Abstract
We present a quantum system composed of infinitely many particles, subject to a nonquadratic Hamiltonian, for which it is possible to investigate the long time behavior of the dynamics and its ergodic properties. We do so both for the KMS states and for a large class of locally normal invariant states, whose very existence is already of some interest.
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REFERENCES
H. Araki and M. Shiraishi, On quasi-free states of the Canonical Commutation Relations I, Publ. Res. Inst. Math. Sci. 7:105–120 (1971-72).
H. Araki, On quasi-free states of the Canonical Commutation Relations II, Publ. Res. Inst. Math. Sci. 7:121–152 (1971-72).
H. Araki and E. J. Woods, Representations of the Canonical Commutation Relations describing a non-relativistic infinite free Bose gas, J. Math. Phys. 4:637-662 (1963).
V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics (Benjamin, New York, 1968).
D. Botvich and H. Maassen, Stability of Bose dynamical systems by branching theory, preprint.
C. Boldrighini, A. Pellegrinotti, and L. Triolo, Convergence to Stationary States for infinite Harmonic Systems, J. Stat. Phys. 30:123–155 (1983).
O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics I (Springer, Berlin/Heidelberg/New York, 1981).
O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics II (Springer, Berlin/Heidelberg/New York, 1981).
P. de Smedt, D. Dürr, J. L. Lebowitz, and C. Liverani, Quantum system in contact with a thermal environment: Rigorous treatment of a simple model, Commun. Math. Phys. 120:195–231 (1988).
N. Dunfurd and J. T. Schwartz, Linear Operators, Parts I, II (Interscience Publisher, New York, 1964).
G. Ford, M. Kac, and P. Mazur, Statistical mechanics of assemblies of coupled oscillators, J. Math. Phys. 6:504–515 (1965).
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration (Wiley-Interscience, New York, 1983).
S. Graffi and A. Martinez, Ergodic properties of infinite quantum harmonic crystals: An analytic approach, J. Math. Phys. 37:5111–5135 (1996).
M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer-Verlag, New York, 1990).
V. Jakšić and C. A. Pillet, On a model for quantum friction III. Ergodic properties of the spin-boson systems, Commun. Math. Phys. 37:627–651 (1996).
R. Haag, R. V. Kadison, and D. Kastler, Nets of C *-algebras and classification of states, Commun. Math. Phys. 16:81–104 (1970).
P. R. Halmos, Lectures in Ergodic Theory (Chelsea, Bronx, New York, 1956).
M. Hübner and H. Spohn, Radiative decay: Nonperturbative approaches, Rev. Math. Phys. 7:363–387 (1995).
N. E. Hurt, Quantum Chaos and Mesoscopic Systems, Mathematics and its applications, Vol. 397 (Kluwer Academic Publ., Dordrecht, London, 1997).
D. Kastler, The C *-algebra of a free Boson field, Commun. Math. Phys. 1:14–48 (1965).
D. Kastler, Equilibrium states of matter and operator algebras, in Symposia Mathematica, Vol. XX (Academic Press, 1976), pp. 49–107.
O. E. Lanford, III and J. L. Lebowitz, Time evolution and ergodic properties of harmonic crystal, in Lecture Notes in Physics, Vol. 38 (1975), pp. 144–177.
M. Lenci, Ergodic properties of the quantum ideal gas in the Maxwell-Boltzmann statistics, J. Math. Phys. 37:5136–5157 (1996).
J. T. Lewis and J. V. Pule, The equilibrium states of the free Boson gas, Commun. Math. Phys. 36:1–18 (1974).
H. Maassen, Return to thermal equilibrium by the solution of a Quantum Langevin Equation J. Stat. Phys. 34(1/2):239–261 (1984).
J. Manuceau and A. Verbeure, Quasi-free states of the CCR-algebra and Bogoliubov transformation, Commun. Math. Phys. 9:293–302 (1968).
C. C. Moore, Group extension and cohomology for locally compact groups. III, Trans. Amer. Math. Soc. 221:1–33 (1976).
H. Narnhofer and W. Thirring, Mixing properties of quantum systems, J. Stat. Phys. 57:811–825 (1989).
P. Nielaba, H. Spohn, and J. L. Valles, Behavior of a quantum particle in contact with a classical heat bath, J. Stat. Phys. 55:745–767 (1989).
M. Pulvirenti, Stability, equilibrium and KMS for an infinite classical system, J. Math. Phys. 18:2099–2103 (1977).
M. Reed and B. Simon, Functional Analysis, revised and enlarged edition (Academic Press, New York, 1983).
D. Ruelle, Natural nonequilibrium states in Quantum Statistical Mechanics, preprint.
P. Sarnak, Arithmetic quantum chaos, Israel Math. Conf. Proc. 8:183–236 (1995).
H. Spohn, Asymptotic completeness for Rayleigh scattering, J. Math. Phys. 38(5):2281–2296 (1997).
H. Spohn and J. L. Lebowitz, Stationary non-equilibrium states of infinite harmonic systems, Commun. Math. Phys. 54:97–120 (1977).
M. Takesaki, Theory of Operator Algebras I (Springer, 1979).
M. Takesaki and M. Winnink, Local normality in quantum statistical mechanics, Commun. Math. Phys. 30:129–152 (1973).
A. Van Daele, Quasi-equivalence of quasi-free states on the Weyl algebra, Commun. Math. Phys. 21:171–191 (1971).
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Fidaleo, F., Liverani, C. Ergodic Properties for a Quantum Nonlinear Dynamics. Journal of Statistical Physics 97, 957–1009 (1999). https://doi.org/10.1023/A:1004662014026
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DOI: https://doi.org/10.1023/A:1004662014026