Lyapunov exponents of stochastic systems---from micro to macro

TL;DR

This paper critically examines the meaning of Lyapunov exponents in stochastic systems, comparing different noise interpretations and their effects on system behavior, with applications to Brownian particles and Dean-Kawasaki dynamics.

Contribution

It introduces a detailed analysis of noise interpretations in stochastic Lyapunov exponents and explores their implications at microscopic and macroscopic levels.

Findings

Different noise assumptions lead to distinct Lyapunov exponent distributions.

The paper provides estimates for the largest Lyapunov exponent in Dean-Kawasaki dynamics.

Limits where noise prescriptions become equivalent are discussed.

Abstract

Lyapunov exponents of dynamical systems are defined from the rates of divergence of nearby trajectories. For stochastic systems, one typically assumes that these trajectories are generated under the "same noise realization". The purpose of this work is to critically examine what this expression means. For Brownian particles, we consider two natural interpretations of the noise: intrinsic to the particles or stemming from the fluctuations of the environment. We show how they lead to different distributions of the largest Lyapunov exponent as well as different fluctuating hydrodynamics for the collective density field. We discuss, both at microscopic and macroscopic levels, the limits in which these noise prescriptions become equivalent. We close this paper by providing an estimate of the largest Lyapunov exponent and of its fluctuations for interacting particles evolving with the Dean-Kawasaki dynamics.