Nonlinear Fokker-Planck equation: stability, distance and corresponding extremal problem in the spatially inhomogeneous case

TL;DR

This paper introduces a generalized distance measure between solutions of the nonlinear Fokker-Planck equation and global Maxwellians, demonstrating local stability in the spatially inhomogeneous case and comparing classical and quantum energy and entropy.

Contribution

It generalizes the Kullback-Leibler distance for the nonlinear Fokker-Planck equation and proves local stability of Maxwellians in inhomogeneous settings.

Findings

Generalized distance measure between solutions and Maxwellians.

Proved local stability of Maxwellians in inhomogeneous case.

Compared classical and quantum energy and entropy.

Abstract

We start with a global Maxwellian $M_{k}$, which is a stationary solution, with the constant total density ($\rho(t)\equiv \wt \rho$), of the Fokker-Planck equation. The notion of distance between the function $M_{k}$ and an arbitrary solution $f$ (with the same total density $\wt \rho$ at the fixed moment $t$) of the Fokker-Planck equation is introduced. In this way, we essentially generalize the important Kullback-Leibler distance, which was studied before. Using this generalization, we show local stability of the global Maxwellians in the spatially inhomogeneous case. We compare also the energy and entropy in the classical and quantum cases.