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The low density limit for an N-level system interacting with a free Bose or Fermi gas. (English) Zbl 0614.46069
It is proved that the reduced dynamics of an N-level system coupled to a free quantum gas converges to a quantum dynamical semigroup in the low density limit. The proof uses a perturbation series of the quantum BBGKY- hierarchy, and the analysis of this series is based on scattering theory. The limiting semigroup contains the full scattering cross section, but it does not depend on the statistics of the reservoir. The dynamics of the semigroup is discussed.
Reviewer: H.Araki

MSC:
46N99 Miscellaneous applications of functional analysis
82B10 Quantum equilibrium statistical mechanics (general)
47A40 Scattering theory of linear operators
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[1] Grad, H.: Principles of the kinetic theory of gases. In: Handbuch der Physik, Vol. 12, Flügge, S. (ed.), Berlin, Heidelberg, New York: Springer 1958
[2] Lanford, O. E.: Time evolution of large classical systems. In: Dynamical Systems, Theory and Applications, Moser, J. (ed.), Berlin, Heidelberg, New York: Springer 1975 · Zbl 0329.70011
[3] Lanford, O. E.: On a derivation of the Boltzmann equation. Astérisque40, Soc. Math. de France (1976) · Zbl 0353.70020
[4] King, F.: Ph.D. Thesis, University of California, Berkeley (1975)
[5] Spohn, H.: The Lorentz process converges to a random flight process. Commun. Math. Phys.60, 277-290 (1978) · Zbl 0381.60099
[6] Lebowitz, J. L., Spohn, H.: Steady state self-diffusion at low density. J. Stat. Phys.29, 39-55 (1982) · Zbl 0511.60098
[7] Boldrighini, C., Bunimovich, L. A., Sinai, Ya. G.: On the Boltzmann equation for the Lorentz gas. Camerino University 1983 (Preprint) · Zbl 0583.76092
[8] Uehling, E. A., Uhlenbeck, G. E.: Transport phenomena in Einstein-Bose and Fermi-Dirac gases. I. Phys. Rev.43, 552-561 (1933) · Zbl 0006.33404
[9] Mori, H., Ono, S.: The quantum-statistical theory of transport phenomena, I. Prog. Theor. Phys.8, 327-340 (1952); Mori, H.: The quantum-statistical theory of transport phenomena, II. Prog. Thoer. Phys.9, 473-491 (1953) Ono, S.: The quantum-statistical theory of transport phenomena, III. Prog. Theor. Phys.12, 113-128 (1954) · Zbl 0048.43402
[10] Ross, J., Kirkwood, J. G.: The statistical-mechanical theory of transport processes. VIII. Quantum theory of transport in gases. J. Chem. Phys.22, 1094-1103 (1954)
[11] Sáenz, A. W.: Transport equation in quantum statistics for spinless molecules. Phys. Rev.105, 546-558 (1957) · Zbl 0077.40207
[12] Mori, H., Ross, J.: Transport equation in quantum gases. Phys. Rev.109, 1877-1882 (1958) · Zbl 0082.45102
[13] Lewis, R. M.: Quantum statistics and the Boltzmann equation. J. Math. Phys.3, 1229-1246 (1962) · Zbl 0114.21601
[14] Hoffman, D. K., Mueller, J. J., Curtiss, C. F.: Quantum-mechanical Boltzmann equation. J. Chem. Phys.43, 2878-2884 (1965)
[15] Kadanoff, L. P., Baym, G.: Quantum statistical mechanics. New York: Benjamin 1962 · Zbl 0115.22901
[16] Prugove?ki, E.: A quantum-mechanical Boltzmann equation for one-particle? s -distribution functions. Physica91A, 229-248 (1978)
[17] Wittwer, P.: Zur Quantenmechanik der Boltzmanngleichung. Diplom Thesis, ETH Zürich (1980)
[18] Palmer, P. F.: The rigorous theory of infinite quantum mechanical systems?Master equations and the dynamics of open systems. D. Phil. Thesis, Oxford University (1976)
[19] Davies, E. B.: Markovian master equations. Commun. Math. Phys.39, 91-110 (1974) · Zbl 0294.60080
[20] Davies, E. B.: Markovian master equations II. Math. Ann.219, 147-158 (1976) · Zbl 0323.60061
[21] Gorini, V., Kossakowski, A.:N-level system in contact with a singular reservoir. J. Math. Phys.17, 1298-1305 (1976)
[22] Frigerio, A., Gorini, V.:N-level system in contact with a singular reservoir. II. J. Math. Phys.17, 2123-2127 (1976)
[23] Hugenholtz, N. M.: Derivation of the Boltzmann equation for a Fermi gas, J. Stat. Phys.32, 231-254 (1983)
[24] Reed, M., Simon, B.: Methods of modern mathematical physics III: Scattering theory, New York: Academic Press 1979 · Zbl 0405.47007
[25] Bloch, C.: Diagram expansions in quantum statistical mechanics. In: Studies in Statistical Mechanics, Vol. III, de Boer, J., Uhlenbeck, G.E. (eds.). Amsterdam: North Holland 1965 · Zbl 0138.22403
[26] Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys.48, 119-130 (1976) · Zbl 0343.47031
[27] Hunzicker, W.: Cluster properties of multiparticle systems. J. Math. Phys.6, 6-10 (1965) · Zbl 0128.21807
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