. Several recent papers studied particularly the OUP without detailed balance[10, 11, 12]. In recent years, taking stochastic process rigorously developed by Kolmogorovas the mathematical representation, stochastic thermodynamics has emerged as the finite-time thermodynamic theory of mesoscopic systems, near and far from equilibrium [13, 14,15, 16]. The fundamental aspects of this new development are the mathematical notion ofstochastic entropy production [17, 18, 19], novel thermodynamic relationships collectivelyknown as nonequilibrium work equalities, and fluctuation theorems [20, 21, 22, 23, 24],and the mathematical concept of non-equilibrium steady-state [25, 26, 27].2undamental to all these advances is the notion of time reversal . Newtonian dynamicequation, in Hamiltonian form: d x i d t = ∂H ( x i , y i ) ∂y i , d y i d t = − ∂H ( x i , y i ) ∂x i , (1)is a canonical example of dynamics with time-reversal symmetry [28]: Under transfor-mation ( t, x i , y i ) −→ ( − t, x i , − y i ) , Eq. (1) is invariant. This invariance requires that H ( x i , y i ) = H ( x i , − y i ) : H is usually a function of y i and terms like ~B · ~y , where ~B changes sign upon time reversal such as a magnetic field with a Lorentz force. Adoptingthis definition to linear stochastic processes, one has a novel definition for time reversibility that is distinctly different from that of Kolmogorov’s, as we shall show below.Consider the linear stochastic differential equation d X ( t ) = − M ( α ) X ( t )d t + ǫ Γd B t , (2)which is an OUP with parameters α and ǫ ; M and Γ are two n × n constant matrices, B t is standard Brownian motion. We further assume that all the eigenvalues of M are strictlypositive and Γ is non-singular. According to the concept of detailed balance, Eq. (2) canbe uniquely written as [29, 30, 31, 32] d X ( t ) = − D n Ξ − + (cid:16) D − M − Ξ − (cid:17)o X d t + ǫ Γd B t , (3a)where D and Ξ( α ) are positive definite matrices: D = ΓΓ T and M Ξ + Ξ M T = 2 D . Ifone identifies the two terms inside {· · · } as dissipative (transient) and conservative (per-petuate) motions, respectively, then a time reversible process should be defined as a sta-tistical equivalence between the probability density of a finite path { X ( t ) = x , X ( t ) = x , · · · , X ( t n ) = x n } in which t < t · · · < t n : f (cid:0) x , x , · · · , x n (cid:1) , and the probability density f X † ( t n ) X † (2 t n − t n − ) ··· X † (2 t n − t ) (cid:0) x n , x n − , · · · , x (cid:1) in which the X † ( t ) follows the adjoint stochastic differential equation [32, 33] d X † ( t ) = − D n Ξ − − (cid:16) D − M − Ξ − (cid:17)o X † d t + ǫ Γd B t , (3b)with initial distribution for X † ( t n ) identical to that of X ( t n ) .Recognizing the underlying circulating, conservative dynamics in Eqs. (3a) and (3b)allows us to connect a Hamiltonian structure with linear stochastic processes, and conse-quently develop a Helmholtz theorem, which historically has served as the fundamental3athematical link between classical Newtonian mechanics and thermodynamics. For highdimensional stochastic processes, variables in the Helmholtz theorem provide the systems’underlying dynamics with a macroscopic picture. An ideal gas-like relation between a setof new, macroscopic variables emerges, confirming the simplicity of the OUP. A work-freeenergy equality in terms of the macroscopic thermodynamic variables, which are fluctuat-ing with the underlying dynamics, captures the nature of the fluctuation in the underlyingstochastic processes. We emphasize that even though the mathematical derivations areessentially the same, the physical meaning of the work relation is closer to the classicalthermodynamics.The paper is structured as follows. In Sec. 2, we first provide the necessary preliminar-ies on the OUP. Sec. 2.1 introduces the conservative dynamics as a part of the stationarybehavior of the OUP. Sec. 2.2 then discusses a long neglected issue of zero energy refer-ence. Secs. 3.1 and 3.2 introduces the stationary free energy function and the dynamic freeenergy functional. Sec. 3.3 studies the novel object of equation of state. It is shown thatthe OUP has a simply, universal ideal thermodynamic behavior. In Sec. 4, we turn to thecirculating dynamics and its relation to classical mechanics as well as stochastic dynamics.Sec. 4.1 focuses on the simplicity of the circulating dynamics as being totally integrable.Sec. 4.2 contains a proof that the stationary probability density of OUP, conditioned onan invariant torus of the underlying conservative dynamics, analogous to a microcanonicalensemble, is an invariant measure of the latter. If the dynamics on an invariant torus isergodic, then the conditional probability is the only, natural invariant measure on the torus.Work equalities and fluctuation theorems are discussed in Sec. 5. Using a macroscopicpresentation of the Jarzynski equality, its relation to Helmholtz theorem is revealed in Sec.5.3. The paper concludes with discussions in Sec. 6. The OUP in Eq. (3a) satisfies the important fluctuation-dissipation relation : D Ξ − = ǫ (cid:0) ΓΓ T (cid:1) × covariance matrix of the stationary OUP. In fact, it has a stationary Gaussiandistribution Z − ( α ) e − ϕ ( x ; α ) /ǫ in which Z ( α ) is a normalization factor and ϕ ( x ; α ) = x T Ξ − ( α ) x . In addition, there is an underlying circulating dynamics d x d t = − (cid:16) M ( α ) − D Ξ − ( α ) (cid:17) x , (4)where the scalar ϕ ( x ; α ) is conserved [11]: dd t ϕ (cid:0) x ( t ); α (cid:1) = − x T Ξ − (cid:16) M − D Ξ − (cid:17) x = − x T (cid:16) Ξ − M − M T Ξ − (cid:17) x = 0 . (5)4n fact, this conservative dynamics can be expressed as [32]: d x d t = − (cid:16) M Ξ − D (cid:17) ∇ x ϕ ( x ; α ) , (6)where ϕ ( x ; α ) = x T Ξ − ( α ) x ; (7)and (cid:0) M ( α )Ξ − D (cid:1) is skew-symmetric.It is of paramount importance to recall that for a Markov process without detailedbalance, its stationary dynamics is quantified by two mathematical objects: a stationaryprobability density and a stationary circulation [25, 34] characterized as a divergence-free,conservative vector field. In general, the latter accounts for the complexity arising from thesystem’s dynamics [35]: how many integrals of motion does it have; whether the conser-vative dynamics is ergodic on an invariant set; etc. Many of the characteristics persist inthe stationary stochastic process, and can be used to classify long time, complex behaviorsin high dimensional systems. On the other hand, the dissipative (transient) dynamics plusnoise drive the system towards the stationary distribution while characterizing “energy”fluctuations.For the OUP in Eq. (3a), the conservative dynamics will be shown to be totally in-tegrable. That is, symmetries would be implied through ⌊ n/ ⌋ first integrals of motions,which are the natural generalizations of the time-reversal symmetries. The remaining part, d X ( t ) = − D ∇ x ϕ ( x ; α )d t + ǫ Γd B t , has a stationary dynamics that is detailed balanced.It is worth noting that any e ϕ ( x ; α ) = ϕ ( x ; α ) + C ( α ) is also a valid substitution for the ϕ ( x ; α ) in Eq. (7). As far as the stochastic dynamical system Eq. (3a) is concerned, thereis no unique e ϕ ( x ; α ) as a function of both dynamic variable x and parameter α . The central object that connects classical Newtonian mechanics with equilibrium thermo-dynamics is the entropy function S ( E, V, N ) , with V and N being the volume and thenumber of particles of a classical mechanical system in a container, and E its total me-chanical energy which is conserved according to Newton’s Second Law of motion. InHamilton’s formulation Eq. (1), E is simply the initial value of the Hamiltonian function H (cid:0) { x i } , { y i } (cid:1) in which x i and y i are the position and momentum of i th particle, respec-tively, ≤ i ≤ N .We recognize that in the classical theory of mechanical motions, replacing H with e H (cid:0) { x i } , { y i } (cid:1) = H (cid:0) { x i } , { y i } (cid:1) + C , where C is a constant, has absolutely no conse-quence to the mathematical theory. Therefore, with parameters contained in the Hamilto-nian function, such as V and N , H (cid:0) { x i } , { y i } ; V, N (cid:1) and H (cid:0) { x i } , { y i } ; V, N (cid:1) + C ( V, N ) C ( V, N ) would cause non-uniqueness in the thermody-namic forces in the relation: d S ( E, V, N ) = (cid:18) ∂S∂E (cid:19)
V,N h d E + p d V − µ d N i , (8)in which p = (cid:18) ∂S∂V (cid:19) E,N (cid:18) ∂S∂E (cid:19)
V,N = − (cid:18) ∂E∂V (cid:19) S,N , µ = (cid:18) ∂E∂N (cid:19) S,V . (9)Corresponding to e H = H + C ( V, N ) one has, for E as the initial values of H (cid:0) { x i } , { y i } ; V, N (cid:1) ,and e E as the initial values of e H (cid:0) { x i } , { y i } ; V, N (cid:1) : e p = − ∂ e E∂V ! S,N = p − (cid:18) ∂C∂V (cid:19) N , e µ = µ + (cid:18) ∂C∂N (cid:19) V . (10)Since pressure p has a mechanical interpretation, one can, by physical principle, uniquelydetermine the form of p as a function of V . The situation for µ is much less clear: Sincethere is not an independent mechanical interpretation of the chemical potential other thanthe thermodynamic one given in Eq. (8), the non-uniqueness is inherent in the mathemat-ical, as well as the physico-chemical theory. The problem has the same origin as Gibbs’paradox [37, 38].In classical chemical thermodynamics, the Hamiltonian function as a function of vary-ing number of particles N , H ( x , · · · , x N , y , · · · , y N ) = P Ni m i y i + V ( x , · · · , x N ) , isuniquely determined via a Kirkwood charging process [36]: V (cid:0) x , · · · , x N , x N +1 (cid:1) = V (cid:0) x , · · · , x N (cid:1) + N X j =1 U j,N +1 (cid:0) x j , x N +1 (cid:1) , (11)where lim | x − y |→∞ U ( x, y ) = 0 . (12)With this convention, the Hamiltonian for a molecular system is uniquely determined inchemical thermodynamics, which yields a consistent chemical potential µ . How to gen-eralize this chemical approach to Hamiltonian dynamics Eq. (1) with no clear separationbetween kinetic and potential parts, however, is unclear.The problem of uniqueness of Hamiltonian function e H is intimately related to theuniqueness of e ϕ ( x ; α ) in Sec. 2.1. As we shall show in the rest of this paper, the zeroenergy reference has deep implications to the theory of stochastic thermodynamics. Theresolution to the problem will be discussed in Sec. 4.3.6 Free energy functions and functional
As the notion of entropy, the definition of free energy is widely varied in the literature. The most general features of free energy, perhaps, are: it is the difference between “internalenergy” and entropy; it is the entropy under a “natural invariant measure”. In this section,we shall present two different types of free energies associated with the OU dynamics inEq. (3a):( i ) Thermodynamic free energy of a stationary dynamics, as a function of mean internalenergy E and parameter α : A ( E, α ) . We identify a “thermodynamic state” as a stateof sustained motion, either for a deterministic conservative dynamics Eq. (4), or for astochastic stationary process defined by Eq. (3a).( ii ) Dynamic free energy functional, Ψ[ f ( x , t )] , for an instantaneous probability distri-bution f ( x , t ) . A ( E, α ) With a particular given ϕ ( x ; α ) , we now introduce two different free energy functions. Thefirst one is defined following the microcanonical ensemble approach; definition of the sec-ond one follows Gibb’s canonical ensemble approach. While the second one is frequentlybeing used in the work-free energy relation (discussed in Sec. 5), the two definitions agreeperfectly in the large dimension limit.The first thermodynamic free energy function, A ( E, α ) , associated with the conserva-tive deterministic motion of Eq. (4) on the surface of ϕ ( x ; α ) = E , is obtained follow-ing the microcanonical ensemble approach through Boltzmann’s entropy function. Letting σ B ( E, α ) correspond to the entropy S and Θ − ( E, α ) correspond to (cid:0) ∂S∂E (cid:1) in Eq. (8), wecan define: σ B ( E, α ) = ln (cid:18)Z ϕ ( x ; α ) ≤ E d x (cid:19) = n E + 12 ln det Ξ( α ) + ln V n , (13a) Θ − ( E, α ) = (cid:18) ∂σ B ∂E (cid:19) α = n E , (13b) A ( E, α ) = E − Θ σ B = 2 En (cid:26) − n E −
12 ln det Ξ( α ) − n π ) + ln Γ (cid:16) n (cid:17) + n (cid:27) , (13c) It has become increasingly clear that the Boltzmann’s entropy for a Hamiltonian dynamics is not unique:There are different geometric characterizations of the level sets of the Hamiltonian that can be acceptablechoices. Neither is Shannon’s entropy in stochastic dynamics unique: other convex functions such as Tsallis’entropy can also be found in the literature. V n = π n/ (cid:0) Γ (cid:0) n + 1 (cid:1)(cid:1) − is the volume of an n -dimensional Euclidean ball withradius 1. Γ( · ) is gamma function. n is the dimension of the OUP in Eq. (3a).The second one, A ( E, α ) , follows Gibbs’ canonical ensemble approach via the “parti-tion function” Z ( α ) : Z ( α ) = Z R n e − ϕ ( x ; α ) /ǫ d x = (cid:16)(cid:0) πǫ (cid:1) n det Ξ( α ) (cid:17) , (14a) A ( E, α ) = − ǫ ln Z ( α )= 2 En (cid:26) − n E −
12 ln det Ξ( α ) − n π ) + n (cid:16) n (cid:17)(cid:27) , (14b)in which mean internal energy E = 1 Z ( α ) Z R n ϕ ( x ; α ) e − ϕ ( x ; α ) /ǫ d x = 12 nǫ . (14c)The two free energy functions A in Eq. (13c) and A in Eq. (14b) are different onlyby a function of n inside the {· · · } . For large n , ln Γ( n + 1) ≈ n ln (cid:0) n (cid:1) − n . Therefore, A and A agree perfectly in the limit of n → ∞ . Ψ[ f α ( x , t )] The thermodynamic free energy A ( E, α ) in Sec. 3.1 sets a universal energy referencepoint for the entire family of stochastic dynamics in Eq. (3a) with different α . For a given α , the time-dependent probability density function f α ( x , t ) follows the partial differentialequation ∂f α ( x , t ) ∂t = ∇ x · (cid:16) ǫ D ∇ x f α ( x , t ) + M ( α ) x f α ( x , t ) (cid:17) . (15)The f α ( x , t ) represents an instantaneous “state” of the probabilistic system, which has afree energy functional Ψ (cid:2) f α ( x , t ) (cid:3) = Z R n ϕ ( x ; α ) f α ( x , t ) d x − (cid:18) − ǫ Z R n f α ( x , t ) ln f α ( x , t ) d x (cid:19) = ǫ Z R n f α ( x , t ) ln (cid:18) f α ( x , t ) Z − ( α ) e − ϕ ( x ; α ) /ǫ (cid:19) d x + A ( α ) . (16)This is a dynamic generalization of the free energy functions in Sec. 3.1. It has twoimportant properties. First, lim t →∞ Ψ (cid:2) f α ( x , t ) (cid:3) = A ( α ) . (17)Second [32, 33], dd t Ψ (cid:2) f α ( x , t ) (cid:3) ≤ , (18)8n which the equality holds if and only if f α ( x , t ) reaches its stationary distribution Z − ( α ) e − ϕ ( x ; α ) /ǫ .The negated rate of change in the dynamic free energy functional, − dΨ / d t , is widely rec-ognized as non-adiabatic entropy production rate.The entropy production rate also has a finite time, stochastic counterpart in terms of thelogarithm of the likelihood ratio: − dd t Ψ (cid:2) f α ( x , t ) (cid:3) = lim s → t E P " | s − t | ln f (cid:0) X ( τ ) | t ≤ τ ≤ s (cid:1) f X † (cid:0) ˆ X ( τ ) | t ≤ τ ≤ s (cid:1) ! , (19)where ˆ X ( τ ) = X ( s − τ + t ) , and the expectation E P (cid:2) · · · (cid:3) is carried out over the diffusionprocess defined by Eq. (3a) and the corresponding Eq. (15): f (cid:0) X ( τ ) | t ≤ τ ≤ s (cid:1) ∝ exp (cid:20) − Z st (cid:16) ˙ X ( τ ) + M X ( τ ) (cid:17) T D (cid:16) ˙ X ( τ ) + M X ( τ ) (cid:17) d τ (cid:21) . (20) With the introduction of the internal energy E and the parameter α , the thermodynamicrelation Eq. (8) - known as the Helmholtz theorem - for the OUP model can be expressedby σ B , α and their conjugate variables. We notice that α enters Eq. (13) only through det Ξ( α ) . If one measures α through e α = det Ξ( α ) , then the Helmholtz theorem writes: d E = Θ( E, e α )d σ B − F e α ( E, e α )d e α = (cid:18) ∂σ B ∂E (cid:19) − d σ − (cid:18) ∂σ B ∂ e α (cid:19) (cid:18) ∂σ B ∂E (cid:19) − d e α. (21)The two conjugate variables, Θ and F e α , correspond to the macroscopic quantities in classi-cal thermodynamics as temperature and force. Following either Boltzmann’s microcanonical or Gibbs’ canonical approach, Sec. 3.1revealed that E = n Θ in which θ = n Θ could be interpreted as an “absolute tempera-ture”. Since the absolute temperature θ is a fluctuating quantity with respect to E and e α , itmay, in general, not bear a simple relationship with the noise strength ǫ . But here in OUP,by comparing the microcanonical approach with the canonical one, we note that the meanabsolute temperature ¯ θ = ǫ n .The thermodynamic conjugate variable of e α , the e α -force: F e α = Θ (cid:18) ∂σ B ( E, e α ) ∂ e α (cid:19) E = nθ e α . (22)A mathematical relation between e α , F e α , and θ is called an equation of state in classicalthermodynamics. The force here should be understood as Onsager’s thermodynamic force: corresponding to a spatialdisplacement is a mechanical force; to a change in number of particles is Gibbs’ chemical potential; to avariation in a parameter through a Maxwell demon then is an informatic force [39, 40]. E being a sole function of temperature θ , and the product ofthermodynamic conjugate variables, e αF e α , equaling to nθ , are hallmarks of thermodynamicbehavior of ideal gas and ideal solution. We thus conclude that the OUP has a universalideal thermodynamic behavior. After discussing the energy function and stationary probability, we now focus on the dy-namic complexity of the system and study the circulating, conservative dynamics. Theuniversal ideal thermodynamic behavior reveals one aspect of the simplicity in OUP; an-other is reflected in the divergence-free motions. For the linear conservative dynamics, Eq.(4), its structure is known to be simple: the vector field is integrable.The conservative dynamics in Eq. (4) can be proved to be purely cyclic (e.g., periodic,or quasi-periodic on an invariant torus). Because the skew-symmetric matrix ( M − D Ξ − ) has only pairs of imaginary eigenvalues { λ ℓ | ≤ ℓ ≤ n } . We can also find real Jordan formof ( M − D Ξ − ) : QJ Q − , where J is block diagonal, with × skew-symmetric blocks: Im (cid:2) λ (2 i − (cid:3) (cid:18) − (cid:19) being the i th block on the diagonal. Natural coordinates for the conservative flow Eq. (4)is therefore: y = Q − x . Poisson bracket {· , ·} can be defined for the linear conservative system as: { ϕ ( x ) , ψ ( x ) } = ∇ ϕ ( x ) T ( M − D Ξ − ) ∇ ψ ( x ) . Then the conservative flow expressed in terms of its Hamil-tonian function ϕ ( x ) is: ˙ x i = (cid:8) x i , ϕ ( x ) (cid:9) . (23)First integrals I i of the conservative flow are: I i = y i − + y i = x T Q − T I (2 i − ∼ (2 i ) Q − x , ≤ i ≤ j n k . (24)Here, I (2 i − ∼ (2 i ) denotes the diagonal matrix with on (2 i − -th to (2 i ) -th diagonalentries, and zero everywhere else.The conservative flow is totally integrable, and can be written in canonical action-anglevariables. Angular coordinates θ i accompanying I i can be found as: θ i = Im (cid:2) λ i − (cid:3) − · arctan (cid:18) y i − y i (cid:19) , ≤ i ≤ j n k . (25)10ence, in the canonical action-angle variables, ϕ = ⌊ n/ ⌋ P i =1 I i , θ ′ i = ∂ϕ∂I i = 1 I ′ i = − ∂ϕ∂θ i = 0 . (26)There are (cid:4) n (cid:5) first integrals, but for the given Poisson bracket, one combination of them isunique, which is the Hamiltonian ϕ that connects to the stationary distribution and gener-ates the conservative flow.In the action-angle variables, it is observable that the system bears the following sym-metries: ( t, θ i , I i ) −→ (cid:0) − t, − θ i , ( − k i I i (cid:1) , where { k i } is a sequence of and . Taking { k i } as a sequence of zeros, we recover the time-reversal invariance in Eq. (1). Hence, forgeneral { k i } , those symmetries are the natural generalizations of the time-reversal symme-try, as displayed in classical Hamiltonian systems. The OUP yields an equilibrium probability density function for X : f eq X ( x ; α ) = Z − ( α ) e − ϕ ( x ; α ) /ǫ .In this section, we calculate the conditional probability density for X restricted on an equalenergy surface D ϕ = E = (cid:8) x | x ∈ R n , ϕ ( x ; α ) = E (cid:9) and prove it to be an invariant mea-sure of the conservative dynamics, Eq. (4), restricted on D ϕ = E . Therefore, in the absenceof fluctuation and dissipation, our definition of “equilibrium free energy” (Eq. (13c)) instochastic thermodynamics retreats to the Boltzmann’s microcanonical ensemble approachin classical mechanics.One can obtain a conditional probability density for X restricted on an equal energysurface D ϕ = E = (cid:8) x | x ∈ R n , ϕ ( x ; α ) = E (cid:9) as: Z ( α ) = Z R n e − ϕ ( x ; α ) /ǫ d x = Z ∞ ϕ min ( α ) exp (cid:18) − Eǫ + S ( E, α ) (cid:19) d E, (27)in which [41] S ( E, α ) = ln (cid:18) ∂∂E Z ϕ ( x ; α ) ≤ E d x (cid:19) α = ln (cid:18)I ϕ ( x ; α )= E dΣ n − k∇ x ϕ ( x ; α ) k (cid:19) . (28)The conditional probability density at x ∈ D ϕ = E is: e − S ( E,α ) k∇ x ϕ ( x ; α ) k = 1 n V n E n − (cid:0) det Ξ (cid:1) − k Ξ − x k . (29)Note this conditional probability is one of the invariant measures of the conservative dy-namics Eq. (4) restricted in D ϕ = E . 11o prove this fact, define the dynamics of the conservative part as: S t , mapping a mea-surable set A → S t ( A ) . Then measure of a set A ⊆ D ϕ = E under d µ = e − S ( E,α ) k∇ x ϕ ( x ; α ) k − dΣ n − is: Z A e − S ( E,α ) k∇ x ϕ ( x ; α ) k dΣ n − = e − S ( E,α ) Z A δ ( E − ϕ ( x ))d x = e − S ( E,α ) Z S − t ( A ) δ ( E − ϕ ( x ))d x = Z S − t ( A ) e − S ( E,α ) k∇ x ϕ ( x ; α ) k dΣ n − , (30)since S − t ( A ) ⊆ D ϕ = E and S t is volume preserving. In general, if the dynamics is er-godic on the entire D ϕ = E , then its invariant measure µ is the physical measure : µ -averageequals time average along a trajectory; if there are other first integrals for the conservativedynamics, then µ can be projected further to lower dimensional invariant sets. Up to now, there are clearly several possibilities to uniquely determine the free additivefunction C ( V, N ) in the Hamiltonian H (cid:0) { x i } , { y i } ; V, N (cid:1) discussed in Introduction.(1) C ( V, N ) is chosen such that the global minimum of H = 0 for each and every V and N . This is widely used, implicitly, in application practices, as in our Eq. (7).(2) C ( V, N ) is chosen according to the “equilibrium free energy”: − ǫ ln Z e − E/ǫ + S ( E,V,N ) d E = 0 . (31)Note that this is precisely the “energy function” in Hatano and Sasa [24].(3) Extra information concerning the fluctuations in V , such as in an isobaric ensemble,and fluctuations in N in grand ensemble, provides an empirically determined basis for thefree energy scale.In terms of the theory of probability, choice (2) uniquely determines the energy ref-erence point according to a conditional probability, and in choice (3) it is uniquely de-termined according to a marginal distribution. How to normalize a probability, which hasalways been considered non-consequential in statistical physics, seems to be a fundamentalproblem in the physics of complex systems. The previous discussions suggest that while a great deal of complexity of a detailed, meso-scopic stationary dynamics is captured by the circulating conservative dynamics, OUP alsohas a macroscopic state of motion that is defined by the internal energy E , or equivalently12he level sets of ϕ ( x ) , D ϕ = E ⊂ R n . Thus, from the macroscopic point of view, a stochasticsystem could be studied through the one-dimensional (1-D) time sequence of fluctuatinginternal energy E , as a function of t , or the change in E due to changes in the parameter α .The celebrated Jarzynski equality connects the mesoscopic fluctuating force with thechange in free energy. We present this result through a projection from n -D phase space to -D function E ( t ) that facilitates experimental verification of the work-free energy relation.This approach reveals a close connection between the Jarzynski equality and the Helmholtztheorem. We start with stating the Jarzynski and Crooks’ equalities, with the mathematicalproofs collected in the Appendix A for readers’ convenience. We then demonstrate thenovel formulation of the Jarzynski equality in the projected space.We have shown that A ( α ) is uniquely determined only up to a particular ϕ ( x ; α ) . Asshown below, the existence of an A ( α ) has a paramount importance in the theories of workequalities, in which the notion of a common energy for a family of stochastic dynamicalsystems with different α has to be given a priori [42]. The macroscopic α -force in the Helmholtz theorem, as a function of E and α , is definedthrough Boltzmann’s entropy σ B . The Jarzynski equality, on the other hand, concerns witha mesoscopic α -force, F α ( x ; α ) = − (cid:18) ∂ϕ ( x ; α ) ∂α (cid:19) x , (32)and the statistical behavior of its corresponding stochastic work W [ X ( τ ) , α ( τ )] = Z t F α (cid:0) X ( τ ); α ( τ ) (cid:1) (cid:18) d α ( τ )d τ (cid:19) d τ. (33)The Jarzynski equality dictates that if the initial distribution of X ( τ ) follows the equilib-rium distribution, then [20] D e − ǫ W [ X ( τ ) ,α ( τ )] E(cid:2) X ( τ ) ,α ( τ ) (cid:3) = e − ǫ ∆ A ( α ) , (34)where the average of a functional over the ensemble of paths is defined as: D G [ X ( τ ) , α ( τ )] E(cid:2) X ( τ ) ,α ( τ ) (cid:3) = Z G [ X ( τ ) , α ( τ )] P [ X ( τ ) , α ( τ )] D [ X ( τ )] , (35)in which P [ X ( τ ) , α ( τ )] D [ X ( τ )] , is an infinite-dimensional probability distribution for theentire paths [ X ( τ )] .It is clear from the proof in Appendix A that the Jarzynski equality is general forMarkov processes with or without detailed balance. Several recent papers have studiedextensively the latter case [43, 44]. 13 .2 Crooks’ approach G. E. Crooks’ approach, when applied to processes without detailed balance [24], con-siders the probability functional of a backward path P [ ˇ X ( t ) | ˇ X (0); ˇ α ( t )] over a forwardone P [ X ( t ) | X (0); α ( t )] , where both the initial and final distribution of X ( τ ) follows theequilibrium distribution: P [ ˇ X ( τ ) , ˇ α ( τ )] P [ X ( τ ) , α ( τ )] = exp (cid:18) Q [ X ( τ ) , α ( τ )] − Q hk [ X ( τ ) , α ( τ )] − ∆ ϕ + ∆ A ǫ (cid:19) = exp (cid:18) − W [ X ( τ ) , α ( τ )] − Q hk [ X ( τ ) , α ( τ )] + ∆ A ǫ (cid:19) , (36)where ˇ X ( τ ) = X ( t − τ ) , ˇ α ( τ ) = α ( t − τ ) ; and Q [ X ( τ ) , α ( τ )] = Z t ∂ϕ∂ X ˙ X d τ,Q hk [ X ( τ ) , α ( τ )] = − Z t ˙ X T D ( α ) − (cid:0) M ( α ) − D ( α )Ξ − ( α ) (cid:1) X d τ (37)are the heat dissipation and the house-keeping heat respectively.If a process describes a physical system in equilibrium, which is expected to be “mi-croscopic reversible” in [22], then (cid:28) P [ ˇ X ( τ ) , ˇ α ( τ )] P [ X ( τ ) , α ( τ )] (cid:29)(cid:2) X ( τ ) ,α ( τ ) (cid:3) = Z D [ X ( τ ) , α ( τ )] P [ ˇ X ( τ ) , ˇ α ( τ )] = 1 . (38)On the other hand, if the system is in detailed balance for each and every α , M ( α ) − D ( α )Ξ − ( α ) = 0 , then the house-keeping heat Q hk [ X ( τ ) , α ( τ )] ≡ . Therefore, path-ensemble average of Eq. (36) gives: e ∆ A /ǫ D e − W [ X ( τ ) ,α ( τ )] /ǫ − Q hk [ X ( τ ) ,α ( τ )] /ǫ E(cid:2) X ( τ ) ,α ( τ ) (cid:3) = e ∆ A /ǫ D e − W [ X ( τ ) ,α ( τ )] /ǫ E(cid:2) X ( τ ) ,α ( τ ) (cid:3) . (39)This is Hatano-Sasa’s result [24]. For systems without detailed balance, Q hk [ X ( τ ) , α ( τ )] measures the magnitude of the divergence-free vector field, or the extent to which the sys-tem is away from detailed balance, even when stationary distribution is attained. At thesame time, (cid:28) P [ ˇ X ( τ ) , ˇ α ( τ )] P [ X ( τ ) , α ( τ )] (cid:29)(cid:2) X ( τ ) ,α ( τ ) (cid:3) measures how much on average the behavior of backward paths is statistically differentfrom forward ones. 14 .2.1 Crooks’ approach through adjoint processes Jarzynski’s approach is based on a mesoscopic α -force; while Crooks’ approach concernswith the stochastic entropy production rate which reflects “heat dissipation”. Therefore,for systems with detailed balance, they are essentially the same result according to the FirstLaw of thermodynamics. For systems without detailed balance, one can again obtained aJarzynski-like equality from the probability P of the forward path over the adjoint proba-bility P † of the backward one, according to the notion of time reversal in Eq. (3b): P † ( X ( τ ) | X ( τ + d τ ); α ( τ )) = P ( X ( τ + d τ ) | X ( τ ); α ( τ )) × (cid:18) f eq X ( X ( τ ); α ( τ )) f eq X ( X ( τ + d τ ); α ( τ )) (cid:19) . (40)Thus, whether a system is in detailed balance or not, one has the Hatano-Sasa equality [24]: (cid:28) P [ X ( τ ) , α ( τ )] P † [ ˇ X ( τ ) , ˇ α ( τ )] (cid:29)(cid:2) X ( τ ) ,α ( τ ) (cid:3) = e ∆ A /ǫ D e − W [ X ( τ ) ,α ( τ )] /ǫ E(cid:2) X ( τ ) ,α ( τ ) (cid:3) . (41) We are now in the position to study the work-free energy relation from a macroscopic view.Essentially, we will consider the stochastic, fluctuating ( E ( t ) , e α ( t )) instead of ( X ( t ); α ( t )) directly. In doing so, we are observing the evolution in the probability distribution of E through a projection from ( X ; α ) to ( E, e α ) . With the projection of the n -dimensional phasespace to the one-dimensional time series E ( t ) , the stationary probability density function f ss E ( E, e α ) of E with e α is also a projection of the original stationary probability densityfunction f ss X ( x ; e α ) in Euclidean space (as discussed in Sec. 4.2): f ss E ( E, e α ) = I ϕ ( x ; e α )= E f ss X ( x ; e α ) dΣ n − ||∇ x ϕ ( x ; e α ) || = 1 Z ( e α ) exp (cid:18) − Eǫ + S ( E, e α ) (cid:19) , (42)in which S ( E, e α ) = ln (cid:18)I ϕ ( x ; e α )= E dΣ n − ||∇ x ϕ ( x ; e α ) || (cid:19) . (43)For the process of ( E ( t ) , e α ( t )) , the total internal energy is no longer E itself. But rather,it would include the “entropic effect”, S ( E, e α ) , caused by the curved space structure, andbecome A ( E, e α ) , as will be discussed in more detail in [45]: A ( E, e α ) = E − ǫ S ( E, e α ) . (44)15he Helmholtz theorem for the new ( E ( t ) , e α ( t )) process reads: d A = e Θ( E, e α )d σ − e F e α ( E, e α )d e α = (cid:18) ∂σ∂E (cid:19) − − ǫ ∂S∂σ ! d σ − (cid:18) ∂σ∂ e α (cid:19) (cid:18) ∂σ∂E (cid:19) − + ǫ ∂S∂ e α ! d e α. (45)Hence, the total force that does the work in this new coordinate is: e F e α ( E, e α ) = − ∂ A ( E, e α ) ∂ e α = 1 n · E e α + ǫ ∂S ( E, e α ) ∂ e α , (46)where ǫ (cid:0) ∂S ( E, e α ) /∂ e α (cid:1) is what chemists called an “entropic force”.Now we define the work that external environment has done to the system through thecontrolled change of e α ( t ) as: W [ E ( τ ) , e α ( τ )] = − Z t e F e α ( E, e α ) ˙ e α d τ. (47)Then the work-free energy relation in macroscopic variables is: D e − W [ E ( τ ) , e α ( τ )] E(cid:2) E ( τ ) , e α ( τ ) (cid:3) = Z (cid:0)e α ( t ) (cid:1) Z (cid:0)e α (0) (cid:1) = exp (cid:18) − ∆ A ( α ) ǫ (cid:19) . (48)Therefore, the averaged minus exponential of work is equal to the minus exponential offree energy difference. Here, we notice that the free energy stays the same through thechange of free variables, as a result of Eq. (27): Z ( α ) = Z R n e − ϕ ( x ; α ) /ǫ d x = Z ∞ ϕ min ( α ) exp (cid:18) − Eǫ + S ( E, α ) (cid:19) d E. In the present work, using the OUP as an example, we have illustrated a possible methodof deriving emergent, macroscopic descriptions of a complex stochastic dynamics from itsmesoscopic law of motion . In recent years, there is a growing awareness of the role of prob-abilistic reasoning as the logic of science [46, 47]. In this framework, prior information,data, and probabilistic deduction are three pillars of a scientific theory. In fields with verycomplex dynamics, statistical inferences focus on the latter two aspects starting with data.In physical sciences that includes chemistry, and cellular biology, the prior plays a funda-mental role as a feasible “mechanism” which enters a scientific model based on “establishedknowledge” — no biochemical phenomena should violate the physical laws of mechanicsand thermodynamics. Indeed, many priors have been rigorously formalized in terms ofmathematical theories. Unfortunately, most of these theories are expressed in terms ofdeterministic mathematics for very simple individual “particles”; obtaining a meaningful16robabilistic prior for a realistic, macroscopic-level system requires a computational taskthat is neither feasible nor meaningful [52, 53]. Nonlinear stochastic dynamical study isthe mathematical deductive process that formulates probabilistic prior based on a givenmechanism.Open systems, when represented in terms of Markov processes, are ubiquitously non-symmetric processes according to Kolmogorov’s terminology. This is one of the lessonswe learned from the open-chemical systems theory. The non-symmetricity can be quanti-fied by entropy production [25]. For discrete-state Markov processes, symmetric processesare equivalent to Kolmogorov’s cycle condition [54]. Interestingly, concepts such as cyclecondition, detailed balance, dissipation and irreversible entropy production had all beenindependently discovered in chemistry: Wegscheider’s relation in 1901 [55], detailed bal-ance by G.N. Lewis in 1925 [56], Onsager’s dissipation function in 1931 [57], and theformulation of entropy production in the 1940s [58, 59].A non-symmetric Markov process implies circulating dynamics in phase space. Suchdynamics is not necessarily dissipative, as exemplified by harmonic oscillators in classicalmechanics. One of us has recently pointed out the important distinction between over-damped thermodynamics and underdamped thermodynamics [32]. The present paper is astudy of OUP in terms of the latter perspective, in which we have identified the unbal-anced circulation as a conservative dynamics, a hallmark of the generalized underdampedthermodynamics [33]. In terms of this conservative dynamics, Boltzmann’s entropy func-tion naturally enters stochastic thermodynamics, and we discover a relation between theHelmholtz theorem [60] and the various work relations.In the past, studies on stochastic thermodynamics with underdamped mechanical mo-tions have always required an explicit identification of even and odd variables. See recent[61] and [32] and the references cited within. One of us has introduced a more generalstochastic formulation of “underdamped” dynamics, with thermodynamics, in which circu-lating motion can be a part of a conservative motion [32] without dissipation. The presentwork is an in-depth study of the OUP within this new framework. It seems to us thateven the term “nonequilibrium” in the literature has two rather different meanings: From aclassical mechanical standpoint, any system with a stationary current is “nonequilibrium”,even though it can be non-dissipative. From a statistical mechanics stand point, on the otherhand, “nonequilibrium”, “irreversible”, and “dissipative” are almost all synonymous.
We now discuss two rather different perspectives on the nature of information theory, ortheories [62, 46].First, in the framework of classical physics in terms of Newtonian mechanics, Boltz-17ann’s law, and Gibbs’ theory of chemical potential, there is a universal First Law of Ther-modynamics based on the function S ( E, V, N, α ) where S is the Boltzmann’s entropy ofa conservative dynamical system at total energy E , e.g., Hamiltonian H (cid:0) { x i } , { y i } (cid:1) = E ,with V and N = ( n , n , · · · , n m ) being the volume and numbers of particles in the chemo-mechanical system, and α = ( α , α , · · · , α ν ) represents controllable parameters of thesystem. Then one has d E = (cid:18) ∂S∂E (cid:19) − V,N,α d S − p d V + µ d N − (cid:18) ∂S∂E (cid:19) − V,N,α (cid:18) ∂S∂α (cid:19)
E,V,N d α, (49)in which ( ∂S/∂E ) − V,N,α is absolute temperature. p and µ are pressure and chemical poten-tial, they are the corresponding thermodynamic forces for changing volume V and numberof particles N , respectively. It is natural to suggest that if an agent is able to manipulate aclassical system through changing α while holding S , V , and N constant, then he or she isproviding to, or extracting from, the classical system non-mechanical, non-chemical work .It will be the origin of a Maxwell’s demon [40].For an isothermal system, one can introduce Helmholtz’s free energy function A = E − T S , then Eq. (49) becomes d A = µ d N − S d T − p d V + F α d α. (50)And for an isothermal, isobaric information manipulation process without chemical reac-tions, one has Gibbs function G = E − T S + pV and d G = µ d N − S d T + V d p + F α d α = F α d α . Note that while the first three terms contain “extensive” quantities N , S , and V , thelast term usually does not. It is nanothermodynamic [63]. Note also that for a feedbacksystem that controls F α , one has Θ = G − F α α and dΘ = − α d F α .Just as µ is a function of temperature T in general, so is F α : It has an entropic part[64, 65]. This is where the “information” in Maxwell’s demon enters thermodynamics.Eqs. (49) and (50), thus, are a grander First Law which now includes feedback informationas a part of the conservation [39] with “informatic energy” F α d α , on a par with heat energy T d S , mechanical energy p d V , and Gibbs’ chemical energy µ d N . Eq. (49) is the theory ofabsolute information in connection to controlling α .In engineering and biological research on complex systems, however, the notion of in-formation often has a more subjective meaning, or meanings, usually hidden in the formof a statistical prior [66, 67]. One of the best examples, perhaps, is in current cellularbiology: Many key biochemical processes inside a living cell are said to be “carryingout cellular signal transduction”. Various biochemical activities and changing molecularconcentrations are “interpreted” as “intracellular signals” that instruct a cell to respond toits environment. Here, two very different, but complementary, mathematical theories areequally valid: Since nearly all cellular biochemical reactions can be considered at constanttemperature and volume, one describes the stochastic biochemical dynamics in terms of18ibbs’ theory based on the µ in Eq. (50). On the other hand, the same stochastic bio-chemical dynamics described in term of the probability theory can also be represented asan information processing machine with communication channels and transmissions of bitsof information, carrying out a myriad of biological functions such as sensing, proofread-ing, timing, adaptation, and amplifications of signal magnitude, detection sensitivity, andresponse specificity [48]. The information flow narratives provide bioscientists a higherlevel of abstraction of a physicochemical reality [49].Such an interpretive information theory, however, will lack the fundamental characterof Eqs. (49) and (50). Still, as a multi-scale, coarse grained theory, some inequalities can beestablished [50, 51]. It is also noted that changing α can always be mechanistically furtherrepresented in terms of changing geometric quantities such as volume and particle numbersvia chemical reactions: The ultimate physical bases of information and its manipulationhave to be matters and known forces.We believe this dual possibility has a fundamental reason, rooted in Kolmogorov’s rig-orous theory of probability: A probability space is an abstract object associated with whichmany different random variables, as measurements, are possible. At this point, it is inter-esting to read the preface of [46] written by E. T. Jaynes, who is considered by many as oneof the greatest information theorists since Shannon: “From many years of experience withits applications in hundreds of real problems, our views on the foundations of probabilitytheory have evolved into something quite complex, which cannot be described in any suchsimplistic terms as ‘pro-this’ or ‘anti-that’. For example, our system of probability couldhardly be more different from that of Kolmogorov, in style, philosophy, and purpose. Whatwe consider to be fully half of probability theory as it is needed in current applications —the principles for assigning probabilities by logical analysis of incomplete information —is not present at all in the Kolmogorov system.”Then in an amazing candidness, Jaynes goes on: “Yet, when all is said and done, wefind ourselves, to our own surprise, in agreement with Kolmogorov and in disagreementwith its critics, on nearly all technical issues.” Acknowledgements.
We thank Professors P. Ao, M. Esposito, and H. Ge for helpfuladvices, Ying Tang, Lowell Thompson, Yue Wang, and Felix Ye for many discussions.
A Derivation of work equalities
A.1 The Jarzynski equality
The basic idea for the derivation is as follows: We first represent a path X ( t ) by a discreteversion with N steps and write the path probability in terms of the product of N transitionprobabilities given by the Q Ni =0 ( · · · ) in Eq. (51). Then the mean-exponential of negative19ork D e − ǫ W [ X ( τ ) ,α ( τ )] E(cid:2) X ( τ ) ,α ( τ ) (cid:3) is [24]: D e − ǫ W [ X , ··· , X N ; α , ··· ,α N ] E(cid:2) X , ··· , X N ; α , ··· ,α N (cid:3) = Z · · · Z N Y i =0 d X i exp − ǫ N X i =1 (cid:16) ϕ ( X i ; α i ) − ϕ ( X i ; α i − ) (cid:17)! × N Y i =1 P (cid:0) X i | X i − ; α i − (cid:1) p (cid:0) X , t ; α (cid:1) , (51)in which the work from state ( X i ; α i ) to state ( X i +1 ; α i +1 ) is defined as the difference inthe global ϕ ( x ; α ) with a common zero reference. This is a consequence of the First Lawof Thermodynamics. Since equilibrium is attained at t , p ( x ; t ) = f eq X ( x ; α ) . With theglobal ϕ ( x ; α ) = − ǫ ln f eq X ( x ; α ) − ǫ ln Z ( α ) , we have: D e − ǫ W [ X , ··· , X N ; α , ··· ,α N ] E(cid:2) X N , ··· , X ; α N , ··· ,α (cid:3) = Z · · · Z N Y i =0 d X i N Y i =1 f eq X ( X i ; α i ) Z ( α i ) f eq X ( X i ; α i − ) Z ( α i − ) · N Y i =1 P ( X i | X i − ; α i − ) f eq X ( X ; α )= Z ( α n ) Z ( α ) Z · · · Z N Y i =0 d X i N Y i =1 P ( X i | X i − ; α i − ) N Y i =1 f eq X ( X i − ; α i − ) , N Y i =1 f eq X ( X i ; α i − )= Z ( α n ) Z ( α ) . (52)Since we have defined in Sec. 3.1 the free energy as: A ( α ) = − ǫ ln Z ( α ) , thus we obtainthe Jarzynski equality: D e − ǫ W [ X ( τ ) ,α ( τ )] E(cid:2) X ( τ ) ,α ( τ ) (cid:3) = e − ǫ ∆ A . (53)In a very similar vein, for the macroscopic thermodynamic variables ( E, e α ) , one definesthe work done to the system by the external environment through controlling e α ( t ) with rate ˙ e α : W (cid:2) E ( τ ) , e α ( τ ) (cid:3) = − Z t e F e α ( E, e α ) ˙ e α d τ. (54)20rite ϕ τ ( e α ) = E ( τ, e α ) . Then the discretized D e − ǫ W [ E ( τ ) , e α ( τ )] E(cid:2) E ( τ ) , e α ( τ ) (cid:3) is: D e − ǫ W [ E , ··· ,E N ; e α , ··· , e α N ] E(cid:2) E , ··· ,E N ; e α , ··· , e α N (cid:3) = Z · · · Z N Y i =0 d E i N Y i =1 P ( E i | E i − ; e α i − ) p ( E , t ; e α ) × exp − N X i =1 ϕ i ( e α i ) − ϕ i ( e α i − ) ǫ + S ( E i , e α i ) − S ( E i , e α i − ) ! , (55)where S ( E, e α ) is defined in Eq. (43). On the other hand, equilibrium probability densityfunction of E at ( E i , e α i ) is: f eq E ( E i , e α i ) = 1 Z ( e α i ) I ϕ ( x ; e α )= E i e − ϕ ( x ; e α i ) /ǫ dΣ n − ||∇ x ϕ ( x ; e α i ) || = Z − ( e α i ) exp (cid:18) − ϕ i ( e α i ) ǫ + S ( E i , e α i ) (cid:19) . (56)Hence, we have D e − ǫ W [ E , ··· ,E N ; e α , ··· , e α N ] E(cid:2) E , ··· ,E N ; e α , ··· , e α N (cid:3) = Z · · · Z N Y i =0 d E i N Y i =1 f eq E ( E i , e α i ) Z ( e α i ) f eq E ( E i , e α i − ) Z ( e α i − ) N Y i =1 P ( E i | E i − , e α i − ) f eq E ( E , e α ) ! = Z ( e α n ) Z ( e α ) Z · · · Z N Y i =0 d E i N Y i =1 P ( E i | E i − , e α i − ) N Y i =1 f eq E ( E i − , e α i − ) , N Y i =1 f eq E ( E i , e α i − )= Z ( e α n ) Z ( e α ) = e − ǫ ∆ A ( α ) . (57)Therefore, the log-mean exponential of minus work is equal to the minus of free energydifference. A.2 Crooks’ approach
Instead of introducing stochastic work functional, G. E. Crooks’ approach recognizes theimportant role of time reversal trajectory ˇ X ( t ) , and the deep relationship between work,energy, and dissipation, e.g., entropy production. Let us consider the path probability ofa backward trajectory P [ ˇ X ( τ ) | ˇ X (0); ˇ α ( τ )] against a forward one P [ X ( τ ) | X (0); α ( τ )] , inwhich (cid:0) ˇ X ( τ ); ˇ α ( τ ) (cid:1) = ( X ( t − τ ); α ( t − τ )) , where the initial and final distribution of X ( τ ) follow the equilibrium distribution. We solve for P [ X ( τ ) | X (0); α ( τ )] from the prob-ability of a Brownian motion whose increments are multivariate Gaussian: ǫ Γ( α i ) − (cid:16) X i +1 − X i − M ( α i ) X i ∆ τ (cid:17) = B t i +1 − B t i . (58)21ence, the probability density functional of a path (cid:2) X , · · · , X N | X ; α , · · · , α N (cid:3) is: P (cid:2) X , · · · , X N | X ; α , · · · , α N (cid:3) = N − Y i =0 P (cid:0) X i +1 | X i ; α i (cid:1) = N − Y i =0 π ∆ τ ) n/ e − ǫ τ ( Γ( α i ) − ( X i +1 − X i ) − Γ( α i ) − M ( α i ) X i ∆ τ ) . (59)Here (cid:0) v ( X , α ) (cid:1) ≡ (cid:0) v ( X , α ) (cid:1) T (cid:0) v ( X , α ) (cid:1) . Then the probability density functional of theinverse path (cid:2) X N , · · · , X | X N ; α N , · · · , α (cid:3) is: P (cid:2) X N , · · · , X | X N ; α N , · · · , α (cid:3) = N Y i =1 P (cid:0) X i − | X i ; α i (cid:1) = N Y i =1 π ∆ τ ) n/ e − ǫ τ ( Γ( α i ) − ( X i − − X i ) − Γ( α i ) − M ( α i ) X i ∆ τ ) . (60)Therefore, offsetting by a normalization factor, an infinite-dimensional functional integral, P (cid:2) X ( τ ) | X (0); α ( τ ) (cid:3) ∝ exp (cid:20) − ǫ Z t (cid:16) Γ( α ) − d X / √ d τ − Γ( α ) − M ( α ) X √ d τ (cid:17) (cid:21) = exp (cid:20) − ǫ Z t (cid:16) Γ( α ) − ˙ X − Γ( α ) − M ( α ) X (cid:17) d τ (cid:21) . (61)Probability of the backward path can be found by substituting τ with t − τ : P (cid:2) ˇ X ( τ ) | ˇ X (0); ˇ α ( τ ) (cid:3) ∝ exp (cid:20) − ǫ Z t (cid:16) − Γ( α ) − ˙ X − Γ( α ) − M ( α ) X (cid:17) d τ (cid:21) . (62)Therefore, we have an equality for heat dissipation: P [ ˇ X ( τ ) | ˇ X (0); ˇ α ( τ )] P [ X ( τ ) | X (0); α ( τ )]= exp (cid:20) ǫ Z t (cid:16) ˙ X T (Γ( α )Γ( α ) T ) − M ( α ) X + X T M ( α ) T (Γ( α )Γ( α ) T ) − ˙ X (cid:17) d τ (cid:21) = exp (cid:20) ǫ Z t (cid:16) ˙ X T D − ( α ) M ( α ) X (cid:17) d τ (cid:21) = exp (cid:20) ǫ Z t ˙ X T Ξ − ( α ) X d τ + 2 ǫ Z t ˙ X T (cid:0) D − ( α ) M ( α ) − Ξ − ( α ) (cid:1) X d τ (cid:21) = exp ( Q (cid:2) X ( τ ) , α ( τ ) (cid:3) ǫ + Q hk (cid:2) X ( τ ) , α ( τ ) (cid:3) ǫ ) . (63)Hatano and Sasa, following Oono and Paniconi, called the term Q hk (cid:2) X ( τ ) , α ( τ ) (cid:3) = − Z t ˙ X T (cid:0) D − ( α ) M ( α ) − Ξ − ( α ) (cid:1) X d τ (64)22ouse-keeping heat [24].Since we start and end with equilibrium distributions with the corresponding e α , p (cid:0) X ; α (cid:1) = f eq X ( X ; α ) = 1 Z ( α ) exp (cid:20) − X T U ( α ) X ǫ (cid:21) ; p (cid:0) X N ; α N (cid:1) = f eq X ( X N ; α N ) = 1 Z ( α N ) exp (cid:20) − X TN U ( α N ) X N ǫ (cid:21) . (65)Therefore, P [ ˇ X ( τ ) , ˇ α ( τ )] P [ X ( τ ) , α ( τ )] = P [ ˇ X ( τ ) | ˇ X (0); ˇ α ( τ )] p ( X N ; α N ) P [ X ( τ ) | X (0); α ( τ )] p ( X ; α )= exp (cid:18) Q [ X ( τ ) , α ( τ )] − Q hk [ X ( τ ) , α ( τ )] − ∆ ϕ + ∆ A ǫ (cid:19) = exp (cid:18) − W [ X ( τ ) , α ( τ )] − Q hk [ X ( τ ) , α ( τ )] + ∆ A ǫ (cid:19) . (66)Now taking ensemble average of the trajectories [ X ( τ ) , α ( τ )] over P [ ˇ X ( τ ) , ˇ α ( τ )] P [ X ( τ ) , α ( τ )] gives: Z D [ X ( τ ) , α ( τ )] P [ ˇ X ( τ ) , ˇ α ( τ )] = (cid:28) P [ ˇ X ( τ ) , ˇ α ( τ )] P [ X ( τ ) , α ( τ )] (cid:29) [ X ( τ ) ,α ( τ )] = e ∆ A D e − W [ X ( τ ) ,α ( τ )] − Q hk [ X ( τ ) ,α ( τ )] E(cid:2) X ( τ ) ,α ( τ ) (cid:3) . (67)When the system is in detailed balance, Crooks’ approach recovers the Jarzynski equality.If one chooses the global energy ϕ with zero reference for each own equilibrium, i.e., ∆ A = 0 for all α , then it recovers the Hatano-Sasa equality. References [1] Landau L D and Lifshitz E M 1980
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