The Fokker-Planck equation for bosons in 2D: well-possedness and asymptotic analysis

TL;DR

This paper proves global existence and exponential convergence to equilibrium for solutions of the 2D Fokker-Planck equation for bosons, using a transformation to relate it to the linear Fokker-Planck equation.

Contribution

It introduces a novel approach using a Hopf-Cole transformation to analyze the 2D bosonic Fokker-Planck equation and establishes convergence results.

Findings

Solutions are globally well-posed in time.

Radially symmetric solutions converge exponentially to equilibrium.

A Csiszár-Kullback inequality for the Bose-Einstein-Fokker-Planck entropy is proved.

Abstract

We show that solutions of the 2D Fokker-Planck equation for bosons are defined globally in time and converge to equilibrium, and this convergence is shown to be exponential for radially symmetric solutions. The main observation is that a variant of the Hopf-Cole transformation relates the 2D equation in radial coordinates to the usual linear Fokker-Planck equation. Hence, radially symmetric solutions can be computed analytically, and our results for general (non radially symmetric) solutions follow from comparison and entropy arguments. In order to show convergence to equilibrium we also prove a version of the Csisz\'ar-Kullback inequality for the Bose-Einstein-Fokker-Planck entropy functional.