Long Term Effects of Small Random Perturbations on Dynamical Systems: Theoretical and Computational Tools
This work provides a computational tool for researchers studying rare events in complex dynamical systems where detailed balance is violated, addressing a known bottleneck in large deviation theory.
The paper reviews theoretical and computational tools for analyzing long-term effects of small random perturbations on dynamical systems, and proposes an algorithm that simplifies the geometric minimum action method. The algorithm is demonstrated on examples from material sciences, fluid dynamics, atmosphere/ocean sciences, and reaction kinetics, handling complex systems with multiplicative or degenerate noise, Markov jump processes, and non-equilibrium dynamics.
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.