No Access

A NOTE ON THE NON-COMMUTATIVE LAPLACE–VARADHAN INTEGRAL LEMMA

W. DE ROECK

Institut für Theoretische Physik, Universität Heidelberg, Germany

Search for more papers by this author

CHRISTIAN MAES

Instituut voor Theoretische Fysica, K. U. Leuven, Belgium

Search for more papers by this author

KAREL NETOČNÝ

Institute of Physics, Academy of Sciences of the Czech Republic, Prague, Czech Republic

Search for more papers by this author

, and

LUC REY-BELLET

Department of Mathematics and Statistics, University of Massachusetts, Amherst, USA

Search for more papers by this author

https://doi.org/10.1142/S0129055X10004089Cited by:3 (Source: Crossref)

Abstract

We continue the study of the free energy of quantum lattice spin systems where to the local Hamiltonian H an arbitrary mean field term is added, a polynomial function of the arithmetic mean of some local observables X and Y that do not necessarily commute. By slightly extending a recent paper by Hiai, Mosonyi, Ohno and Petz [10], we prove in general that the free energy is given by a variational principle over the range of the operators X and Y. As in [10], the result is a non-commutative extension of the Laplace–Varadhan asymptotic formula.

Keywords:

AMSC: 82B10

References

H. Araki and P. D. F. Ion, Comm. Math. Phys. 35, 1 (1974), DOI: 10.1007/BF01646450. Crossref, Web of Science, ADS, Google Scholar
O. Brattelli and D. W. Robinson , Operator Algebras and Quantum Statistical Mechanics: 2 , 2nd edn. ( Springer-Verlag , Berlin , 1996 ) . Google Scholar
J.-B. Bru and W. de Siqueira Pedra, Equilibrium states of Fermi systems with long range interactions, in preparation . Google Scholar
R. Heylen, D. Bollé and N. S. Skantzos, Phys. Rev. E 74, 056111 (2006). Google Scholar
A. Dembo and O. Zeitouni , Large Deviations Techniques and Applications ( Springer , Berlin , 1993 ) . Google Scholar
F. den Hollander , Large Deviations , Field Institute Monographs 14 ( Amer. Math. Soc. , 2000 ) . Google Scholar
J. D. Deuschel and D. W. Stroock , Large Deviations , Pure and Applied Mathematics 137 ( Academic Press , Boston , 1989 ) . Google Scholar
R. S. Ellis , Entropy, Large Deviations, and Statistical Mechanics ( Springer , 2005 ) . Google Scholar
H.-O. Georgii , Gibbs Measures and Phase Transitions , De Gruyter Studies in Mathematics 9 ( De Gruyter , 1988 ) . Crossref, Google Scholar
F. Hiaiet al., Rev. Math. Phys. 20, 335 (2008), DOI: 10.1142/S0129055X08003298. Link, Web of Science, Google Scholar
F. Hiai, M. Mosonyi and O. Tomohiro, J. Math. Phys. 48(12), 123301 (2007), DOI: 10.1063/1.2812417. Crossref, Web of Science, ADS, Google Scholar
R. B. Israel , Convexity in the Theory of Lattice Gases , Princeton Series in Physics ( Princeton University Press , 1979 ) . Google Scholar
M. Lenci and L. Rey-Bellet, J. Stat. Phys. 119, 715 (2005), DOI: 10.1007/s10955-005-3015-3. Crossref, Web of Science, ADS, Google Scholar
K. Netočný and F. Redig, J. Stat. Phys. 117, 521 (2004), DOI: 10.1007/s10955-004-3452-4. Crossref, Web of Science, ADS, Google Scholar
Y. Ogata, Large deviations in quantum spin chain , arXiv:0803.0113 . Google Scholar
S. Olla, Probab. Theory Related Fields 77, 343 (1988), DOI: 10.1007/BF00319293. Crossref, Web of Science, Google Scholar
D. Petz, G. A. Raggio and A. Verbeure, Comm. Math. Phys. 121, 271 (1989), DOI: 10.1007/BF01217806. Crossref, Web of Science, ADS, Google Scholar
C.-E. Pfister , In and Out of Equilibrium, Probability with a Physics Flavor 1 , ed. V. Sidoravicius ( Birkhäuser , 2002 ) . Google Scholar
W. De Roeck, C. Maes and K. Netočný, J. Math. Phys. 47, 073303 (2006), DOI: 10.1063/1.2217810. Crossref, Web of Science, Google Scholar
B. Simon , The Statistical Mechanics of Lattice Gases ( Princeton University Press , Princeton , 1993 ) . Crossref, Google Scholar
E. J. Stormer, Funct. Anal. 3, 48 (1969), DOI: 10.1016/0022-1236(69)90050-0. Crossref, Google Scholar
S. R. S. Varadhan, Comm. Pure Appl. Math. 19, 261 (1966), DOI: 10.1002/cpa.3160190303. Crossref, Web of Science, Google Scholar
S. R. S. Varadhan , Large Deviations and Applications ( Society for Industrial and Applied Mathematics , 1984 ) . Crossref, Google Scholar