Diffusion in the Markovian limit of the spatio-temporal colored noise

TL;DR

This paper investigates the diffusion process under non-Markovian spatio-temporal noise, deriving a Fokker-Planck equation in the Markovian limit and analyzing its effects on escape rates over potential barriers.

Contribution

It introduces a novel approach to derive a Fokker-Planck equation in the Markovian limit of non-Markovian noise, revealing how systematic reduction of potential affects escape rates.

Findings

Derived a Fokker-Planck equation in the Markovian limit.

Identified renormalization of barrier height and prefactor.

Found a maximum escape rate at a specific scaling limit.

Abstract

We explore the diffusion process in the non-Markovian spatio-temporal noise.%the escape rate problem in the non-Markovian spatio-temporal random noise. There is a non-trivial short memory regime, i.e., the Markovian limit characterized by a scaling relation between the spatial and temporal correlation lengths. In this regime, a Fokker-Planck equation is derived by expanding the trajectory around the systematic motion and the non-Markovian nature amounts to the systematic reduction of the potential. For a system with the potential barrier, this fact leads to the renormalization of both the barrier height and collisional prefactor in the Kramers escape rate, with the resultant rate showing a maximum at some scaling limit.