Spectral decomposition approach to macroscopic parameters of Fokker-Planck flows: Part 2

TL;DR

This paper explores the spectral decomposition of the Fokker-Planck equation to derive macroscopic parameters that obey hydrodynamic conservation laws, revealing negligible additional terms at large times and limitations of potential velocity assumptions.

Contribution

It introduces a spectral decomposition method focusing on low damping terms to connect Fokker-Planck flows with continuum mechanics, extending previous potential velocity models.

Findings

Macroscopic parameters satisfy hydrodynamic conservation laws.

Additional stress and velocity terms are negligible at large times.

Zero degree potential velocity theory cannot specify initial velocities independently.

Abstract

In this paper we proceed with investigation of connections between Fokker - Planck equation and continuum mechanics. We base upon expressions from our work [2], based upon the spectral decomposition of Fokker - Planck equation solution. In this decomposition we preserve only terms with the smallest degrees of damping. We find, that macroscopic parameters of Fokker-Planck flows, obtained in this way, satisfy the set of conservation laws of classic hydrodynamics. The expression for stresses (30) contains additional term - this term is negligible in big times limit. We proved also, that the velocities field alone satisfy Burgers equation without mass forces - but with some additional term. This term is also negligible in big times limit. For the zero degree theory, considered in [1], there are no additional terms. But this theory is valid only for the potential velocities field, fully deductible from density - the potential is proportional to density logarithm. In this theory we can not specify initial conditions for velocities independently from density. Taking in account of the next degree terms could partly solve this problem, but result in some loss of exactness.