Transitions de phase dynamiques : des automates cellulaires probabilistes aux réseaux d'applications couplées
(2009)
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- Authors
- Supervisors
- Bricmont, Jean
- Abstract
- (en) The asymptotic behaviour of a dynamical system is described by probability measures that are invariant under the dynamic. For a chaotic dynamical system, there are many invariant measures but some of them have a stronger physical meaning, especially the SRB measures, which are invariant measures with some regularity properties as the absolute continuity with respect to the Lebesgue measure. Under very general assumptions, it can be proven that finite dimensional chaotic dynamical systems have only one SRB measure. In the case of infinite dimensional systems, the picture is radically different: weakly coupled systems tends to have only one SRB measure, and strongly coupled systems may have several. This phenomenon is known as a phase transition. This thesis will be devoted to the study of phase transitions for some infinite dimensional chaotic dynamical systems called coupled map lattices. We will define a coupled map lattice inspired by Stavskaya's probabilistic cellular automata for which it is possible to prove a phase transition at the level of SRB measures. We will also prove that a wide class of measures converge exponentially fast toward the invariant measure that appears at strong coupling. This will imply that this SRB measure has the exponential decay of correlations property both in space and time.
- Affiliations
Citations
de Maere d’Aertrycke, A. (2009). Transitions de phase dynamiques : des automates cellulaires probabilistes aux réseaux d’applications couplées. https://hdl.handle.net/2078.5/128713