Generalised Kramers model

TL;DR

This paper extends the classical Kramers model by incorporating position- and time-dependent stochastic forces, analyzing stationary solutions under different external forcing regimes to understand particle dynamics in complex potentials.

Contribution

It introduces a generalized Kramers model with state-dependent noise and analyzes its stationary solutions in weak and strong forcing limits.

Findings

Stationary solution resembles an increased potential in weak forcing.

Non-zero probability flux appears in strong forcing with broken symmetry.

Model captures more realistic particle behaviors in complex environments.

Abstract

We study a particular generalisation of the classical Kramers model describing Brownian particles in the external potential. The generalised model includes the stochastic force which is modelled as an additive random noise that depends upon the position of the particle, as well as time. The stationary solution of the Fokker-Planck equation is analysed in two limits: weak external forcing, where the solution is equivalent to the increase of the potential compared to the classical model, and strong external forcing, where the solution yields a non-zero probability flux for the motion in a periodic potential with a broken reflection symmetry.