Emergent second law for non-equilibrium steady states

Abstract

The Gibbs distribution universally characterizes states of thermal equilibrium. In order to extend the Gibbs distribution to non-equilibrium steady states, one must relate the self-information $\mathcal{I}(x)=-\log (P_{\text{ss}}(x))$ of microstate $x$ to measurable physical quantities. This is a central problem in non-equilibrium statistical physics. By considering open systems described by stochastic dynamics which become deterministic in the macroscopic limit, we show that changes $\Delta \mathcal{I}=\mathcal{I}(x_t)-\mathcal{I}(x_0)$ in steady state self-information along deterministic trajectories can be bounded by the macroscopic entropy production $\Sigma$. This bound takes the form of an emergent second law $\Sigma +k_b\Delta \mathcal{I}\ge 0$, which contains the usual second law $\Sigma \ge 0$ as a corollary, and is saturated in the linear regime close to equilibrium. We thus obtain a tighter version of the second law of thermodynamics that provides a link between the deterministic relaxation of a system and the non-equilibrium fluctuations at steady state. In addition to its fundamental value, our result leads to novel methods for computing non-equilibrium distributions, providing a deterministic alternative to Gillespie simulations or spectral methods.

arXiv commentNew section with a tighter version of the bound given by the emergent second law

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