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4: Pictorial representation of an isolated system Λ = Λ 1 Λ 2 formed by a small system Λ 1 immersed in a thermal bath Λ 2 .
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L'objectif de ce thèse est l'étude des systèmes dynamiques avec interaction à longue portée. La complexité de leur dynamique met en évidence des propriétés contre-intuitives et inattendues, comme l'existence d'états stationnaires hors-équilibre (QSS). Dans le QSS on peut observer des propriétés particulières: chaleur spécifique négative, inéquivale...
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... the curves shown in figure (6.14) were obtained for l = 1000, which, as previously mentioned, is the maximum value attainable with our computing facilities. To be sure of their convergence to the asymptotic shape in the limit L → ∞, we can observe the scaling of two test curves m (L) reported in figure (6.17), for different values of α. ...
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... In the considered systems, we either had in the mean-field case no space dependent interaction, and in the α-HMF model, the spatial interaction was set up on a onedimensional lattice with a power-law decreasing coupling constant. The simplest generalizations that come to mind are twofold, what happens when the interaction actually depends on the distance and in a phase space that allows "collisions" [47], the other way is to keep the lattice and focus on its topology and the way the constant of motion is set up, for instance studying the case of diluted complex networks and see how and in which settings the aforementioned features arise [48]. Work is in progress for these two proposed directions, if the results display as well selforganized regularity, one may then tackle systems with higher embedded dimensions, before considering more realistic physical models and potential applications in for instance hot non-collisional plasmas or self-gravitating systems. ...
Dynamics of many-body long-range interacting systems are investigated. Using the XY-Hamiltonian mean-field model as a case study, we show that regular trajectories, associated with invariant tori of the single-particle dynamics emerge as the number of particles is increased. Moreover, the construction of stationary solutions as well as studies of the maximal Lyapunov exponent of the systems show the same trend towards integrability. This feature provides a dynamical interpretation of the emergence of long-lasting out-of-equilibrium regimes observed generically in long-range systems. Extensions beyond the mean-field system are considered and display similar features.
