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Arnold, Anton; López, José L.; Markowich, Peter A.; Soler, Juan

An analysis of quantum Fokker-Planck models: a Wigner function approach. (English) Zbl 1062.35097

Rev. Mat. Iberoam. 20, No. 3, 771-814 (2004).
Summary: The analysis of dissipative transport equations within the framework of open quantum systems with Fokker-Planck-type scattering is carried out from the perspective of a Wigner function approach. In particular, the well-posedness of the self-consistent whole-space problem in 3D is analyzed: existence of solutions, uniqueness and asymptotic behavior in time, where we adopt the viewpoint of mild solutions in this paper. Also, the admissibility of a density matrix formulation in Lindblad form with Fokker-Planck dissipation mechanisms is discussed. We remark that our solution concept allows to carry out the analysis directly on the level of the kinetic equation instead of on the level of the density operator.

Cited in 29 Documents

MSC:

35Q40 PDEs in connection with quantum mechanics
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
35S10 Initial value problems for PDEs with pseudodifferential operators
82C70 Transport processes in time-dependent statistical mechanics

Keywords:

Wigner-Poisson-Fokker-Planck system; local-in-time solutions; uniqueness; stability; large-time behavior
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\textit{A. Arnold} et al., Rev. Mat. Iberoam. 20, No. 3, 771--814 (2004; Zbl 1062.35097)
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