Equivalence of nonequilibrium ensembles in turbulence models

TL;DR

This paper investigates the conditions under which different turbulence models, including reversible and irreversible ensembles, produce equivalent macroscopic properties, especially as systems become more chaotic.

Contribution

It demonstrates the equivalence of reversible and irreversible turbulence models in a multiscale shell model, linking microscopic dynamics to macroscopic observables.

Findings

Reversible and irreversible ensembles show equivalent mean macroscopic properties.

Equivalence improves as the turbulence system becomes more chaotic.

Error in equivalence diminishes with increasing chaos.

Abstract

Understanding under what conditions it is possible to construct equivalent ensembles is key to advancing our ability to connect microscopic and macroscopic properties of non-equilibrium statistical mechanics. In the case of fluid dynamical systems, one issue is to test whether different models for viscosity lead to the same macroscopic properties of the fluid systems in different regimes. Such models include, besides the standard choice of constant viscosity, cases where the time symmetry of the evolution equations is exactly preserved, as it must be in the corresponding microscopic systems, when available. Here a time-reversible dynamics is obtained by imposing the conservation of global observables. We test the equivalence of reversible and irreversible ensembles for the case of a multiscale shell model of turbulence. We verify that the equivalence is obeyed for the mean values of macroscopic observables, up to an error that vanishes as the system becomes more and more chaotic.