Steady-State Thermodynamics of Langevin Systems

Abstract

We study Langevin dynamics describing nonequilibirum steady states. Employing the phenomenological 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. The second law of thermodynamics describes the fundamental limitation on possible transitions between equilibrium 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 nonequilibrium steady state systems is still primitive. The broad goal with which we are concerned in this paper is to establish the connection between the phenomena displayed by nonequilibrium steady states and thermodynamic laws. We expect that a unified framework that describes both equilibrium 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 [1,2,3,4]. Among them, a phenomenological framework proposed by Oono and Paniconi seems most sophistcated, and their framework has been named 'steady state thermodynamics' (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 "house-keeping 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:Here Q tot and Q hk denote the total heat and the housekeeping 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, Q ex reduces to the total heat Q tot , because in this case Q hk = 0. Because any proper formulation of SST should reduce to equilibrium thermodynamics in the appropriate limit, Q ex 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]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 ∆S between two steady states can be measured as −Q ex /T resulting from a slow process connecting these two states. This allows us to define the generalized entropy of nonequilibrium steady states, because using it we can determine the generalized entrop...

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