On the relaxation to nonequilibrium steady states

TL;DR

This paper investigates the relaxation processes of classical particle systems to equilibrium and nonequilibrium steady states, establishing conditions like T-mixing for ensemble and observational independence, and deriving Prigogine's principle microscopically.

Contribution

It introduces the concepts of {\Omega}T-mixing and weak T-mixing as necessary and sufficient conditions for relaxation, linking dynamical properties to physical measurements and ergodicity.

Findings

{\Omega}T-mixing is necessary and sufficient for relaxation of ensemble averages.

Weak T-mixing ensures relaxation independence from initial ensemble.

Results enable microscopic derivation of Prigogine's minimum entropy production principle.

Abstract

The issue of relaxation has been addressed in terms of ergodic theory in the past. However, the application of that theory to models of physical interest is problematic, especially when dealing with relaxation to nonequilibrium steady states. Here, we consider the relaxation of classical, thermostatted particle systems to equilibrium as well as to nonequilibrium steady states, using dynamical notions including decay of correlations. We show that the condition known as {\Omega}T-mixing is necessary and sufficient to prove relaxation of ensemble averages to steady state values. We then observe that the condition known as weak T-mixing applied to smooth observables is sufficient for relaxation to be independent of the initial ensemble. Lastly, weak T-mixing for integrable functions makes relaxation independent of the ensemble member, apart from a negligible set of members enabling the result to be applied to observations from a single physical experiment. The results also allow us to give a microscopic derivation of Prigogine's principle of minimum entropy production in the linear response regime. The key to deriving these results lies in shifting the discussion from characteristics of dynamical systems, such as those related to metric transitivity, to physical measurements and to the behaviour of observables. This naturally leads to the notion of physical ergodicity.