A shortcut way to the Fokker-Planck equation for the non-Markovian dynamics

TL;DR

This paper introduces a simplified method to derive the Fokker-Planck equation for non-Markovian Langevin dynamics with various complex forces, enhancing analytical tools in stochastic thermodynamics.

Contribution

The paper presents a novel shortcut approach to derive the Fokker-Planck equation applicable to a broad class of linear non-Markovian systems with colored noise and additional forces.

Findings

Method is applicable to systems with harmonic potential and magnetic fields.

Simplifies derivation of Fokker-Planck equations for complex non-Markovian dynamics.

Potential to advance research in stochastic thermodynamics.

Abstract

Using a shortcut way we have derived the Fokker-Planck equation for the Langevin dynamics with a generalized frictional memory kernel and time-dependent force field. Then we have shown that this method is applicable for the non-Markovian dynamics with additional force from harmonic potential or magnetic field or both of them. The simplicity of the method in these complex cases is highly noticeable and it is applicable to derive the Fokker-Planck equation for any kind of linear Langevin equation of motion which describes additive colored noise driven non Markovian dynamics with or without frictional memory kernel. For example, one may apply the method even for the linear system with an additional colored Gaussian noise which is not related to the damping strength. With these the present study may get strong attention in the field of stochastic thermodynamics which is now at early stage to consider the non-Markovian dynamics.