Evolution to a singular measure and two sums of Lyapunov exponents

TL;DR

This paper investigates conditions under which the infinite time evolution of densities in dissipative flows results in singular measures, using Lyapunov exponents and Green-Kubo formulas to analyze entropy production fluctuations.

Contribution

It introduces a criterion based on Lyapunov exponents and Green-Kubo formulas to determine when the evolved measure becomes singular in dissipative dynamical systems.

Findings

Derived a condition for measure singularity using Lyapunov exponents.

Connected sums of Lyapunov exponents to entropy production fluctuations.

Provided examples of computing sums for specific velocity fields.

Abstract

We consider dissipative dynamical systems represented by a smooth compressible flow in a finite domain. The density evolves according to the continuity (Liouville) equation. For a general, non-degenerate flow the result of the infinite time evolution of an initially smooth density is a singular measure. We give a condition for the non-degeneracy which allows to decide for a given flow whether the infinite time limit is singular. The condition uses a Green-Kubo type formula for the space-averaged sum of forward and backward-in-time Lyapunov exponents. We discuss how the sums determine the fluctuations of the entropy production rate in the SRB state and give examples of computation of the sums for certain velocity fields.