Accélération stochastique dans un gaz de Lorentz inélastique

Emilie Soret 1, 2
1 MEPHYSTO - Quantitative methods for stochastic models in physics
Inria Lille - Nord Europe, ULB - Université Libre de Bruxelles [Bruxelles], LPP - Laboratoire Paul Painlevé - UMR 8524
Abstract : In this thesis, we study the dynamics of a particle in an inelastic environment composed of scatterer which is commonly known as inelastic Lorentz gas. In the inert case, the environment is not affected by the particle. The kinetic energy of the latter grows with the time and this phenomenon is called « stochastic acceleration ». We approximate the particle's motion by a Markov chain where each step corresponds to a unique collision of the particle with a scatterer. We show that the particle's averaged kinetic energy grows with the time with the exponent 2/5. The result is proved by using probabilistic arguments, bringing into weak convergence theorems of Markov chain as well as the weak convergence of the chain, correctly rescaled in time and space, to a Bessel process. We thus obtain a convergence result for the velocity vector. Under a different rescaling that the one used for the kinetic energy, the latter converges weakly to a spherical brownian motion. In the dynamical case, the evolution of the degrees of freedom of the Lorentz gas is affected by the particle and the dynamical system considered is constitued of the particle and the environment. In such a system, the stochastic acceleration phenomenon cannot be observed. However, we show that the velocity distribution admits a stationnary state.
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Emilie Soret. Accélération stochastique dans un gaz de Lorentz inélastique. Probabilités [math.PR]. Université Lille 1, 2015. Français. ⟨tel-01236109⟩

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