Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2012-01-03
Physics
Condensed Matter
Statistical Mechanics
PhD thesis discussed on January 5th 2010 at the Universit\'e libre de Bruxelles. 131 pages, the table of contents is clickable
Scientific paper
Gravitational and electrostatic interactions are fundamental examples of systems with long-range interactions. Equilibrium properties of simple models with long-range interactions are well understood and exhibit exotic behaviors : negative specific heat and inequivalence of statistical ensembles for instance. The understanding of the dynamical evolution in the case of long-range interacting systems still represents a theoretical challenge. Phenomena such as out-of-equilibrium phase transitions or quasi-stationary states have been found even in simple models. The purpose of the present thesis is to investigate the dynamical properties of systems with long-range interactions, specializing on one-dimensional models. The appropriate kinetic description for these systems is the Vlasov equation. A numerical simulation tool for the Vlasov equation is developed. A detailed study of the out-of-equilibrium phase transition occuring in the Free-Electron Laser is performed and the transition is analyzed with the help of Lynden-Bell's theory. Then, the presence of stretching and folding in phase space for the Hamiltonian Mean-Field model is studied and quantified from the point of view of fluid dynamics. Finally, a system of uncoupled pendula for which the asymptotic states are similar to the ones of the Hamiltonian Mean-Field model is introduced. Its asymptotic evolution is predicted via both Lynden-Bell's theory and an exact computation. This system displays a fast initial evolution similar to the violent relaxation found for interacting systems. Moreover, an out-of-equilibrium phase transition is found if one imposes a self-consistent condition on the system. In summary, the present thesis discusses original results related to the occurence of quasi-stationary states and out-of-equilibrium phase transitions in 1D models with long-range interaction.
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