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This paper considers the space homogenous Boltzmann equation with Maxwellmolecules and arbitrary angular distribution. Following Kac's program, emphasisis laid on the the associated conservative Kac's stochastic $N$-particlesystem, a Markov process with binary collisions conserving energy and totalmomentum. An explicit Markov coupling (a probabilistic, Markovian coupling oftwo copies of the process) is constructed, using simultaneous collisions, andparallel coupling of each binary random collision on the sphere of collisionaldirections. The euclidean distance between the two coupled systems is almostsurely decreasing with respect to time, and the associated quadratic couplingcreation (the time variation of the averaged squared coupling distance) iscomputed explicitly. Then, a family (indexed by $\delta > 0$) 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^{-\delta} $, with constants depending on moments of the velocitydistributions strictly greater than $2(1 + \delta)$. The case of order $4$moment is treated explicitly, achieving Kac's program without any chaospropagation analysis. Finally, two counter-examples are suggested indicatingthat the method: (i) requires the dependance on $>2$-moments, and (ii) cannotprovide contractivity in quadratic Wasserstein distance in any case.

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