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A $N$-uniform quantitative Tanaka's theorem for the conservative Kac's $N$-particle system with Maxwell molecules

This paper considers the space homogenous Boltzmann equation with Maxwell molecules and arbitrary angular distribution. Following Kac's program, emphasis is laid on the the associated conservative Kac's stochastic $N$-particle system, a Markov process with binary collisions conserving energy and total momentum. An explicit Markov coupling (a probabilistic, Markovian coupling of two copies of the process) is constructed, using simultaneous collisions, and parallel coupling of each binary random collision on the sphere of collisional directions. The euclidean distance between the two coupled systems is almost surely decreasing with respect to time, and the associated quadratic coupling creation (the time variation of the averaged squared coupling distance) is computed explicitly. Then, a family (indexed by $δ> 0$) of $N$-uniform ''weak'' coupling / coupling creation inequalities are proven, that leads to a $N$-uniform power law trend to equilibrium of order ${\sim}_{ t \to + \infty} t^{-δ} $, with constants depending on moments of the velocity distributions strictly greater than $2(1 + δ)$. The case of order $4$ moment is treated explicitly, achieving Kac's program without any chaos propagation analysis. Finally, two counter-examples are suggested indicating that the method: (i) requires the dependance on $>2$-moments, and (ii) cannot provide contractivity in quadratic Wasserstein distance in any case.