The linear Boltzmann equation as the low density limit of a random Schrödinger equation

Reviews in Mathematical Physics

Volumn 17, Issue 6, 2005, Pages 669-743

  Eng, David ^a   Erdos, László ^b  

^a NEW YORK UNIVERSITY   (United States)

^b UNIVERSITY OF MUNICH   (Germany)

Author keywords

Anderson model; Boltzmann Grad limit; Lorentz gas; Quantum Boltzmann equation

Indexed keywords

EID: 23744437682     PISSN: 0129055X     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0129055X0500242X     Document Type: Article

References (21)

1. M. Aizenman, Localization at weak disorder: Some elementary bounds, Rev. Math. Phys. 6 (1994) 1163-1182.
2. M. Aizenman and S. Molchanov, Localization at large disorder and at extreme energies: An elementary derivation, Commun. Math. Phys. 157 (1993) 245-278.
3. P. Anderson, Absences of diffusion in certain random lattices, Phys. Rev. 109 (1958) 1492-1505.
4. C. Boldrighini, L. Bunimovich and Y. Sinai, On the Boltzmann equation for the Lorentz gas, J. Stat. Phys. 32 (1983) 477-501.
5. T. Chen, Localization lengths and Boltzmann limit for the Anderson model at small disorders in dimension 3, preprint, xxx.lanl.gov/math-ph/0305051.
6. r convergence a random Schrödinger to a linear Boltzmann evolution, preprint, xxx.lanl.gov/math-ph/0407037.
7. D. Dürr, S. Goldstein and J. Lebowitz, Asymptotic motion of a classical particle in a random potential in two dimensions: Landau model, Commun. Math. Phys. 113 (1987) 209-230.
8. H. von Dreifus and A. Klein, Localization for random Schrödinger operators with correlated potentials, Commun. Math. Phys. 140 (1991) 133-147.
9. L. Erdos and H.-T. Yau, Linear Boltzmann equation as the weak coupling limit of the random Schrödinger equation, Comm. Pure Appl. Math. LIII (2000) 667-735.
10. L. Erdos and H.-T. Yau, Linear Boltzmann equation as the scaling limit of quantum Lorentz gas, Advances in Differential Equations and Mathematical Physics, Contemporary Mathematics 217 (1998) 137-155.
11. J. Fröhlich and T. Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Commun. Math. Phys. 88 (1983) 151-184.
12. J. Fröhlich, F. Martinelli, S. Scoppola and T. Spencer, Constructive proof of localization in the Anderson tight binding model, Commun. Math. Phys. 101 (1985) 21-46.
13. G. Gallavotti, Rigorous theory of the Boltzmann equation in the Lorentz gas, Nota, inteerna n. 358, Univ. di Roma (1970).
14. 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 (1992) 209-298.
15. H. Kesten and G. Papanicolaou, A limit theorem for stochastic acceleration, Commun. Math. Phys. 78 (1980) 19-63.
16. A. Klein, Absolutely continuous spectrum in the Anderson model on the Bethe lattice, Math. Res. Lett. 1 (1994) 399-407.
17. A. Klein, Spreading of wave packets in the Anderson model on the Bethe lattice, Commun. Math. Phys. 177 (1996) 755-773.
18. L. J. Landau, Observation of quantum particles on a large space-time scale, J. Stat. Phys. 77 (1994) 259-309.
19. M. Reed and B. Simon, Methods of modern mathematical physics, Scattering Theory, Vol. 3 (Academic Press, 1980).
20. H. Spohn, Derivation of the transport equation for electrons moving through random impurities, J. Stat. Phys. 17 (1977) 385-412.
21. H. Spohn, The Lorentz process converges to a random flight process, Commun. Math. Phys. 60 (1978) 277-290.