Steady state fluctuation relation and time-reversibility for non-smooth chaotic maps

TL;DR

This paper demonstrates that steady state fluctuation relations can hold in non-smooth, non-fully chaotic maps that violate traditional assumptions like time-reversibility and the Anosov property, broadening their applicability.

Contribution

It constructs and analyzes a simple non-smooth map showing fluctuation relations hold despite violations of typical chaotic system conditions.

Findings

Fluctuation relations are valid even in non-smooth, irreversible maps.

Standard assumptions like time-reversibility are not necessary for fluctuation relations.

Discontinuous invariant measures do not prevent the validity of fluctuation relations.

Abstract

Steady state fluctuation relations for dynamical systems are commonly derived under the assumption of some form of time-reversibility and of chaos. There are, however, cases in which they are observed to hold even if the usual notion of time reversal invariance is violated, e.g. for local fluctuations of Navier-Stokes systems. Here we construct and study analytically a simple non-smooth map in which the standard steady state fluctuation relation is valid, although the model violates the Anosov property of chaotic dynamical systems. Particularly, the time reversal operation is performed by a discontinuous involution, and the invariant measure is also discontinuous along the unstable manifolds. This further indicates that the validity of fluctuation relations for dynamical systems does not rely on particularly elaborate conditions, usually violated by systems of interest in physics. Indeed, even an irreversible map is proved to verify the steady state fluctuation relation.