Analysis of slow transitions between nonequilibrium steady states

Abstract

Transitions between nonequilibrium steady states obey a generalized Clausius inequality, which becomes an equality in the quasistatic limit. For slow but finite transitions, we show that the behavior of the system is described by a response matrix whose elements are given by a far-fromequilibrium Green-Kubo formula, involving the decay of correlations evaluated in the nonequilibrium steady state. This result leads to a fluctuation-dissipation relation between the mean and variance of the nonadiabatic entropy production, ∆sna. Furthermore, our results extend -to nonequilibrium steady states -the thermodynamic metric structure introduced by Sivak and Crooks for analyzing minimal-dissipation protocols for transitions between equilibrium states.Classical thermodynamics is a macroscopic theory built around the concept of the equilibrium state [1], whereas statistical mechanics is a microscopic theory that represents equilibrium states in terms of known statistical ensembles [2]. Both frameworks accurately describe the equilibrium properties of matter and the constraints that must be satisfied when systems undergo transitions between equilibrium states. No comparably general theories exist for systems away from equilibrium, not even for steady states. We have no a priori microscopic representations of nonequilibrium steady states; neither do we have a macroscopic, empirical thermodynamic theory to compare to.A phenomenological thermodynamic theory of nonequilibrium steady states was suggested by Oono and Paniconi [3], who introduced the crucial concept of housekeeping heat, which is the heat transfer necessary to maintain a system in a given steady state. The excess heat Q ex is then the renormalization of the total heat Q by the housekeeping heat Q hk :In the context of Markovian dynamics, these notions were modified somewhat and given quantitative expressions by Hatano and Sasa [4], Speck and Seifert [5] and Ge and Qian [6, 7], leading to derivations of a generalized Clausius inequality for transitions between steady states [4, 6]:Here ∆S denotes the net change in the entropy of the system, and Q ex is the ensemble-averaged rate of excess heat transfer to a reservoir at inverse temperature β(t).(See also Bertini et al [8,9] for a related inequality for renormalized work.) Under equilibrium dynamics Eq. 2 reduces to the usual Clausius inequality, as Q hk = 0 and hence Q ex = Q.Eq. 2 suggests that there may exist a steady-state thermodynamic framework that closely parallels equilibrium thermodynamics, with steady states and excess heat in the roles traditionally assigned to equilibrium states and heat, respectively. In this Letter we further develop these parallels, by analyzing finitely slow transitions from one nonequilibrium steady state to another. At leading order in a perturbative expansion in the driving speed, Eq. 2 becomes an equality and hysteresis vanishes, suggesting that quasistatic steady-state processes are natural counterparts of reversible thermodynamic processes [3]. At the next order, a response matr...

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