A short review on the derivation of the nonlinear quantum Boltzmann equations

Communications in Mathematical Sciences

Volumn , Issue SUPPL. 1, 2007, Pages 55-71

  Benedetto, D ^a   Castella, F ^b   Esposito, R ^c   Pulvirenti, M ^a  

^a SAPIENZA UNIVERSITY OF ROME   (Italy)

^b UNIVERSITÉ DE RENNES 1   (France)

^c UNIVERSITY OF L'AQUILA   (Italy)

Author keywords

Kinetic equations; Low density; Quantum systems; Stationary phase; Weak coupling

Indexed keywords

EID: 34247523869     PISSN: 15396746     EISSN: None     Source Type: Journal    
DOI: 10.4310/cms.2007.v5.n5.a5     Document Type: Review

References (19)

1. R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics, John Wiley & Sons, New-York, 1975.
2. D. Benedetto, F. Castella, R. Esposito and M. Pulvirenti, Some considerations on the derivation of the nonlinear Quantum Boltzmann equation, J. Stat. Phys., 116(114), 381-410, 2004.
3. D. Benedetto, F. Castella, R. Esposito and M. Pulvirenti, On the weak-coupling limit for Bosons and Fermions, Math. Mod. Meth. Appl. Sci., 15(12), 1811-1843, 2005.
4. D. Benedetto, F. Castella, R. Esposito and M. Pulvirenti, Some considerations on the derivation of the nonlinear Quantum Boltzmann equation II: the low-density regime, J. Stat. Phys., 124(2-4), 951-996, 2006.
5. F. Castella, From the von Neumann equation to the Quantum Boltzmann equation in a deterministic framework, J. Stat. Phys., 104(1/2), 387-447, 2001.
6. F. Castella, From the von Neumann equation to the Quantum Boltzmann equation II: identifying the Born series, J. Stat. Phys., 106(5/6), 1197-1220, 2002.
7. S. Chapman and T. G. Cowling, The Mathematical Theory of Non-uniform Gases, Cambridge Univ. Press, Cambridge, England, 1970.
8. C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Appl. Math. Sci., Springer-Verlag, New York, 106, 1994.
9. L. Erdös, M. Salmhofer and H.-T. Yau, On the quantum Boltzmann equation, J. Stat. Phys., 116(1-4), 367-380, 2004.
10. M. N. Hugenholtz, Derivation of the Boltzmann equation for a Fermi gas, J. Stat. Phys., 32, 231-254, 1983.
11. N. T. Ho and L. J. Landau, Fermi gas in a lattice in the van Hove limit, J. Stat. Phys., 87, 821-845, 1997.
12. R. Illner and M. Pulvirenti, Global validity of the Boltzmann equation for a two-dimensional rare gas in the vacuum, Comm. Math. Phys., 105, 189-203, 1986, Erratum and improved result, Comm. Math. Phys., 121, 143-146, 1989.
13. O. Lanford III, Time evolution of large classical systems, Lecture Notes in Physics, 38, 1-111, E. J. Moser ed., Springer-Verlag, 1975.
14. P. L. Lions and T. Paul, Sur les mesures de Wigner, Revista Matematica ibero americana, 9(3), 553-618, 1993.
15. M. Reed and B. Simon, Methods of Modern Mathematical Physics III. Scattering Theory, Academic Press, New York-London, 1979.
16. H. Spohn, Large Scale Dynamics of Interacting Particles, Springer, 1991.
17. H. Spohn, Quantum kinetic equations, in "On Three Levels: Micro-Meso and Macro Approaches in Physics", M. Fannes, C. Maes, A Verbeure eds, Nato Series B: Physics, 324, 1-10, 1994.
18. E. A. Uehling and G. E. Uhlenbeck, Transport phenomena in Einstein-Bose and Fermi-Dirac gases. I, Phys. Rev., 43, 552-561, 1933.
19. E. P. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev., 40, 749C759, 1932.