Emergent second law for non-equilibrium steady states

Abstract

AbstractThe 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 $${{{{{{{\mathcal{I}}}}}}}}(x)=-\!\log ({P}_{{{{{{{{\rm{ss}}}}}}}}}(x))$$ I ( x ) = − log ( P 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 $${{\Delta }}{{{{{{{\mathcal{I}}}}}}}}={{{{{{{\mathcal{I}}}}}}}}({x}_{t})-{{{{{{{\mathcal{I}}}}}}}}({x}_{0})$$ Δ I = I ( x t ) − I ( x 0 ) in steady state self-information along deterministic trajectories can be bounded by the macroscopic entropy production Σ. This bound takes the form of an emergent second law $${{\Sigma }}+{k}_{b}{{\Delta }}{{{{{{{\mathcal{I}}}}}}}}\,\ge \,0$$ Σ + k b Δ I ≥ 0 , which contains the usual second law Σ ≥ 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.