Semiclassical expansion for the thermodynamic limit of the ground state energy of Kac's operator

Helffer, Bernard; Ramond, Thierry

Helffer, Bernard ; Ramond, Thierry

Journées équations aux dérivées partielles (2000), article no. 13, 17 p.

Résumé
Abstract

Nous poursuivons l’étude de l’opérateur de Kac semiclassique initiée par B. Helffer. Ce type d’opérateurs a été par exemple obtenu par M. Kac lors de l’étude d’un modèle de spins 2D par la méthode dite «de l’opérateur de transfert». On s’intéresse ici à la limite thermodynamique $Λ (h)$ de l’énergie de l’état fondamental de cet opérateur. Pour le modèle de spins de Kac, $Λ (h)$ s’avère être l’énergie libre par spin, et le régime semiclassique correspond à l’approximation de champ moyen. Sous des hypothèses sur le potentiel qui recouvrent les exemples physiques, nous montrons que $Λ (h)$ possède un développement asymptotique formel en puissances de $h$ , et nous en déduisons différentes estimations. Notre approche repose de manière essentielle sur la notion de fonction standard introduite par J. Sjöstrand pour l’étude de questions similaires concernant l’opérateur de Schrödinger. Ceci est un travail en collaboration avec Bernard Helffer.

We continue the study started by the first author of the semiclassical Kac Operator. This kind of operator has been obtained for example by M. Kac as he was studying a 2D spin lattice by the so-called “transfer operator method”. We are interested here in the thermodynamical limit $Λ (h)$ of the ground state energy of this operator. For Kac’s spin model, $Λ (h)$ is the free energy per spin, and the semiclassical regime corresponds to the mean-field approximation. Under suitable assumptions, which are satisfied by the physical examples we have in mind, we construct a formal asymptotic expansion for $Λ (h)$ in powers of $h$ , from which we derive precise estimates on $Λ (h)$ . We work in the settings of Standard Functions introduced by J. Sjöstrand for the study of similar questions in the case of Schrödinger operators.

MR Zbl

@incollection{JEDP_2000____A13_0,
     author = {Helffer, Bernard and Ramond, Thierry},
     title = {Semiclassical expansion for the thermodynamic limit of the ground state energy of {Kac's} operator},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {13},
     pages = {1--17},
     publisher = {Universit\'e de Nantes},
     year = {2000},
     mrnumber = {2001i:81087},
     zbl = {01808703},
     language = {en},
     url = {http://archive.numdam.org/item/JEDP_2000____A13_0/}
}

TY  - JOUR
AU  - Helffer, Bernard
AU  - Ramond, Thierry
TI  - Semiclassical expansion for the thermodynamic limit of the ground state energy of Kac's operator
JO  - Journées équations aux dérivées partielles
PY  - 2000
SP  - 1
EP  - 17
PB  - Université de Nantes
UR  - http://archive.numdam.org/item/JEDP_2000____A13_0/
LA  - en
ID  - JEDP_2000____A13_0
ER  -

%0 Journal Article
%A Helffer, Bernard
%A Ramond, Thierry
%T Semiclassical expansion for the thermodynamic limit of the ground state energy of Kac's operator
%J Journées équations aux dérivées partielles
%D 2000
%P 1-17
%I Université de Nantes
%U http://archive.numdam.org/item/JEDP_2000____A13_0/
%G en
%F JEDP_2000____A13_0

Helffer, Bernard; Ramond, Thierry. Semiclassical expansion for the thermodynamic limit of the ground state energy of Kac's operator. Journées équations aux dérivées partielles (2000), article  no. 13, 17 p. http://archive.numdam.org/item/JEDP_2000____A13_0/

Bibliographie
Cité par

[E1] R. Ellis : Entropy, Large Deviations, and Statistical Mechanics, Grundlehren der math. Wiss. 271, Springer. | MR | Zbl

[He1] B. Helffer : Around a stationary phase theorem in large dimension, Journal of Functional Analysis, Vol. 119, n.1, 1994, p. 217-252. | MR | Zbl

[He2] B. Helffer : Spectral Properties of the Kac Operator in Large Dimension, Proceedings on Mathematical Quantum Theory II : Schrödinger operators (August 1993), J. Feldman, R. Froese and L.M. Rosen (ed.), CRM Proc. Lect. Notes. 8, 179-211 (1995). | Zbl

[He3] B. Helffer : Recent results and open problems on Schrödinger operators, Laplace integrals and transfer operators in large dimension. Michael Demuth (ed.) et al., Schrödinger operators, Markov semigroups, wavelet analysis, operator algebras. Berlin : Akademie Verlag. Math. Top. 11, 11-162 (1996). | MR | Zbl

[He4] B. Helffer : Semiclassical analysis for the transfer operator : WKB constructions in large dimension, Comm. in Mathematical Physics 187 (1997), p. 81-113. | MR | Zbl

[He-Sj] B. Helffer, J. Sjöstrand : Semiclassical expansions of the thermodynamic limit for a Schrödinger equation. I : The one well case. D. Robert (ed.), Methodes semi-classiques, Volume 2. Colloque international (Nantes, juin 1991), Astérisque. 210, 135-181 (1992). | Zbl

[Ka] M. Kac : Mathematical mechanisms of phase transitions, Brandeis lectures (1966). New-York : Gordon and Breach, 1966. | Zbl

[Mø] J. Møller : The low-temperature limit of transfer operator in fixed dimension. Prépublication de l'Université Paris Sud 2000-2035. | Zbl

[Ru] D. Ruelle : Probability estimates for continuous spin systems, Comm. in Mathematical Physics 50 (1976), p. 189-194. | MR

[Sj1] J. Sjöstrand : Potential wells in high dimensions I, Ann. de l'Institut Henri Poincaré, Phys. Théor., Vol. 58 n.1 (1993), p. 1-41. | Numdam | MR | Zbl

[Sj2] J. Sjöstrand : Potential wells in high dimensions II, Ann. de l'Institut Henri Poincaré, Phys. Théor., Vol. 58 n.1 (1993), p. 43-55. | Numdam | Zbl

[Sj3] J. Sjöstrand : Evolution equations in a large number of variables, Math. Nachr. 166 (1994), p. 17-53. | MR | Zbl