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...

In systems driven away from equilibrium, the velocity correlation function and the linear-response function to a small perturbation force do not satisfy the fluctuation-response relation (FRR) due to the lack of detailed balance in contrast to equilibrium systems. In this Letter, an equality between an extent of the FRR violation and the rate of energy dissipation is proved for Langevin systems under nonequilibrium conditions. This equality enables us to calculate the rate of energy dissipation by quantifying the extent of the FRR violation, which can be measured experimentally.

The present paper reports our attempt to search for a new universal framework in nonequilibrium physics. We propose a thermodynamic formalism that is expected to apply to a large class of nonequilibrium steady states including a heat conducting fluid, a sheared fluid, and an electrically conducting fluid. We call our theory steady state thermodynamics (SST) after Oono and Paniconi's original proposal. The construction of SST is based on a careful examination of how the basic notions in thermodynamics should be modified in nonequilibrium steady states. We define all thermodynamic quantities through operational procedures which can be (in principle) realized experimentally. Based on SST thus constructed, we make some nontrivial predictions, including an extension of Einstein's formula on density fluctuation, an extension of the minimum work principle, the existence of a new osmotic pressure of a purely nonequilibrium origin, and a shift of coexistence temperature. All these predictions may be checked experimentally to test SST for its quantitative validity.

Starting from microscopic mechanics, we derive thermodynamic relations for heat conducting nonequilibrium steady states. The extended Clausius relation enables one to experimentally determine nonequilibrium entropy to the second order in the heat current. The associated Shannon-like microscopic expression of the entropy is suggestive. When the heat current is fixed, the extended Gibbs relation provides a unified treatment of thermodynamic forces in linear nonequilibrium regime.PACS numbers: 05.70. Ln, 05.60.Cd Thermodynamics (TD) is a theoretical framework that describes universal quantitative laws obeyed by macroscopic systems in equilibrium. A core of TD is the Clausius relation ∆S = Q/T , which relates the entropy with the heat transfer caused by a change in the system. Combined with the energy conservation, the Clausius relation leads to the Gibbs relation T dS = dU + i f i dν i , where ν i is a controllable parameter and f i the corresponding generalized force. The Gibbs relation is particularly useful since it represents the forces as gradients of suitable thermodynamic potentials. It also played a key role when Gibbs constructed equilibrium statistical mechanics.Here we wish to address the fundamental question whether TD can be extended to nonequilibrium steady states (NESS) which, like equilibrium states, lack macroscopic time-dependence. We shall call the possible extension Steady State Thermodynamics (SST). The possibility of SST is far from trivial since NESS exhibit many properties which are very different from equilibrium states. First of all a naive extension of the Clausius relation to NESS is never possible since the heat transfer Q generally diverges linearly in time. It is also a deep theoretical question whether the long range correlation universally observed in NESS [1] is consistent with SST. In addition to such abstract interests, there are nonequilibrium phenomena which may be better understood using SST. An interesting example is the force exerted on a small rigid body placed in a heat conducting fluid [2]. This force may be understood as a thermodynamic force in SST (see [3] for related ideas).NESS sufficiently close to equilibrium can be characterized by the linear response theory. But this theory, which requires an ensemble of trajectories in space-time, does not lead us directly to SST. It is also clear that the theory only gives the result up to the first order in the "degree of nonequilibrium."Although an extension of TD to NESS (or, equivalently, a construction of SST) may sound as a formidably difficult task, there are at least two branches of studies which are encouraging. One is the series of works which reveal deep implications on NESS of the microscopic time-reversal symmetry. It has been shown that the simple symmetry (1) leads to various nontrivial results including the Green-Kubo relation, Kawasaki's non-linear response relation, and the fluctuation theorem [4,5]. Although none of these works directly treat extensions of TD, techniques for characterizing NESS and energy ...

When the process of a system in contact with a heat bath is described by classical Langevin equation, the method of stochastic energetics [K. Sekimoto, J. Phys. Soc. Jpn. 66 (1997) 1234] enables to derive the form of Helmholtz free energy and the dissipation function of the system. We prove that the irreversible heat Qirr and the time lapse ∆t of an isothermal process obey the complementarity relation, Qirr ∆t ≥ kBT Smin, where Smin depends on the initial and the final values of the control parameters, but it does not depend on the pathway between these values.