I have just uploaded on arXiv a joint work with Stéphane Mischler on “Kac’s Program in Kinetic Theory”. In this paper, we answer a set of questions raised by Mark Kac in his seminal paper Foundations of kinetic theory (Proc. Third Berkeley Symp. Math. Stat. & Prob., 1956) about the derivation of Boltzmann equations from many-particle jump processes.
UPDATE: We have just uploaded an announcement note, to appear shortly in the Comptes-rendus de l’Académie des Sciences de Paris.
In order to gain insight into one of the main challenges in statistical mechanics, namely the mathematical derivation of the irreversible Boltzmann equation from Hamiltonian microscopic dynamics, Kac proposed the problem of deriving the spatially homogeneous Boltzmann equation from a many-particle jump process, and introduced a rigorous notion of molecular chaos to this purpose. He then raised a series of open questions: (1) prove the propagation of chaos for realistic collision processes; (2) relate the time scales of relaxation of the limit nonlinear equation with those of the many-particle jump process; (3) prove the convergence of the many-particle entropy towards the Boltzmann entropy in order to provide a microscopic justification of the H-theorem of Boltzmann in this context.
We answer these questions as follows: We prove the propagation of chaos with quantitative and uniform in time estimates for the two main collision kernels, namely hard spheres and Maxwell molecules (without any Grad’s angular cutoff). Even not taking into account the uniformity in time of our estimate, this improves significantly over previous important results by H. McKean, A. Grünbaum and A.-S. Sznitman. Moreover these are the first uniform in time estimates of chaoticity for many-particle collision processes to our knowledge. We then introduce some “time interpolation” method in order to deduce some time scales of relaxation independent of the number of particles. This method uses the uniform in time chaos estimate in order to interpolate between the relaxation rates in the limit nonlinear PDE and the spectral gap estimates in setting for the many-particle jump processes obtained in the breakthrough papers of E. Janvresse Spectral of Kac’s model of Boltzmann equation (Ann. Probab., 2001), E. Carlen, M. Carvalho, M. Loss Determination of the spectral gap for Kac’s master equation and related stochastic evolutions (Acta. Math., 2003), and D. Maslen The eigenvalues of Kac’s master equation (Math. Z., 2003). Finally we prove the propagation of entropic chaos, thus answering the point (3) above as well as an open question raised in a recent paper by E. Carlen, M. Carvalho, J. Leroux, C. Villani Entropy and chaos in the Kac model (Kinet. Relat. Models, 2010).
This preprint includes, significantly improves on, and supersedes our previous preprint on this topic.