VOLUME 86, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 16 APRIL 2001 Steady-State Thermodynamics of Langevin Systems Takahiro Hatano and Shin-ichi Sasa Department of Pure and Applied Sciences, University of Tokyo, Komaba, Tokyo 153-8902, Japan (Received 26 October 2000) We study Langevin dynamics describing nonequilibirum steady states. Employing the phenomenologi- cal framework of steady-state thermodynamics constructed by Oono and Paniconi [Prog. Theor. Phys. Suppl. 130, 29 (1998)], we find that the extended form of the second law which they proposed holds for transitions between steady states and that the Shannon entropy difference is related to the excess heat produced in an infinitely slow operation. A generalized version of the Jarzynski work relation plays an important role in our theory. DOI: 10.1103/PhysRevLett.86.3463 PACS numbers: 05.70.Ln, 05.40.Jc The second law of thermodynamics describes the funda- mental limitation on possible transitions between equilib- rium states. In addition, it leads to the definition of entropy, in terms of which the heat capacity and equations of state can be treated in a unified way. In contrast to equilibrium systems, with their elegant theoretical framework, the understanding of nonequilib- rium steady-state systems is still primitive. The broad goal with which we are concerned in this paper is to estab- lish the connection between the phenomena displayed by nonequilibrium steady states and thermodynamic laws. We expect that a unified framework that describes both equi- librium and nonequilibrium phenomena can be obtained by extending the second law to the state space consisting of equilibrium and nonequilibrium steady states. There have been several attempts to construct such a framework [1���4]. Among them, a phenomenological framework pro- posed by Oono and Paniconi seems most sophisticated, and their framework has been named ���steady-state thermody- namics��� (SST) [4]. Oono and Paniconi focused on transitions between steady states and distinguished steadily generated heat, which is generated even when the system remains in a single state in the state space and the total heat. They call the former the ���housekeeping heat.��� Subtracting the house- keeping heat from the total heat defines the excess heat, which reflects the change of the system in the state space: Qex Qtot 2 Qhk . (1) Here Qtot and Qhk denote the total heat and the housekeep- ing heat, respectively. By convention, we take the sign of heat to be positive when it flows from the system to the heat bath. For equilibrium systems, Qex reduces to the total heat Qtot, because in this case Qhk 0. Because any proper formulation of SST should reduce to equilibrium thermo- dynamics in the appropriate limit, Qex should correspond to the change of a generalized entropy S within the SST. Here we treat systems in contact with a single heat bath whose temperature is denoted by T, so that the second law of SST reads [4] TDS $ 2Qex . (2) The equality here holds for an infinitely slow operation in which the system is in a steady state at each time during a transition. (We call such a process a ���slow process.���) That is, the generalized entropy difference DS between two steady states can be measured as 2Qex T resulting from a slow process connecting these two states. This al- lows us to define the generalized entropy of nonequilib- rium steady states experimentally, by measuring the excess heat obtained in a slow process between any nonequilib- rium steady state and an equilibrium state, whose entropy is known. These are phenomenological considerations and they should ultimately be confirmed through experiments. As a preliminary step toward this confirmation, in this Letter, we find support for the validity of the above discussion by studying a simple stochastic model. With the same moti- vation, Sekimoto and Oono considered a simple Langevin system and defined the quantity Qex [5,6]. However, this nonequilibrium system reduces to an equilibrium system through a suitable transformation of variables and hence lacks generality. Also, one of the present authors has found that the minimum work principle holds for certain types of transitions between steady states [7], with some assump- tion regarding the steady-state measure. In this Letter, we derive the inequality (2) in a more general context and show that the equality holds for slow processes. This result relates the excess heat to the generalized entropy. We consider the dynamics of a Brownian particle in a circuit driven by an external force. These dynamics are described by the Langevin equation gx 2 ���U x l ���x 1 f 1 j t , (3) where j t represents Gaussian white noise whose inten- sity is 2gkBT. We employ periodic boundary conditions, and thus the particle flows due to the nonconservative force f. This simple nonequilibrium system was investigated by Kurchan with regard to the fluctuation theorem [8]. Tran- sitions between steady states are realized by changing the parameters l and f. We assume that if the system is left unperturbed, it eventually reaches a steady state which is uniquely determined by the parameter values. Although 0031-9007 01 86(16) 3463(4)$15.00 �� 2001 The American Physical Society 3463

VOLUME 86, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 16 APRIL 2001 we consider an explicitly one-dimensional system for sim- plicity, multidimensional cases, including many-particle systems, are essentially the same. We write the steady-state probability distribution func- tion as rss x a , where a denotes the set of control pa- rameters of the system, l and f. Then we manipulate the system by changing the value of a during the inter- val from t 0 to t t . We assume that the system is initially in a steady state, and after the completion of the manipulation it converges to a new steady state. Let t de- note the time at which the system reaches the new steady state 0 , t , t . We discretize 0, t as t0, t1, . . . , tN . We denote the value of a at the ith time step by ai. This value changes at each time step from time t0 until time tM t , while after this time it remains fixed: ai aM for i $ M. We also write x ti as xi. We consider the limit of an infinitely fine discretization by keeping t and t fixed and taking N ! `. Let us introduce a new quantity f x a defined by f x a 2 logrss x a , (4) where rss x a is the probability distribution function of the steady state corresponding to a. Let P x0 j x a be the transition probability from x to x0 in one time step (whose length is Dt t N) for a given value of a. Note that by definition Z dx0 P x j x0 a rss x0 a rss x a . (5) Then for a given sequence x0, x1, . . . , xN , which is col- lectively denoted by x , the average of a quantity g x is written as g Z N Y i 0 dxi ��� N21 Y i 0 P xi11 j xi ai ! rss x0 a0 g x , (6) where the symbol expresses that this is an approximate equality that becomes exact in an appropriate, infinitely fine discretization limit of N ! `. Now, in order to derive Eq. (2) for the system described by Eq. (3), we utilize a Jarzynski-type equality. For tran- sitions between isothermal equilibrium states, it is known that the following equality holds between the work done to the system W and the equilibrium Helmholtz free energy difference DF [9]: e2bW c e2bDF. (7) Here b 1 kBT and ? c denotes the average over all possible histories with respect to equilibrium fluctuations. Note that the minimum work principle W c $ DF im- mediately follows from this relation, due to the Jensen in- equality ex $ e x . In a similar way, we now set out to derive Eq. (2) through the somewhat generalized version of Eq. (7) exp 2bQex 2 Df 1 , (8) where Df f xN aN 2 f x0 a0 . We start with the identity *" N21 Y i 0 rss xi11 ai11 rss xi11 ai #+ 1 . (9) This follows from Eqs. (5) and (6). Rewriting Eq. (9) using f, we have * exp " N21X i 0 2f xi11 ai11 1 f xi11 ai #+ 1 . (10) Taking the limit N ! `, Eq. (10) becomes �� exp ��� 2 Z t 0 dt a ���f x a ���a ����� 1 . (11) It can be easily seen that Eq. (11) reduces to the equilibrium Jarzynski equality (7) when we set f 2b F 2 U . Now we express the left-hand side of Eq. (11) in terms of heat, so that we can find the correspondence with Eq. (8). First, for Langevin systems, the total heat flowing into the heat bath, Qtot, is defined by Qtot Z t 0 dt gx t 2 j t x t . (12) Note that the products of x t and the other quantities are of the Stratonovich type. This interpretation of the heat was proposed and investigated by Sekimoto [10]. In addition, we note that bQtot satisfies the fluctuation theorem if the system remains in a steady state [8]. Next we rewrite Eq. (3) as gx b x 2 b21 ���f x a ���x 1 j t , (13) where b x f 2 ���U x a ���x 1 b21 ���f x a ���x . (14) Equation (13) corresponds to the decomposition of the flux x into an irreversible part b x and a reversible part ���f ���x, in the sense of Refs. [11,12]. Multiplying Eq. (13) by x t dt and integrating with respect to t from t 0 to t t, we get bQtot Z t 0 dt bb x x t 2 Df 1 Z t 0 dt ���f x a ���a a t . (15) Here we define the housekeeping heat as [13] Qhk Z t 0 dt b x x t . (16) We discuss a physical meaning of this expression later e.g., see Eq. (27). Using Qex Qtot 2 Qhk, we can rewrite Eq. (11) as the generalized Jarzynski equality (8). 3464