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| HAL: hal-00526815, version 1 |
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| The Wigner-Fokker-Planck equation: Stationary states and large time behavior |
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| Anton Arnold 1Irene Gamba 2 |
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| (2010-10-15) |
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| 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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| 1: | Institut für Analysis und Scientific Computing |
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Technische Universität Wien |
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| 2: | Departement of Mathematics (UT, USA) |
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University of Texas at Austin |
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| 3: | CEntre de REcherches en MAthématiques de la DEcision (CEREMADE) |
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CNRS : UMR7534 – Université Paris IX - Paris Dauphine |
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| 4: | DPMMS/CMS |
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University of Cambridge |
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| 5: | Department of Mathematics, Statistics, and Computer Science (UIC) |
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University of Illinois at Chicago |
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| Subject | : | Mathematics/Analysis of PDEs Physics/Mathematical Physics Mathematics/Mathematical Physics Physics/Quantum Physics |
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| Wigner transform – Fokker Planck operator – Spectral gap – Stationary solution – Large time behavior |
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| Attached file list to this document: | ||||||||||
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| hal-00526815, version 1 | |
| http://hal.archives-ouvertes.fr/hal-00526815 | |
| oai:hal.archives-ouvertes.fr:hal-00526815 | |
| From: Clément Mouhot | |
| Submitted on: Friday, 15 October 2010 20:17:17 | |
| Updated on: Monday, 18 October 2010 09:57:32 | |
