Rate description of Fokker-Planck processes with time dependent parameters

TL;DR

This paper develops a framework for approximating continuous Markov processes with metastable states by discrete rate processes with time-dependent rates, especially under slow parameter variations, and demonstrates how to recover long-term behavior from this reduced model.

Contribution

It introduces a quantitative criterion for when a kinetic description with frozen, time-dependent rates is valid and shows how to reconstruct continuous process behavior from the discrete model.

Findings

Validates the rate description for processes with slow parameter changes

Provides a method to recover long-term dynamics from discrete rates

Applies theory to a periodically driven bistable Brownian oscillator

Abstract

The reduction of a continuous Markov process with multiple metastable states to a discrete rate process is investigated in the presence of slow time dependent parameters such as periodic external forces or slowly fluctuating barrier heights. A quantitative criterion is provided under which condition a kinetic description with time dependent frozen rates applies. Finally it is shown how the long time behavior of the underlying continuous process can be retrieved from the knowledge of the discrete process by means of an appropriate random decoration of the discrete states. As a particular example of the presented theory an over-damped bistable Brownian oscillator with periodic driving is discussed.