[Formalisme thermodynamique et grandes déviations dans les systèmes à dynamique markovienne]
Le formalisme thermodynamique, qui a d'abord été développé dans le cadre des systèmes dynamiques puis appliqué aux processus de Markov, s'avère également pertinent pour les dynamiques de Markov en temps continu, à condition toutefois d'interpréter les définitions en jeu de façon appropriée. Ce formalisme peut être reformulé en termes de fonction génératrice d'une observable, puis étendu à d'autres observables. En particulier, l'observable K donnant le nombre d'événements ayant lieu dans un intervalle de temps donné, bien que très simple, contient déjà la signature de transitions de phases dynamiques.
Pour les modèles de champ moyen à l'équilibre, et dans la limite des grands systèmes, le formalisme peut s'appliquer simplement et montre comment les transitions de phase thermodynamiques peuvent affecter les propriétés dynamiques de ces systèmes. Cela est illustré sur le cas du modèle de Potts en champ moyen, et il s'avère que le cas d'Ising diffère qualitativement des autres cas.
The thermodynamic formalism, which was first developed for dynamical systems and then applied to discrete Markov processes, turns out to be well suited for continuous time Markov processes as well, provided the definitions are interpreted in an appropriate way. Moreover, it can be reformulated in terms of the generating function of an observable, and then extended to other observables. In particular, the simple observable K giving the number of events occurring over a given time interval turns out to contain already the signature of dynamical phase transitions.
For mean-field models in equilibrium, and in the limit of large systems, the formalism is rather simple to apply and shows how thermodynamic phase transitions may modify the dynamical properties of the systems. This is exemplified with the q-state mean-field Potts model, for which the Ising limit
Publié le :
Mots-clés : Formalisme thermodynamique, Grandes déviations, Chaos
Vivien Lecomte 1 ; Cécile Appert-Rolland 2 ; Frédéric van Wijland 1
@article{CRPHYS_2007__8_5-6_609_0,
author = {Vivien Lecomte and C\'ecile Appert-Rolland and Fr\'ed\'eric van Wijland},
title = {Thermodynamic formalism and large deviation functions in continuous time {Markov} dynamics},
journal = {Comptes Rendus. Physique},
pages = {609--619},
publisher = {Elsevier},
volume = {8},
number = {5-6},
year = {2007},
doi = {10.1016/j.crhy.2007.05.005},
language = {en},
}
TY - JOUR AU - Vivien Lecomte AU - Cécile Appert-Rolland AU - Frédéric van Wijland TI - Thermodynamic formalism and large deviation functions in continuous time Markov dynamics JO - Comptes Rendus. Physique PY - 2007 SP - 609 EP - 619 VL - 8 IS - 5-6 PB - Elsevier DO - 10.1016/j.crhy.2007.05.005 LA - en ID - CRPHYS_2007__8_5-6_609_0 ER -
%0 Journal Article %A Vivien Lecomte %A Cécile Appert-Rolland %A Frédéric van Wijland %T Thermodynamic formalism and large deviation functions in continuous time Markov dynamics %J Comptes Rendus. Physique %D 2007 %P 609-619 %V 8 %N 5-6 %I Elsevier %R 10.1016/j.crhy.2007.05.005 %G en %F CRPHYS_2007__8_5-6_609_0
Vivien Lecomte; Cécile Appert-Rolland; Frédéric van Wijland. Thermodynamic formalism and large deviation functions in continuous time Markov dynamics. Comptes Rendus. Physique, Work, dissipation, and fluctuations in nonequilibrium physics, Volume 8 (2007) no. 5-6, pp. 609-619. doi : 10.1016/j.crhy.2007.05.005. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2007.05.005/
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