[Généralisation du modèle de Curie-Weiss et Pression quadratique en théorie ergodique]
We explain the Curie-Weiss model in statistical mechanics within an ergodic viewpoint. More precisely, we simultaneously define in , on the one hand a generalized Curie-Weiss model within the statistical mechanics viewpoint and on the other hand, the quadratic free energy and quadratic pressure within the ergodic theory viewpoint. We show that there are finitely many invariant measures that maximize the quadratic free energy. They are all dynamical Gibbs measures. Moreover, the probabilistic Gibbs measures for the generalized Curie-Weiss model converge to a determined combination of the (dynamical) conformal measures associated with these dynamical Gibbs measures. The standard Curie-Weiss model is a particular case of our generalized Curie-Weiss model. An ergodic viewpoint over the Curie-Weiss-Potts model is also given.
On explique ici un modèle généralisé de Curie-Weiss (champ moyen) en utilisant le vocabulaire de la théorie ergodique. On introduit le concept de pression quadratique en théorie ergodique et on montre que pour tout potentiel Hölder dans le sous-shift unilatère , il n’y a qu’un nombre fini de mesures invariantes qui maximisent la pression quadratique et, que ce sont toutes des mesures d’équilibre pour un multiple du potentiel. On montre que la limite thermodynamique des mesures de Gibbs associées à l’Hamiltonien en champ moyen convergent vers une combinaison des mesures conformes associées à chaque mesure qui maximise la pression quadratique. Le cas standard de Curie-Weiss s’obtient pour un exemple particulier de potentiel. Enfin, le modèle de Curie-Weiss-Potts est également expliqué avec le vocabulaire de la théorie ergodique.
Révisé le :
Accepté le :
Publié le :
DOI : 10.24033/bsmf.2779
Keywords: thermodynamic formalism, equilibrium states, Curie-Weiss model, Curie-Weiss-Potts model, Gibbs measure, phase transition
Mots-clés : formalisme thermodynamique, état d’équilibre, modèles de Curie-Weiss et de Curie-Weiss-Potts, mesure de Gibbs, transition de phase
Leplaideur, Renaud 1, 2 ; Watbled, Frédérique 3
@article{BSMF_2019__147_2_197_0,
author = {Leplaideur, Renaud and Watbled, Fr\'ed\'erique},
title = {Generalized {Curie-Weiss} model and quadratic pressure in ergodic theory},
journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
pages = {197--219},
year = {2019},
publisher = {Soci\'et\'e math\'ematique de France},
volume = {147},
number = {2},
doi = {10.24033/bsmf.2779},
mrnumber = {3982275},
zbl = {1430.37007},
language = {en},
url = {https://www.numdam.org/articles/10.24033/bsmf.2779/}
}
TY - JOUR AU - Leplaideur, Renaud AU - Watbled, Frédérique TI - Generalized Curie-Weiss model and quadratic pressure in ergodic theory JO - Bulletin de la Société Mathématique de France PY - 2019 SP - 197 EP - 219 VL - 147 IS - 2 PB - Société mathématique de France UR - https://www.numdam.org/articles/10.24033/bsmf.2779/ DO - 10.24033/bsmf.2779 LA - en ID - BSMF_2019__147_2_197_0 ER -
%0 Journal Article %A Leplaideur, Renaud %A Watbled, Frédérique %T Generalized Curie-Weiss model and quadratic pressure in ergodic theory %J Bulletin de la Société Mathématique de France %D 2019 %P 197-219 %V 147 %N 2 %I Société mathématique de France %U https://www.numdam.org/articles/10.24033/bsmf.2779/ %R 10.24033/bsmf.2779 %G en %F BSMF_2019__147_2_197_0
Leplaideur, Renaud; Watbled, Frédérique. Generalized Curie-Weiss model and quadratic pressure in ergodic theory. Bulletin de la Société Mathématique de France, Tome 147 (2019) no. 2, pp. 197-219. doi: 10.24033/bsmf.2779
Thermodynamic formalism and substitutions, ArXiv e-prints (2015) | arXiv
Renormalization, freezing phase transitions and Fibonacci quasicrystals, Ann. Sci. Éc. Norm. Supér. (4), Volume 48 (2015), pp. 739-763 | MR | Zbl | Numdam | DOI
Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin, 1975, i+108 pages (2nd ed. - 2008 by JR Chazottes) | MR | Zbl | DOI
Transformations dilatantes de l’intervalle et théorèmes limites, Astérisque (1996), pp. 1-109 (Études spectrales d’opérateurs de transfert et applications) | MR | Numdam | Zbl
Complete analysis of phase transitions and ensemble equivalence for the Curie-Weiss-Potts model, J. Math. Phys., Volume 46 (2005), p. 063301, 25 | MR | Zbl | DOI
Low-temperature phase transitions in the quadratic family., Adv. Math., Volume 248 (2013), pp. 453-494 | MR | Zbl | DOI
Large deviations techniques and applications, Springer, 2010, xvi + 396 pages | Zbl | MR | DOI
Entropy, large deviations, and statistical mechanics, Classics in Mathematics, Springer-Verlag, Berlin, 2006, xiv+364 pages (Reprint of the 1985 original) | MR | Zbl | DOI
Limit theorems for sums of dependent random variables occurring in statistical mechanics, Z. Wahrsch. Verw. Gebiete, Volume 44 (1978), pp. 117-139 | MR | Zbl | DOI
The statistics of Curie-Weiss models, J. Statist. Phys., Volume 19 (1978), pp. 149-161 | MR | DOI
Asymptotic expansions, Dover Publications, Inc., New York, 1956, vi+108 pages | MR | Zbl
Limit theorems for the empirical vector of the Curie-Weiss-Potts model., Stochastic Processes Appl., Volume 35 (1990), pp. 59-79 | Zbl | MR | DOI
Gibbs measures and phase transitions, de Gruyter Studies in Mathematics, 9, Walter de Gruyter & Co., 2011, xiv+545 pages | MR | Zbl | DOI
Limit theorems for Markov chains and stochastic properties of dynamical systems by quasi-compactness, Lecture Notes in Mathematics, 1766, Springer, 2001, viii+145 pages | MR | Zbl | DOI
Calculation of Partition Functions, Phys. Rev. Lett., Volume 3 (1959), pp. 77-78 http://link.aps.org/doi/10.1103/PhysRevLett.3.77 | DOI
Théorie ergodique pour des classes d’opérations non complètement continues, Ann. of Math. (2), Volume 52 (1950), pp. 140-147 | DOI | MR | Zbl
Equilibrium states in ergodic theory., Cambridge University Press, 1998, ix + 178 pages | Zbl | MR | DOI
Chaos: butterflies also generate phase transitions, J. Stat. Phys., Volume 161 (2015), pp. 151-170 | MR | Zbl | DOI
On thermodynamics of rational maps. II. Non-recurrent maps, J. London Math. Soc. (2), Volume 67 (2003), pp. 417-432 | MR | Zbl | DOI
Large deviations for the empirical field of Curie-Weiss models, Stochastics, Volume 25 (1988), pp. 3-14 | MR | Zbl | DOI
Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187-188, SMF, 1990 | MR | Zbl | Numdam
A course on large deviations with an introduction to Gibbs measures, Graduate Studies in Mathematics, 162, American Mathematical Society, 2015, xiv+318 pages | MR | DOI
Thermodynamic formalism, Cambridge Mathematical Library, Cambridge University Press, 2004, xx+174 pages (The mathematical structures of equilibrium statistical mechanics) | MR | Zbl | DOI
Statistical mechanics of a one-dimensional lattice gas., Commun. Math. Phys., Volume 9 (1968), pp. 267-278 | MR | Zbl | DOI
Statistical mechanics, World Scientific Publishing Co. Inc., River Edge, NJ, 1999, xvi+219 pages (Rigorous results, Reprint of the 1989 edition) | MR | Zbl
Continuous phase transitions for dynamical systems, Comm. Math. Phys., Volume 267 (2006), pp. 631-667 | MR | Zbl | DOI
Gibbs measures in ergodic theory,, Uspehi Mat. Nauk, Volume 27 (1972), pp. 21-64 | MR | Zbl
Theory of phase transitions: rigorous results, International Series in Natural Philosophy, 108, Pergamon Press, 1982, viii+150 pages (Translated from the Russian by J. Fritz, A. Krámli, P. Major and D. Szász) | MR | Zbl
A method for the computation of quantum distribution functions, Dokl. Akad. Nauk SSSR (N.S.), Volume 115 (1957), pp. 1097-1100 | MR | Zbl
Estimates of the invariant densities of endomorphisms with indifferent fixed points, Israel J. Math., Volume 37 (1980), pp. 303-314 | MR | Zbl | DOI
An introduction to ergodic theory. Paperback ed., Springer, 2000, ix + 250 pages | Zbl
Cité par Sources :
