Abstract
We study here a very popular 1D jump model introduced by Kac: it consists of N velocities encountering random binary collisions at which they randomly exchange energy. We show the uniform (in N) exponential contractivity of the dynamics in a non-standard Monge-Kantorovich-Wasserstein: precisely the MKW metric of order 2 on the energy. The result is optimal in the sense that for each N, the contractivity constant is equal to the \(L^2\) spectral gap of the generator associated to Kac’s dynamic. As a corollary, we get an uniform but non optimal contractivity in the MKW metric of order 4. We use a simple coupling that works better that the parallel one. The estimates are simple and new (to the best of our knowledge).
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Acknowledgments
The author would like to thank Matthias Rousset and Alexandre Gaudillière for very stimulating discussion about that problem: in particular the proof of Corollary 1 is due to the latter; and also Max Fathi for pointing out reference [8], which uses a similar argument to prove the uniform contractivity in MKW distance of order 2 of the Kac model on SO(n) (but with a mixing time that is quadratic in the number of jump, instead of linear here).
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Hauray, M. Uniform Contractivity in Wasserstein Metric for the Original 1D Kac’s Model. J Stat Phys 162, 1566–1570 (2016). https://doi.org/10.1007/s10955-016-1476-1
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DOI: https://doi.org/10.1007/s10955-016-1476-1