Distribution of Fluctuational Paths in Noise-Driven Systems

TL;DR

This paper develops a variational framework to analyze the distribution of optimal fluctuational paths in noise-driven systems, including colored Gaussian noise, and examines the shape and width of these path distributions.

Contribution

It introduces a variational formulation for optimal paths in systems with colored Gaussian noise and analyzes the distribution shape and width of fluctuational paths.

Findings

Formulated variational problem for optimal paths in colored noise systems

Derived linear equations for the distribution width of fluctuational paths

Solved the equations in the limiting case to characterize path distribution

Abstract

Dynamics of a system that performs a large fluctuation to a given state is essentially deterministic: the distribution of fluctuational paths peaks sharply at a certain optimal path along which the system is most likely to move. For the general case of a system driven by colored Gaussian noise, we provide a formulation of the variational problem for optimal paths. We also consider the prehistory problem, which makes it possible to analyze the shape of the distribution of fluctuational paths that arrive at a given state. We obtain, and solve in the limiting case, a set of linear equations for the characteristic width of this distribution.