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THE WIGNER-FOKKER-PLANCK EQUATION: STATIONARY STATES AND LARGE TIME BEHAVIOR
 

Summary: THE WIGNER-FOKKER-PLANCK EQUATION: STATIONARY
STATES AND LARGE TIME BEHAVIOR
ANTON ARNOLD, IRENE M. GAMBA, MARIA PIA GUALDANI, ST´EPHANE MISCHLER,
CL´EMENT MOUHOT, AND CHRISTOF SPARBER
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 ad-
mits 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 L2­spaces. 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.
1. Introduction
This work is devoted to the study of the Wigner-Fokker-Planck equation (WFP),
considered in the following dimensionless form (where all physical constants are
normalized to one for simplicity):
(1.1)
tw + · xw + [V ]w = w + 2 div (w) + xw,
w t=0

  

Source: Arnold, Anton - Institut für Analysis und Scientific Computing, Technische Universität Wien
Gamba, Irene M.- Department of Mathematics, University of Texas at Austin
Gualdani, Maria Pia - Department of Mathematics, University of Texas at Austin

 

Collections: Mathematics