Abstract
In a previous paper [Ca1], the author studied a low density limit in the periodic von Neumann equation with potential, modified by a damping term. The model studied in [Ca1], considered in dimensions d≥3, is deterministic. It describes the quantum dynamics of an electron in a periodic box (actually on a torus) containing one obstacle, when the electron additionally interacts with, say, an external bath of photons. The periodicity condition may be replaced by a Dirichlet boundary condition as well. In the appropriate low density asymptotics, followed by the limit where the damping vanishes, the author proved in [Ca1] that the above system is described in the limit by a linear, space homogeneous, Boltzmann equation, with a cross-section given as an explicit power series expansion in the potential. The present paper continues the above study in that it identifies the cross-section previously obtained in [Ca1] as the usual Born series of quantum scattering theory, which is the physically expected result. Hence we establish that a von Neumann equation converges, in the appropriate low density scaling, towards a linear Boltzmann equation with cross-section given by the full Born series expansion: we do not restrict ourselves to a weak coupling limit, where only the first term of the Born series would be obtained (Fermi's Golden Rule).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
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 Lorentz gas, J. Stat. Phys. 32:47–501 (1983).
J. Bourgain, F. Golse, and B. Wennberg, On the distribution of free path lengths for the periodic Lorentz gas, Comm. Math. Phys. 190:49–508 (1998).
R. W. Boyd, Nonlinear Optics (Academic Press, 1992).
D. Calecki, Lecture (University Paris 6, 1997).
F. Castella, From the von Neumann equation to the quantum Boltzmann equation in a deterministic framework, to appear in J. Stat. Phys. (2001).
F. Castella, Résultats de convergence et de non-convergence de l'équation de von Neumann périodique vers l'équation de Boltzmann quantique, Séminaire E.D.P. (Ecole Polytechnique, exposé N. XXI, 199–2000).
F. Castella, On the derivation of a Quantum Boltzmann Equation from the periodic von Neumann equation, Mod. Math. An. Num. 33:32–349 (1999).
F. Castella and P. Degond, Convergence de l'équation de von Neumann vers l'équation de Boltzmann Quantique dans un cadre déterministe, C. R. Acad. Sci. Sér. I 329:23–236 (1999).
F. Castella and A. Plagne, A distribution result for slices of sums of squares, To appear in Math. Proc. Camb. Philos. Soc.
F. Castella, A. Plagne, Non-derivation of the Quantum Boltzmann equation from the periodic Schrödinger equation, Preprint (2001).
F. Castella, L. Erdös, F. Frommlet, and P. A. Markowich, Fokker-Planck equations as scaling limits of reversible quantum systems, J. Statist. Phys. 100:54–601 (2000).
C. Cohen-Tannoudji, B. Diu, and F. Laloë, Mécanique Quantique, I et II, Enseignement des Sciences 16 (Hermann, 1973).
M. Combescot, Is there a generalized Fermi Golden Rule (Preprint Université Paris VI, 1999).
R. Dümcke, The low density limit for an N-level system interacting with a free Bose or Fermi gas, Comm. Math. Phys. 97:33–359 (1985).
L. Erdös and H. T. Yau, Linear boltzmann equation as scaling limit of quantum lorentz gas advances, in Differential Equations and Mathematical Physics (Atlanta, GA, 1997), pp. 13–155; Contemp. Math., Vol. 217 (Amer. Math. Soc., Providence, RI, 1998).
L. Erdös and H. T. Yau, Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation, Comm. Pure Appl. Math. 53:66–735 (2000).
M. V. Fischetti, Theory of electron transport in small semiconductor devices using the Pauli master equations, J. Appl. Phys. 83:27–291 (1998).
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. 5:20–298 (1993).
L. Hörmander, The Analysis of Linear Partial Differential Operators (Springer-Verlag, Berlin, 1994).
K. Huang, Statistical Mechanics (Wiley and Sons, 1963).
J. B. Keller, G. Papanicolaou, and L. Ryzhik, Transport equations for elastic and other waves in random media, Wave Motion 24:32–370 (1996).
W. Kohn and J. M. Luttinger, Phys. Rev. 108:590 (1957).
W. Kohn and J. M. Luttinger, Phys. Rev. 109:1892 (1958).
H. J. Kreuzer, Nonequilibrium Thermodynamics and its Statistical Foundations (Oxford Science Publications, Monographs on Physics and Chemistry of Materials, 1983).
R. Kubo, J. Phys. Soc. Jap. 12 (1958).
L. J. Landau, Observation of Quantum Particles on a Large Space-Time Scale, J. Stat. Phys. 77:25–309 (1994).
G. Lindblad, On the generators of Quantum dynamical semigroups, Comm. Math. Phys. 48:11–130 (1976).
R. Loudon, The Quantum Theory of Light (Clarendon Press, Oxford, 1991).
P. A. Markowich, C. Ringhofer, and C. Schmeiser, Semiconductor Equations (Springer-Verlag, Vienna, 1990).
A. C. Newell and J. V. Moloney, Nonlinear Optics, Advanced Topics in the Interdisciplinary Mathematical Sciences (Addison-Wesley Publishing Company, 1992).
F. Nier, Asymptotic Analysis of a scaled Wigner equation and Quantum Scattering, Transp. Theor. Stat. Phys. 24:59–629 (1995).
F. Nier, A semi-classical picture of quantum scattering, Ann. Sci. Ec. Norm. Sup. Sér. 4 29:14–183 (1996).
W. Pauli, Festschrift zum 60. Geburtstage A. Sommerfelds (Hirzel, Leipzig, 1928), p. 30.
F. Poupaud and A. Vasseur, Classical and Quantum Transport in Random Media (Preprint, University of Nice, 2001).
I. Prigogine, Non-Equilibrium Statistical Mechanics (Interscience, New York, 1962).
M. Reed and B. Simon, Methods of Modern Mathematical Physics. III. Scattering theory (Academic Press, New York/London, 1979).
M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Addison-Wesley, 1977).
H. Spohn, Derivation of the transport equation for electrons moving through random impurities, J. Stat. Phys. 17:38–412 (1977).
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.
N. G. Van Kampen, Stochastic processes in physics and chemistry, Lecture Notes in Mathematics, Vol. 888 (North-Holland, 1981).
R. Zwanzig, Quantum Statistical Mechanics, P. H. E. Meijer, ed. (Gordon and Breach, New York, 1966).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Castella, F. From the von Neumann Equation to the Quantum Boltzmann Equation II: Identifying the Born Series. Journal of Statistical Physics 106, 1197–1220 (2002). https://doi.org/10.1023/A:1014098122698
Issue Date:
DOI: https://doi.org/10.1023/A:1014098122698