American Institute of Mathematical Sciences

Advanced
May  2009, 8(3): 845-870. doi: 10.3934/cpaa.2009.8.845

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  March 2008 Revised  December 2008 Published  February 2009

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.
Keywords: Fokker-Planck operator, quantum transport, Classical limit, Hartree equation, Wigner measure, open quantum system..
Mathematics Subject Classification: 81Q15, 82C10, 35Q40, 35S10, 47J35, 47H06, 47H20, 81V70, 81Q99, 81V9.
Citation: Roberta Bosi. Classical limit for linear and nonlinear quantum Fokker-Planck systems. Communications on Pure & Applied Analysis, 2009, 8 (3) : 845-870. doi: 10.3934/cpaa.2009.8.845
[1]

Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250
[2]

Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks & Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028
[3]

Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017
[4]

Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic & Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485
[5]

Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic & Related Models, 2018, 11 (2) : 337-355. doi: 10.3934/krm.2018016
[6]

José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401
[7]

Patrick Cattiaux, Elissar Nasreddine, Marjolaine Puel. Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case. Kinetic & Related Models, 2019, 12 (4) : 727-748. doi: 10.3934/krm.2019028
[8]

Zeinab Karaki. Trend to the equilibrium for the Fokker-Planck system with an external magnetic field. Kinetic & Related Models, 2020, 13 (2) : 309-344. doi: 10.3934/krm.2020011
[9]

Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic & Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056
[10]

Andreas Denner, Oliver Junge, Daniel Matthes. Computing coherent sets using the Fokker-Planck equation. Journal of Computational Dynamics, 2016, 3 (2) : 163-177. doi: 10.3934/jcd.2016008
[11]

Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028
[12]

Li Chen, Xiu-Qing Chen, Ansgar Jüngel. Semiclassical limit in a simplified quantum energy-transport model for semiconductors. Kinetic & Related Models, 2011, 4 (4) : 1049-1062. doi: 10.3934/krm.2011.4.1049
[13]

Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215
[14]

Martin Burger, Ina Humpert, Jan-Frederik Pietschmann. On Fokker-Planck equations with In- and Outflow of Mass. Kinetic & Related Models, 2020, 13 (2) : 249-277. doi: 10.3934/krm.2020009
[15]

Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic & Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165
[16]

Ling Hsiao, Fucai Li, Shu Wang. Combined quasineutral and inviscid limit of the Vlasov-Poisson-Fokker-Planck system. Communications on Pure & Applied Analysis, 2008, 7 (3) : 579-589. doi: 10.3934/cpaa.2008.7.579
[17]

Ludovic Dan Lemle. $L^1(R^d,dx)$-uniqueness of weak solutions for the Fokker-Planck equation associated with a class of Dirichlet operators. Electronic Research Announcements, 2008, 15: 65-70. doi: 10.3934/era.2008.15.65
[18]

Joseph G. Conlon, André Schlichting. A non-local problem for the Fokker-Planck equation related to the Becker-Döring model. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1821-1889. doi: 10.3934/dcds.2019079
[19]

Simon Plazotta. A BDF2-approach for the non-linear Fokker-Planck equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2893-2913. doi: 10.3934/dcds.2019120
[20]

Ruikuan Liu, Tian Ma, Shouhong Wang, Jiayan Yang. Thermodynamical potentials of classical and quantum systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1411-1448. doi: 10.3934/dcdsb.2018214

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences