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| An explicit Markov coupling of collisional conservative Kac's N-particle system with Maxwell collisions is constructed. Parallel geometric coupling of simultaneous collisions is used. In agreement with Tanaka's dissipation of the space homogenous Maxwell-Boltzmann in Wasserstein distance, the resulting coupling is almost surely decreasing, and the L2-coupling creation is computed explicitly. Some quasi-contractive and uniform in N coupling / coupling creation inequalities are then proved, relying on 2 + a-moments (a > 0) of velocity distributions; upon N-uniform propagation of moments of the particle system, it yields a N-scalable a-power law trend to equilibrium. The case of order 4 moment yields a simple solution of Kac's program for convergence to equilibrium of collisional N-particle systems. |
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