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

ANTON ARNOLD, IRENE M. GAMBA, MARIA PIA GUALDANI, STÉPHANE MISCHLER, CLEMENT MOUHOT, CHRISTOF SPARBER

Research output: Contribution to journalArticlepeer-review

14 Scopus citations

Abstract

We consider the linear WignerFokkerPlanck 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 FokkerPlanck 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. © 2012 World Scientific Publishing Company.
Original languageEnglish (US)
Pages (from-to)1250034
JournalMathematical Models and Methods in Applied Sciences
Volume22
Issue number11
DOIs
StatePublished - Sep 10 2012
Externally publishedYes

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