Rate description of Fokker-Planck processes with time-periodic parameters

TL;DR

This paper analyzes the large-time behavior of periodically driven Fokker-Planck processes with multiple metastable states, deriving exact equations for key functions and demonstrating their application to a bistable Brownian oscillator across various driving frequencies.

Contribution

It introduces exact equations for occupation probabilities, probability densities, and localizing functions in time-periodic Fokker-Planck processes, providing new tools for analyzing metastable dynamics.

Findings

Derived exact equations for metastable state functions

Demonstrated methods on a bistable Brownian oscillator

Analyzed dynamics across different driving frequencies

Abstract

The large time dynamics of a periodically driven Fokker-Planck process possessing several metastable states is investigated. At weak noise transitions between the metastable states are rare. Their dynamics then represent a discrete Markovian process characterized by time dependent rates. Apart from the occupation probabilities, so-called specific probability densities and localizing functions can be associated to each metastable state. Together, these three sets of functions uniquely characterize the large time dynamics of the conditional probability density of the original process. Exact equations of motion are formulated for these three sets of functions and strategies are discussed how to solve them. These methods are illustrated and their usefulness is demonstrated by means of the example of a bistable Brownian oscillator within a large range of driving frequencies from the slow semiadiabatic to the fast driving regime.