Long Term Effects of Small Random Perturbations on Dynamical Systems: Theoretical and Computational Tools

TL;DR

This paper reviews theoretical and computational methods for analyzing the long-term effects of small random perturbations on dynamical systems, introducing a simplified algorithm for computing the action functional in complex stochastic models.

Contribution

It proposes a simplified geometric minimum action algorithm and demonstrates its application to diverse complex stochastic systems across multiple scientific fields.

Findings

Algorithm effectively computes large deviations in complex models.

Applications include systems with multiplicative noise and non-equilibrium dynamics.

Method outperforms traditional approaches in challenging scenarios.

Abstract

Small random perturbations may have a dramatic impact on the long time evolution of dynamical systems, and large deviation theory is often the right theoretical framework to understand these effects. At the core of the theory lies the minimization of an action functional, which in many cases of interest has to be computed by numerical means. Here we review the theoretical and computational aspects behind these calculations, and propose an algorithm that simplifies the geometric minimum action method to minimize the action in the space of arc-length parametrized curves. We then illustrate this algorithm's capabilities by applying it to various examples from material sciences, fluid dynamics, atmosphere/ocean sciences, and reaction kinetics. In terms of models, these examples involve stochastic (ordinary or partial) differential equations with multiplicative or degenerate noise, Markov jump processes, and systems with fast and slow degrees of freedom, which all violate detailed balance, so that simpler computational methods are not applicable.