Niveau: Supérieur, Doctorat, Bac+8
MATHEMATICS OF GRANULAR MATERIALS CEDRIC VILLANI Abstract. This is a short and somewhat informal review on the most mathe- matical parts of the kinetic theory of granular media, intended for physicists and for mathematicians outside the field. Contents Introduction 1 1. Modelling 2 2. Maxwellian toolbox 11 3. Gradient flow structure 19 4. One-dimensional rigidity 21 5. True inelastic hard spheres 24 6. The future of inelastic kinetic theory? 31 References 32 Introduction Granular materials are a very trendy subject nowadays, and the number of pub- lications devoted to it has grown tremendously since the beginning of the nineties. These contributions deal with experiments, modelling, numerical simulations, in- dustrial design as well as theoretical work. Some of the most spectacular effects appearing in the dynamics of granular gases are reviewed in a short and pedagogic survey by Barrat, Trizac and Ernst [3]; they include clustering, spontaneous loss of homogeneity, inverse Maxwell Demons, modification of Fourier's law, violation of equipartition of energy, and non-Gaussian equilibrium kinetic distributions. There is also a recent textbook on the subject by Brilliantov and Poschel [18]. This field constitutes a potential whole new area of applications opening up for mathematicians; yet the relevant mathematical literature is still restricted, due to the extreme theoretical complexity of the subject. The present survey deals with one of the (relatively) most advanced parts of the theory, in which kinetic models are used for granular gases, and interactions are described by inelastic collisions.
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