The Wigner-Fokker-Planck equation: Stationary states and large-time behavior

TL;DR

This paper analyzes the Wigner-Fokker-Planck equation with confining potentials, proving existence and uniqueness of stationary states, spectral gap properties, and exponential convergence of solutions to equilibrium.

Contribution

It establishes the existence of a unique stationary solution for perturbed harmonic potentials and demonstrates exponential convergence to this steady state.

Findings

Existence of a unique stationary solution in weighted Sobolev spaces.

Spectral gap results for Fokker-Planck operators.

Exponential convergence of solutions to the steady state.

Abstract

We consider the linear Wigner-Fokker-Planck equation subject to confining potentials which are smooth perturbations of the harmonic oscillator potential. For a certain class of perturbations we prove that the equation admits a unique stationary solution in a weighted Sobolev space. A key ingredient of the proof is a new result on the existence of spectral gaps for Fokker-Planck type operators in certain weighted $L^2$-spaces. In addition we show that the steady state corresponds to a positive density matrix operator with unit trace and that the solutions of the time-dependent problem converge towards the steady state with an exponential rate.