Long time behavior of Fokker-Planck equations for bosons and fermions

TL;DR

This paper investigates the long-term behavior of quantum Fokker-Planck equations for bosons and fermions, proving exponential decay to equilibrium without assuming initial proximity to equilibrium, using an $L^2$-hypocoercivity approach.

Contribution

It introduces a novel $L^2$-hypocoercivity method to prove exponential decay for space-inhomogeneous quantum Fokker-Planck equations with nonlinear terms.

Findings

Solutions decay exponentially to equilibrium

No close-to-equilibrium assumption needed

Uses a new Lyapunov functional based on entropy

Abstract

This paper is concerned with space inhomogeneous quantum Fokker-Planck equations posed on a classical kinetic phase space. The nonlinear factor $f(1\pm f)$ appears both in the transport term and in the collison part of the Fokker-Planck operator, accounting for the inclusion principle of bosons and the exclusion principle of fermions. Assuming that global solutions exist, we prove exponential decay of the solutions to the global equilibrium in a weighted $L^2$-space without a close-to-equilibrium assumption. Our analysis is in the spirit of an $L^2$-hypocoercivity method. Our main Lyapunov functional is constructed from a logarithmic relative entropy and the (nonlinear) projection of the solution to the manifold of local-in-$x$ equilibria.