Spectral decomposition approach to macroscopic parameters of Fokker-Planck flows: Part 1

TL;DR

This paper explores the spectral decomposition of Fokker-Planck equations to derive macroscopic parameters, revealing connections to continuum mechanics and classical hydrodynamics, including potential velocity fields and diffusion behavior.

Contribution

It introduces a spectral decomposition method focusing on minimal damping terms to connect Fokker-Planck solutions with hydrodynamic equations and Burgers equation.

Findings

Velocity fields are potential and proportional to the density logarithm.

Derived macroscopic parameters satisfy classical hydrodynamics equations.

Potential velocity fields satisfy Burgers equation without mass forces.

Abstract

In this paper we proceed with investigation of connections between Fokker - Planck equation and continuum mechanics. In spectral decomposition of Fokker - Planck equation solution we preserve only terms with the smallest degree of damping. We find, that macroscopic parameters of Fokker-Planck flows, obtained in this way, have following properties: velocities field possess potential, its potential is proportional to density logarithm and satisfy diffusion equation. We proved, that such a pair of density and velocities field satisfy the set of classic hydrodynamics equations for isothermal compressible fluid with friction mass force, proportional to velocity. We proved also, that the potential velocities field alone, with potential, which satisfy diffusion equation, satisfy Burgers equation without mass forces.