# Probability and Statistics Seminar

### Reaching the best possible rate of convergence to equilibrium of Boltzmann-equation solutions

This talk concerns a definitive answer to the problem of quantifying the relaxation to equilibrium of the solutions to the spatially homogeneous Boltzmann equation for Maxwellian molecules. Under really mild conditions on the initial datum - closed to necessity - and a weak, physically consistent, angular cutoff hypothesis, the main result states that the total variation distance (i.e. the ${L}^{1}$-norm in the absolutely continuous case) between the solution and the limiting Maxwellian distribution admits an upper bound of the form $C\mathrm{exp}\left(-{\Lambda }_{b}^{*}t\right)$, ${\Lambda }_{b}^{*}$ being the spectral gap of the linearized collision operator and $C$ a constant depending only on the initial datum. Hilbert hinted at the validity of this quantification in 1912, which was explicitly formulated as a conjecture by McKean in 1966. The main line of the new proof is based on an analogy between the problem of convergence to equilibrium and the central limit theorem of probability theory, as suggested by McKean.