Non-equilibrium Dynamics and Random Matrices

In this talk we will explain several results surrounding global stability problem for the Boltzmann equation 1872 with the physically important collision kernels derived by Maxwell 1867 for the full range of inverse power intermolecular potentials, \(r^{-(p-1)}\) with \(p > 2\) and more generally. This is a problem which had remained open for quite a long time. Specifically, we now have global solutions that are perturbations of the Maxwellian equilibrium states, and which decay rapidly in time to equilibrium. This proof is facilitated by our sharp geometric understanding of the diffusive nature of the non cut-off collision operator. Furthemore, since the work of Ukai-Asano in 1982, it has been a longstanding open problem to determine the optimal large time decay rates for the soft potential Boltzmann equation in the whole space, with or without the angular cut-off assumption. We prove that our solutions converge to the global Maxwellian with the optimal large-time decay rates. We furthermore prove the optimal decay rates for the high \(k\)-th order derivatives. Much of this is joint work with P. Gressman, other results are joint with V. Sohinger.