Abstract
The aim of this paper is to give a review of some recent results concerning the problem of constructing and studying time evolution for quantum systems with infinitely many degrees of freedom. The first rigorous results in this direction are due to O. E. Lanford and D. W. Robinson [17–20], see also [13], Ch. 5.3. An approach to this problem has been proposed by O. Bratteli and D. W. Robinson [12], see also [13], Ch. 6.3. Among recent papers we refer to [1–3], [10–11], [16], [24].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Anshelevich V. V. First integrals and stationary states of quantum spin dynamics of Heisenberg. (Rus-sian) Teoret. Mat. Fiz. 43, no. 1, 107–110 (1980).
Anshelevich V. V., Gusev E. V. First integrals of one dimensional quantum Ising model with diametrical magnetic field. (Russian) Teoret. Mat. Fiz. 47, no. 2, 230–242 (1981).
Araki H., Barouch E. On the dynamics and ergodic prop-erties of the X-Y model. J. Stat. Phys. 31, no. 2, 327–346 (1983).
Arnold V. I. Integrals of quickly oscillating functions and singularities of projections of Lagrange manifolds. (Russian) Funktsional. Anal, i Prilozhen. 6, no. 3, 61–62 (1972).
Arnold V. I. Normal forms of functions near degene-rated critical points, Weyl groups A^, D^, E^ and Lagrange singularities. (Russian) Funktsional. Anal, i Prilozhen. 6, no. 4, 3–25 (1972).
Arnold V. I. Remarks on stationary phase method and Coxeter numbers. (Russian) Uspekhi Mat. Nauk 28, no. 5, 17–44 (1973).
Arnold V. I., Varchenko A. N., Gussein-Zadeh S. M. Singularities of differentiable mappings, vol. I. ( Russian) “Nauka”, Moscow, 1982.
Arnold V. I., Varchenko A. N., Gussein-Zadeh S. M. Singularities of differentiable mappings, vol. II. ( Russian) “Nauka”, Moscow, 1984.
Boldrighini C., Pellegrinotti A., Triolo L. Convergence to stationary states for infinite harmonic systems. J. Stat. Phys. 30, no. 1, 123–155 (1983).
Botvich D. D. Spectral properties of fermion dynamical systems. (Russian). Diss. Thesis. Moscow State Univer-sity (M. V. Lomonosov), 1983.
Botvich D. D., Malyshev V. A. Unitary equivalence of temperature dynamics for ideal and locally perturbed Fermi-gas. Commun. Math. Phys. 91, 301–312 (1983).
Bratelli 0., Robinson D. W., Green’s functions, Hamil tonians and modular automorphisms. Commun. Math. Phys. 50, 133–156 (1976).
Bratelli 0., Robinson D. W. Operator algebras and quantum Statistical mechanics, vol. II. Springer- Verlag, New York - Heidelberg - Berlin, 1981.
Colin de Verdiere Y. Nombre de points entiers dans une famille homothetique de domaines de R*1. Ann. Scient. Ecole Norm. Sup. 4eme ser. 10 no. 4, 559–576 (1977).
Duistermaat J. T. Oscillatory integrals, Lagrange im-mersions and unfoldings of singularities. Commun. Pure Appl. Math. 27, no. 2~, 207–281 (1974).
Gusev E. V. First integrals of some stochastic operators of Statistical physics. (Russian) Diss. Thesis, Moscow State University (M. V. Lomonosov ), 1982.
Lanford 0. E., Robinson D. W. Statistical mechanics of quantum spin systems III. Commun. Math. Phys. 327–338 (1968).
Lanford 0. E., Robinson D. W. Approach to equilibrium of free quantum systems. Commun. Math. Phys. 24, 193 — 210 (1972).
Robinson D. W. Statistical mechanics of quantum spin systems. Commun. Math. Phys. 6, 151–160 (1967).
Robinson D. W. Statistical mechanics of quantum spin systems II. Commun. Math. Phys. 7, 337–348 (1968).
Shuhov A. G., Suhov Yu. M. Ergodic properties of groups of Bogoliubov transformations of CAR C*-al-gebras. (Russian) Izv. Akad. Nauk SSSR, ser. Mat. (to appear).
Shuhov A. G., Suhov Yu. M. in preparation.
Suhov Yu. M. Convergence to equilibrium State for one dimensional quantum system of hard rods. (Russian) Izv Akad. Nauk SSSR, ser. Mat. 46, no. 6, 1274–1315 (1982)
Suhov Yu. M. Convergence to equilibrium for free Fermi -gas. (Russian) Teoret. M.t. Fiz. 55, no. 2, 282–290 (1983).
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1985 Springer Science+Business Media New York
About this chapter
Cite this chapter
Shuhov, A.G., Suhov, Y.M. (1985). Linear and Related Models of Time Evolution in Quantum Statistical Mechanics. In: Fritz, J., Jaffe, A., Szász, D. (eds) Statistical Physics and Dynamical Systems. Progress in Physics, vol 10. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-6653-7_6
Download citation
DOI: https://doi.org/10.1007/978-1-4899-6653-7_6
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4899-6655-1
Online ISBN: 978-1-4899-6653-7
eBook Packages: Springer Book Archive