Abstract
The paper reexamines the treatment of irreversible quantum systems by master equations. Shortcomings of the conventional theory of quantum Markov processes pointed out by Talkner are analyzed. It is shown that a frequently used quantum regression hypothesis is not correct, in general. A new generalized master equation determining the relaxation to equilibrium is derived by means of time-dependent projection operator techniques. It is shown that this master equation also determines the time evolution of equilibrium correlations and response functions. The Markovian approximation is discussed, and a new type of Markovian limit, the Brownian motion limit, is introduced besides the weak coupling limit. The shortcomings of the conventional approach are resolved by deriving new formulae for the time evolution of the correlation and response functions of a quantum Markov process. The symmetries of the process are emphasized, and it is shown how the fluctuation-dissipation theorem and the detailed balance symmetry emerge from the master equation approach.
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References
Mori, H.: Progr. Theor. Phys.33, 423 (1965)
Robertson, B.: Phys. Rev.144, 151 (1966)
Grabert, H.: J. Stat. Phys.19, 479 (1978)
Grabert, H.: Projection operator techniques in nonequilibrium statistical mechanics, Springer Tracts in Modern Physics. Vol. 95. Berlin, Heidelberg, New York: Springer 1982
Green, M.S.: J. Chem. Phys.20, 1281 (1952)
Pauli, W.: Festschrift zum 60. Geburtstage A. Sommerfeld. p. 30. Leipzig: Hirzel 1928
Wangsness, R.K., Bloch, F.: Phys. Rev.89, 728 (1953)
Redfield, A.G.: IBM J. Res. Develop.1, 19 (1957)
Argyres, P.N., Kelley, P.L.: Phys. Rev.134, A 98 (1964)
Haken, H., Weidlich, W.: In: Proceedings of the International School of Physics “Enrico Fermi”. Vol.42, p. 630. New York: Academic Press 1969
Haken, H.: in Handbuch der Physik. Vol. XXV/2c. Berlin, Heidelberg, New York: Springer 1969
Haake, F.: Springer Tracts in Modern Physics. Vol. 66, p. 98. Berlin, Heidelberg, New York: Springer 1973
Davies, E.B.: Quantum theory of open systems. London: Academic Press 1976
Gorini, V., Kossakowski, A., Sudarshan, E.C.G.: J. Math. Phys.17, 821 (1976)
Spohn, H.: Rev. Mod. Phys.53, 569 (1980)
Talkner, P.: Dissertation, Univ. Stuttgart (1979)
Haken, H., Weidlich, W.: Z. Phys.205, 96 (1967)
Einstein, A.: Ann. Phys. (Leipzig)17, 549 (1905)
Onsager, L.: Phys. Rev.37, 405,38, 2265 (1931)
Lax, M.: Phys. Rev.172, 350 (1968)
Kadanoff, L.P., Baym, G.: Quantum statistical mechanics. New York: Benjamin 1962
Callen, H.B., Welton, T.A.: Phys. Rev.83, 34 (1951)
Kubo, R.: Rep. Prog. Phys. (London)29, 255 (1966)
Lindblad, G.: J. Math. Phys.20, 2081 (1979)
Lewis, J.T.: Phys. Rep. C77, 339 (1981)
Jaynes, E.T.: In: Statistical physics. Brandeis Lectures. Vol. 3. New York: Benjamin 1962
Grabert, H.: Phys. Lett.57 A, 105 (1976)
Zwanzig, R.: Phys. Rev.124, 983 (1961)
Grabert, H., Talkner, P., Hänggi, P.: Z. Phys. B — Condensed Matter26, 389 (1977)
Grabert, H., Talkner, P., Hänggi, P., Thomas, H.: Z. Phys. B — Condensed Matter29, 273 (1978)
Messiah, A.: Quantum mechanics. Vol. 2. Amsterdam: North Holland 1968
Grabert, H., Talkner, P.: (in preparation)
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Grabert, H. Nonlinear relaxation and fluctuations of damped quantum systems. Z. Physik B - Condensed Matter 49, 161–172 (1982). https://doi.org/10.1007/BF01314753
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DOI: https://doi.org/10.1007/BF01314753