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...e [40, 50, 22, 56] which solves the problem (note however that since the L2 norm does not “tensorize” correctly in high dimension, it is not possible to pass to the limit in these estimates), and see =-=[74, 21, 58]-=- for some progress in the “nonlinear” case. 3. Negative results at the functional level 3.1. Counterexample in the Maxwell molecules case with only mass, energy and entropy bounds. The explicit soluti...

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...he two systems of particles is similar. 4 NICOLAS FOURNIER, DAVID GODINHO See also Peyre [16] who gives large deviations estimates for the Boltzmann equation for Maxwell molecules and Mischler-Mouhot =-=[15]-=- who give results of chaos propagation with quantitative estimates for the Boltzmann equation for hard spheres and for Maxwell molecules. 1.5. Comments. We managed to obtain some bounds uniform in tim...

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...-dependence. But, in contrast to the master equation, it is posed on the low dimensional space (zj,ρj) ∈ {−1,1} × R+. Propogation of chaos can be proved in model cases, such as the Boltzmann equation =-=[27, 30, 31]-=-, its caricature proposed by Ka˘c [26] and models of swarming behavior [11, 12] (see also [41]). The second and last model reduction is to take the limit of an infinite number of cells, i.e. taking th...

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... Hard-Sphere dynamics and hard potentials (see also [23, 24, 40] and [19, 20, 21, 34, 41]). A new approach yielding global-in-time results has been recently developed by Mischler, Mouhot and Wennberg =-=[36, 37, 38]-=-. Kac proposed a caricature of the Boltzmann equation leading to the Kac kinetic equation [26]. Propagation of chaos for the Kac master equation and the related question of gap estimates have received...

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... according to the definition of Kac, but also give precise error bounds in terms of the number of particles, and a detailed information about the limiting equation. The results are proven in [38] and =-=[37]-=-. An important ingredient in the abstract formulation is the de Finetti (or Hewitt Savage) theorem, which is also presented in these notes, following the lectures by P.L. Lions [34]. 18 BERNT WENNBERG...

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...pendence. But, in contrast to the master equation, it is posed on the low dimensional space (zj , ρj) ∈ {−1, 1} × R+. Propogation of chaos can be proved in model cases, such as the Boltzmann equation =-=[27, 30, 31]-=-, its caricature proposed by Kac̆ [26] and models of swarming behavior [11, 12] (see also [41]). The second and last model reduction is to take the limit of an infinite number of cells, i.e. taking th...

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...pendence. But, in contrast to the master equation, it is posed on the low dimensional space (zj , ρj) ∈ {−1, 1} × R+. Propogation of chaos can be proved in model cases, such as the Boltzmann equation =-=[27, 30, 31]-=-, its caricature proposed by Kac̆ [26] and models of swarming behavior [11, 12] (see also [41]). The second and last model reduction is to take the limit of an infinite number of cells, i.e. taking th...