Abstract
We consider a non-linear map on the space of density matrices, which we call the Boltzmann map τ. It is the composition of a doubly stochastic mapT on the space ofn-body states, and the conditional expectation onto the one-body space. WhenT is ergodic, then the iterates of τ take any initial state to the uniform distribution. If the energy levels are equally spaced, andT conserves energy and is ergodic on each energy shell, then iterates of τ take any initial state of finite energy to a canonical distribution.
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Streater, R.F.: Convergence of the iterated Boltzmann map (to appear in Publ. R.I.M.S., 1984)
Dixmier, J.: Les algèbres d'operateurs sur l'espaces hilbertiennes. Paris: Gauthier-Villars 1969
Manuceau, J. et al.: Entropy and normal states. Commun. Math. Phys.27, 327 (1972)
Alberti, P.M., Uhlmann, A.: Stochasticity and partial order. Dordrecht: Reidel 1982
Alicki, R., Messer, J.: Non-linear quantum dynamic semigroups for many-body open systems. J. State Phys.32, 299 (1983)
Ruelle, D.: Statistical mechanics. New York: Benjamin 1969
Lanford, O.E., Robinson, D.W.: Mean entropy of states in quantum-statistical mechanics. J. Math. Phys.9, 1120 (1968)
Lieb, E.H., Ruskai, M.B.: Proof of the strong subadditivity of quantum-mechanical entropy. J. Math. Phys.14, 1938 (1973) (Appendix by B. Simon)
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Communicated by J. L. Lebowitz
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Streater, R.F. Convergence of the quantum Boltzmann map. Commun.Math. Phys. 98, 177–185 (1985). https://doi.org/10.1007/BF01220506
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DOI: https://doi.org/10.1007/BF01220506