Mathematics

Thermalization is the process through which a physical system evolves from an out-of-equilibrium state to a thermal state, where it can be described by statistical mechanics. In general, the thermalization process is rather complicate and it is not yet fully understood. This problem is known since the seminal work of Fermi, Pasta, Ulam, who studied numerically a simple one-dimensional model of a nonlinear crystal. This experiment opened the avenue to the discover of slow thermalization processes and it is the first historical example of prethermalization. Prethermalization describes the transient states observed in the early stages of a system’s evolution towards thermal equilibrium. These states appear to be quasi-stationary and exhibit properties different from both the initial and the final thermal states. In experiments, these are used to construct time crystals, floquet and quasi-floquet phases of matter. In this course I would like to cover basic notions about thermalization and to explain one of the rigorous proof of prethermalization available in the literature. In particular, i plan to cover the following topics:1 - Dynamical foundations of statistical mechanicsa. Ergodic theory: Space and time averages, ergodic theory, the microcanonical measure. Mixing and thermalization.b. Canonical and grand-canonical ensambles.c. The FPUT problem: a prethermal perspective and the vicinity of integrable systemsd. Obstructions to thermalization for small energies: the role of KAM and Nekhoroshev theorems2 - Prethermalization in quantum mechanicsa. Thermalization of local observables. Local integrals of motion.b. Floquet systems and the existence of a prethermal state.