Classical limit for linear and nonlinear quantum Fokker-Planck systems

Roberta Bosi

doi:10.3934/cpaa.2009.8.845

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2009, Volume 8, Issue 3: 845-870. Doi: 10.3934/cpaa.2009.8.845

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Classical limit for linear and nonlinear quantum Fokker-Planck systems

Roberta Bosi^1,

1.
Institut für Analysis und Scientific Computing, Wiedner Hauptstr. 8, A 1040 Wien, Österreich

Received: February 29, 2008

Revised: November 30, 2008

Published: April 2009

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Abstract

We study the classical limit of some linear and nonlinear Quantum Fokker-Planck systems. In the nonlinear case we consider an Hartree-type potential. By the use of the Wigner transform and compactness methods, we prove the convergence of the system to a linear and nonlinear Vlasov Fokker- Planck equation respectively. The physical case with a Poisson coupling in three dimensions is included.

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Mathematics Subject Classification: 81Q15, 82C10, 35Q40, 35S10, 47J35, 47H06, 47H20, 81V70, 81Q99, 81V99.

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