Diffusion in a Time-dependent External Field

TL;DR

This paper investigates diffusion processes in time-dependent external fields using a generalized master equation, focusing on the quasi Fokker-Planck approximation and introducing a new collision integral for particle systems.

Contribution

It extends the generalized master equation framework to include time-dependent fields and introduces a novel collision integral for systems with moving and resting particles.

Findings

Validation of the phenomenological approach through kinetic equation solutions

Derivation of diffusion behavior in time-dependent external fields

Introduction of a new collision integral for particle state transitions

Abstract

The problem of diffusion in a time-dependent (and generally inhomogeneous) external field is considered on the basis of a generalized master equation with two times, introduced in [1,2]. We consider the case of the quasi Fokker-Planck approximation, when the probability transition function for diffusion (PTD-function) does not possess a long tail in coordinate space and can be expanded as a function of instantaneous displacements. The more complicated case of long tails in the PTD will be discussed separately. We also discuss diffusion on the basis of hydrodynamic and kinetic equations and show the validity of the phenomenological approach. A new type of "collision" integral is introduced for the description of diffusion in a system of particles, which can transfer from a moving state to the rest state (with some waiting time distribution). The solution of the appropriate kinetic equation in the external field also confirms the phenomenological approach of the generalized master equation.