Abstract
In this paper we analyze a system of Nidentical quantum particles in a weak-coupling regime. The time evolution of the Wigner transform of the one-particle reduced density matrix is represented by means of a perturbative series. The expansion is obtained upon iterating the Duhamel formula. For short times, we rigorously prove that a subseries of the latter, converges to the solution of the Boltzmann equation which is physically relevant in the context. In particular, we recover the transition rate as it is predicted by Fermi's Golden Rule. However, we are not able to prove that the quantity neglected while retaining a subseries of the complete original perturbative expansion, indeed vanishes in the limit: we only give plausibility arguments in this direction. The present study holds in any space dimension d≥2.
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REFERENCES
N. W. Ashcroft and N. D. Mermin, Solid State Physics(Saunders, Philadelphia, 1976).
A. Bohm, Quantum Mechanics, Texts and monographs in Physics (Springer-Verlag, 1979).
C. Boldrighini, L. A. Bunimovich, and Ya. Sinai, On the Boltzmann equation for the Lorentzgas, J. Stat. Phys. 32:477–501 (1983).
J. Bourgain, F. Golse, and B. Wennberg, On the distribution of free path lengths for the periodic Lorentzgas, Comm. Math. Phys. 190:491–508 (1998).
F. Castella, From the von Neumann equation to the Quantum Boltzmann equation in a deterministic framework, J. Stat. Phys. 104:387–447 (2001).
F. Castella, From the von Neumann equation to the Quantum Boltzmann equation II: Identifying the Born series, J. Stat. Phys. 106:1197–1220 (2002).
F. Castella and A. Plagne, A distribution result for slices of sums of squares, Math. Proc. Cambridge Philos. Soc. 132:1–22 (2002).
F. Castella and A. Plagne, Non-derivation of the Quantum Boltzmann equation from the periodic Schrödinger equation, to appear in Indiana Univ. Math. J.(2003).
S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases(Cambridge Univ. Press, Cambridge, England, 1970).
C. Cercignani, R. Illner, and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, Vol. 106 (Springer-Verlag, New York, 1994).
S. L. Chuang, Physics of Optoelectronic Devices, Wiley Series in Pure and Applied Optics (New York, 1995).
C. Cohen-Tannoudji, B. Diu, and F. Laloë, Mécanique Quantique, I et II, Enseignement des Sciences, Vol. 16 (Hermann, 1973).
M. Combescot, On the generalized golden rule for transition probabilities, Phys. A: Math. Gen. 34:6087–6104 (2001).
R. Dümcke, The low density limit for an N-level system interacting with a free Bose or Fermi gas, Comm. Math. Phys. 97:331–359 (1985).
D. Dürr, S. Goldstein, and J. L. Lebowitz, Asymptotic motion of a classical particle in a random potential in two dimension: The Landau model, Comm. Math. Phys. 113:209–230 (1987).
L. Erdös, M. Salmhofer, and H.-T. Yau, On the Quantum Boltzmann Equation, Mathematical Physics Archive, 03–56 (University of Texas, 2003).
L. Erdös and H. T. Yau, Linear Boltzmann equation as scaling limit of quantum Lorentz gas, in Advances in Differential Equations and Mathematical Physics (Atlanta, GA, 1997), Contemp. Math., Vol. 217 (Amer. Math. Soc., Providence, RI, 1998), pp. 137–155.
L. Erdös and H. T. Yau, Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation, Comm. Pure Appl. Math. 53:667–735 (2000).
R. Esposito, M. Pulvirenti, and A. Teta, The Boltzmann equation for a one-dimensional quantum Lorentzgas, Comm. Math. Phys. 204:619–649 (1999); Erratum, Comm. Math. Phys. 214:493-494 (2000).
M. V. Fischetti, Theory of electron transport in small semiconductor devices using the Pauli master equations, J. Appl. Phys. 83:270–291 (1998).
G. Gallavotti, Time Evolution Problems in Classical Statistical Mechanics and the Wind-Tree-Model, Cargese Lectures in Physics, Vol. IV, D. Kastler, ed. (Gordon & Breach, Paris, 1970); Divergences and approach to equilibrium in the Lorenzand the wind-tree models, Phys. Rev. 185:308-322 (1969). See also the book Statistical Mechanics, Appendix A2 to Ch. 1 (Springer-Verlag, 1999).
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. 5:209–298 (1993).
L. Hörmander, The Analysis of Linear Partial Differential Operators(Springer-Verlag, Berlin, 1994).
K. Huang, Statistical Mechanics(Wiley, 1963).
R. Illner and M. Pulvirenti, Global Validity of the Boltzmann equation for a two dimensional rare gas in vacuum, Comm. Math. Phys. 105:189–203 (1986); Erratum and improved result, Comm. Math. Phys. 121:143-146 (1989).
J. B. Keller, G. Papanicolaou, and L. Ryzhik, Transport equations for elastic and other waves in random media, Wave Motion 24:327–370 (1996).
W. Kohn and J. M. Luttinger, Phys. Rev. 108:590 (1957).
W. Kohn and J. M. Luttinger, Phys. Rev. 109:1892 (1958).
R. Kubo, J. Phys. Soc. Jap. 12:000–000 (1958).
O. Lanford, III, Time Evolution of Large Classical Systems, Lecture Notes in Physics, Vol. 38, E. J. Moser, ed. (Springer-Verlag, 1975), pp. 1–111.
L. J. Landau, Observation of quantum particles on a large space-time scale, J. Stat. Phys. 77:259–309 (1994).
P. A. Markowich, C. Ringhofer, and C. Schmeiser, Semiconductor Equations(Springer-Verlag, Vienna, 1990).
F. Nier, Asymptotic analysis of a scaled Wigner equation and quantum scattering, Transp. Theor. Stat. Phys. 24:591–629 (1995).
F. Nier, A semi-classical picture of quantum scattering, Ann. Sci. Ec. Norm. Sup., 4. Sér. 29:149–183 (1996).
F. Poupaud and A. Vasseur, Classical and Quantum Transport in Random Media, preprint (University of Nice, 2001).
W. Pauli, Festschrift zum 60. Geburtstage A. Sommerfelds(Hirzel, Leipzig, 1928), p. 30.
M. Reed and B. Simon, Methods of Modern Mathematical Physics III . Scattering Theory(Academic Press, New York/London, 1979).
E. Rosencher and B. Vinter, Optoelectronique(Dunod, 2002).
H. Spohn, Derivation of the transport equation for electrons moving through random impurities, J. Stat. Phys. 17:385–412 (1977).
H. Spohn, Large Scale Dynamics of Interacting Particles(Springer, 1991).
H. Spohn, Kinetic equations from Hamiltonian dynamics: Markovian limits, Rev. Modern Phys. 52:569–615 (1980).
H. Spohn, Quantum kinetic equations, in On Three Levels: Micro-Meso and Macro Approaches in Physics, Vol. 324, M. Fannes, C. Maes, and A. Verbeure, eds., Nato ASI Series B: Physics (1994), pp. 1–10.
L. Van Hove, Physica 21:517 (1955).
L. Van Hove, Physica 23:441 (1957).
L. Van Hove, in Fundamental Problems in Statistical Mechanics, E. G. D. Cohen, ed. (1962), p. 157.
R. Zwanzig, Quantum Statistical Mechanics, P. H. E. Meijer, ed. (Gordon & Breach, New York, 1966).
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Benedetto, D., Castella, F., Esposito, R. et al. Some Considerations on the Derivation of the Nonlinear Quantum Boltzmann Equation. Journal of Statistical Physics 116, 381–410 (2004). https://doi.org/10.1023/B:JOSS.0000037205.09518.3f
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DOI: https://doi.org/10.1023/B:JOSS.0000037205.09518.3f