Stochastic Instability of Quasi-Isolated Systems

TL;DR

This paper investigates how small stochastic perturbations affect the stability of dynamical systems, revealing conditions under which systems become stochastically unstable and linking this instability to the irreversibility of time.

Contribution

It introduces a new framework for analyzing stochastic instability in quasi-isolated systems using local stability indices and explores the implications for time irreversibility.

Findings

Existence of stochastic perturbations causing instability.

Non-commutativity of stability limits indicates instability.

Stochastic instability relates to the irreversibility of time.

Abstract

The stability of solutions to evolution equations with respect to small stochastic perturbations is considered. The stability of a stochastic dynamical system is characterized by the local stability index. The limit of this index with respect to infinite time describes the asymptotic stability of a stochastic dynamical system. Another limit of the stability index is given by the vanishing intensity of stochastic perturbations. A dynamical system is stochastically unstable when these two limits do not commute with each other. Several examples illustrate the thesis that there always exist such stochastic perturbations which render a given dynamical system stochastically unstable. The stochastic instability of quasi-isolated systems is responsible for the irreversibility of time arrow.