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Local mean field and energy transport in non-equilibrium systems

Abstract : Chains of oscillator systems enable to model microscopically a solid, in order to study energy transport and prove Fourier’s law. In this thesis, we introduce two new models of chains of oscillators with local mean field mechanical interaction and stochastic collisions that preserve the system’s total energy. The first model is a model with stochastic velocity exchanges of Kac type. The second one is a model with random flips of velocities, where the sign of the particles’ velocities is changed at random times.As we consider local mean field models, particles are not indistinguishable, and the conservative stochastic exchanges in our first model are an additional difficulty for the proof of a Vlasov limit. We first derive a quantitative mean field limit, that we then use to prove that energy evolves diffusively at a given timescale for the model with long-range exchanges and for a restricted class of anharmonic potentials. At the same timescale, we also prove that there is no evolution of energy for the model with flips of velocities.For harmonic interactions, we then compute thermal conductivity via Green-Kubo formula for both models, to highlight that the timescale at which energy evolves for the model with velocity flips is longer and therefore that the mechanisms at play for energy transport are different.
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Submitted on : Wednesday, October 28, 2020 - 3:48:08 PM
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Alejandro Fernandez Montero. Local mean field and energy transport in non-equilibrium systems. Probability [math.PR]. Institut Polytechnique de Paris, 2020. English. ⟨NNT : 2020IPPAX044⟩. ⟨tel-02982380⟩

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