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arXiv · 1010.2791

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

Anton Arnold·Irene M. Gamba·Maria Pia Gualdani·Stephane Mischler·Clement Mouhot·Christof Sparber
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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.

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Anton Arnold, Irene M. Gamba, Maria Pia Gualdani, Stephane Mischler, Clement Mouhot, Christof Sparber. 2012-01-20. The Wigner-Fokker-Planck equation: Stationary states and large-time behavior. https://arxiv.org/abs/1010.2791

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