Abstract
For a large class of quantum models of mean-field type the thermodynamic limit of the free energy density is proved to be given by the Gibbs variational principle. The latter is shown to be equivalent to a non-commutative version of Varadhan's asymptotic formula.
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Araki, H.: Relative hamiltonian for faithful normal states of a von Neumann algebra. Publ. Res. Inst. Math. Sci.9, 165–209 (1973)
Araki, H.: Golden-Thompson and Peierls-Bogoliubov inequalities for a general von Neumann algebra. Commun. Math. Phys.34, 167–178 (1973)
Araki, H.: On uniqueness of KMS states of one-dimensional quantum lattice systems. Commun. Math. Phys.44, 1–7 (1975)
Araki, H.: Relative entropy of states of von Neumann algebras. II. Publ. Res. Inst. Math. Sci.13, 173–192 (1977)
Araki, H., Masuda, T.: Positive cones andL p-spaces for von Neumann algebras. Publ. Res. Inst. Math. Sci.18, 339–411 (1982)
Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics. II. Berlin, Heidelberg, New York: Springer 1981
Cegla, W., Lewis, J.T., Raggio, G.A.: The free energy of quantum spin systems and large deviations. Commun. Math. Phys.118, 337–354 (1988)
Cramér, H.: On a new limit theorem in the theory of probability. In: Colloquium on the theory of probability. Paris: Hermann 1937
Ellis, R.S.: Entropy, large deviations and statistical mechanics. Berlin, Heidelberg, New York: Springer 1985
Fannes, M., Spohn, H., Verbeuere, A.: Equilibrium states for mean field models. J. Math. Phys.21, 355–358 (1980)
Petz, D.: Quasi-entropies for finite quantum systems. Rep. Math. Phys.23, 57–65 (1986)
Petz, D.: A variational expression for the relative entropy. Commun. Math. Phys.114, 345–349 (1988)
Phelps, R.R.: Lectures on Choquet's theorem. New York, Toronto, London, Melbourne: Van Nostrand 1966
Quaegebeur, J., Verbeuere, A.: Stability for mean field models. Ann. Inst. Henri Poincaré A22, 343–349 (1980)
Segal, I.E., Kunze, R.A.: Integrals and operators. Berlin, Heidelberg, New York: Springer 1978
Størmer, E.: Symmetric states of infinite tensor products ofC*-algebras. J. Funct. Anal.3, 48–68 (1969)
Takesaki, M.: Theory of operator algebras I. Berlin, Heidelberg, New York: Springer 1979
Umegaki, H.: Conditional expectations in an operator algebra IV (Entropy and information). Kodai Math. Sem. Rep.14, 59–85 (1962)
Varadhan, S.R.S.: Asymptotic probabilities and differential equations. Commun. Pure Appl. Math.19, 261–286 (1966)
Varadhan, S.R.S.: Large deviations and applications. Philadelphia, PA: Society for Industrial and Applied Mathematics 1984
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Communicated by H. Araki
On leave from the Mathematical Institute HAS, Budapest, Hungary
On leave from the Dublin Institute for Advanced Studies, Dublin, Ireland
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Petz, D., Raggio, G.A. & Verbeure, A. Asymptotics of Varadhan-type and the Gibbs variational principle. Commun.Math. Phys. 121, 271–282 (1989). https://doi.org/10.1007/BF01217806
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DOI: https://doi.org/10.1007/BF01217806