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

Anton Arnold (1) , Irene Gamba (2) , Maria Pia Gualdani (2) , Stéphane Mischler (3) , Clément Mouhot (4) , Christof Sparber (5)

1 Institut für Analysis und Scientific Computing
2 Departement of Mathematics [Austin]
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
4 DPMMS - Department of Pure Mathematics and Mathematical Statistics
5 UIC - Department of Mathematics, Statistics and Computer Science [Chicago]

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.

Mots clés: Wigner transform; Fokker Planck operator; Spectral gap; Stationary solution; Large time behavior; Wigner transform

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Physique mathématique [math-ph]; Physique Quantique [quant-ph]

Anton Arnold, Irene Gamba, Maria Pia Gualdani, Stéphane Mischler, Clément Mouhot, et al.. The Wigner-Fokker-Planck equation: Stationary states and large time behavior. Mathematical Models and Methods in Applied Sciences, 2012, 22 (11), pp.1250034. ⟨10.1142/S0218202512500340⟩. ⟨hal-00526815⟩