On the derivation of a quantum Boltzmann equation from the periodic Von-Neumann equation
Castella, François
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 33 (1999), p. 329-349 / Harvested from Numdam
Publié le : 1999-01-01
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     author = {Castella, Fran\c cois},
     title = {On the derivation of a quantum Boltzmann equation from the periodic Von-Neumann equation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {33},
     year = {1999},
     pages = {329-349},
     mrnumber = {1700038},
     zbl = {0954.82023},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_1999__33_2_329_0}
}
Castella, François. On the derivation of a quantum Boltzmann equation from the periodic Von-Neumann equation. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 33 (1999) pp. 329-349. http://gdmltest.u-ga.fr/item/M2AN_1999__33_2_329_0/

[1] J.R. Barker, Fundamental aspects of Quantum Transport, in Handbook on Semiconductors, T.S. Moss Ed., North-Holland (1982).

[2] N. Ben Abdallah and P. Degond, On a hierarchy of macroscopic models for semi-conductors. J. Math. Phys. 37 (1996) 3306-3333. | MR 1401227 | Zbl 0868.45006

[3] J. Bourgain, F. Golse and B. Wennberg, On the distribution of free path lengths for the periodic Lorentz gas. Comm. Math. Phys. 190 (1998) 491-508. | MR 1600299 | Zbl 0910.60082

[4] A.O. Caldeira and A.J. Leggett, Path Integral Approach to Brownian Motion. Physica A 121 (1983) 587-616. | MR 726154 | Zbl 0585.60082

[5] D. Calecki, Lecture, University of Paris 6 (1997).

[6] F. Castella, Ph.D. thesis, University of Paris VI, France (1997).

[7] F. Castella and P. Degond, The Von-Neumann equation with deterministic potential converges towards the Quantum Boltzmann equation. Preprint (1999).

[8] F. Castella, L. Erdös, F. Frommlet and P.A. Markowich, Caldeira-Leggett Master equation. Work in progress (1999).

[9] C. Cohen-Tannoudji, B. Diu and F. Laloë, Mécanique Quantique, I et II, Enseignement des Sciences, Vol. 16. Hermann (1973).

[10] L. Erdös and H.T. Yau, Linear Boltzmann equation as scaling limit of quantum Lorentz gas. Preprint (1998). | Zbl 0894.35027

[11] R. Esposito, M. Pulvirenti and A. Teta, The Boltzmann Equation for a one-dimensional Quantum Lorentz gas. Preprint (1998). | Zbl 0940.35167

[12] T.G. Ho, L.J. Landau and A.J. Wilkins, On the weak coupling limit for a Fermi gas in a random potential. Rev. Math, Phys,5 (1993) 209-298. | Zbl 0816.46079

[13] R. Jancel, Foundations of Classical and Quantum Statistical Mechanics. Pergamon, Braunschweig (1969).

[14] W. Kohn and J.M. Luttinger, Phys. Rev. 108 (1957) 590-611. | Zbl 0092.21901

[15] W. Kohn and J.M. Luttinger, Phys. Rev. 109 (1958) 1892. | Zbl 0092.21902

[16] H.J. Kreuzer, Nonequilibrium thermodynamics and its statistical foundations. Monographs on Physics and Chemistry of Materials. Oxford Science Publications (1983). | MR 602695

[17] R. Kubo, J. Phys. Soc. Jap. 12 (1957) 570. | MR 98482

[18] P.L. Lions and T. Paul, Sur les mesures de Wigner. Revista Matematica ibero americana 9 (1993) 553-618. | MR 1251718 | Zbl 0801.35117

[19] J.M. Luttinger, Mathematical Methods in Solid State and Superfluid Theory, Oliver and Boyd Eds. (1968) 157.

[20] A. Majorana and S.A. Marano, Space homogeneous solutions to the Cauchy problem for the semi-conductor Boltzmann equation. SIAM J. Math. Anal. 28 (1997) 1294-1308. | MR 1474215 | Zbl 0896.45006

[21] P.A. Markowich, C. Ringhoffer and C. Schmeiser, Semiconductor equations. Springer-Verlag, Wien (1990). | MR 1063852 | Zbl 0765.35001

[22] P.A. Markowich and C. Schmeiser, The drift-diffus ion limit for electron-phonon interaction in semiconductors. Math. Mod. Meth. Appl. Sci. 7 (1997) 707-729. | MR 1460701 | Zbl 0884.45006

[23] F.J. Mustieles, Global existence of solutions for the non-linear Boltzmann equation of semiconductors physics. Rev. Mat. Iberoam. 6 (1990) 43-59. | MR 1086150 | Zbl 0728.45010

[24] F. Nier, Asymptotic Analysis of a scaled Wigner equation and Quantum Scattering. Transp. Theor. Stat. Phys. 24 (1995) 591-629. | MR 1321368 | Zbl 0870.45003

[25] F. Nier, A semi-classical picture of quantum scattering. Ann. Sci. Ec. Norm. Sup. 29 (1996) 149-183. | Numdam | MR 1373932 | Zbl 0858.35106

[26] W. Pauli, Festschrift zum 60 Geburtstage A. Sommerfelds. Hirzel, Leipzig (1928) 30.

[27] I. Prigogine, Non-Equilibrium Statistical Mechanics. Interscience, New-York (1962). | MR 187841 | Zbl 0106.43301

[28] H. Spohn, Derivation of the transport equation for electrons moving through random impurities. J. Stat. Phys. 17 (1997) 385-412. | MR 471824 | Zbl 0964.82508

[29] L. Van Hove, Physica 21 (1955) 517-540. | MR 71346 | Zbl 0065.19505

[30] L. Van Hove, Physica 23 (1957) 441. | MR 89576 | Zbl 0079.19405

[31] L. Van Hove, in Fundamental Problems in Statistical Mechanics, E.G.D. Cohen Ed. (1962) 157.

[32] N.G. Van Kampen, Stochastic processes in physics and chemistry, Lecture Notes in Mathematics. North-Holland 888 (1981). | MR 648937 | Zbl 0511.60038

[33] R. Zwanzig, Quantum Statistical Mechanics, P.H.E. Meijer Ed., Gordon and Breach, New-York (1966).