Tuesday 21/1 room 136 h 9-11;
Wednesday 22/1 room 134 h 9-11;
Tuesday 28/1 room 136 h 9-11;
Wednesday 29/1 room 134 h 9-11;
Monday 03/2 room 134 h 9-11 [note unusual day];
Tuesday 04/2 room 136 h 9-11;
[Wednesady 05/2 room 134 h 16-18 - Seminar] on the FPUT problem by prof. Giancarlo Benettin (Università di Padova);
Tuesday 18/2 room 136 h 9-11;
Wednesday 19/2 room 136 h 9-11;
Tuesday 25/2 room 136 h 9-11;
Wednesday 26/2 room 136 h 9-11;
[Wednesday 26/2 room 136 h 16-18 - Seminar] Mathematical Physics Seminar on Prethermalization
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 mechanics
a. 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 systems
d. Obstructions to thermalization for small energies: the role of KAM and Nekhoroshev theorems
2 - Prethermalization in quantum mechanics
a. Thermalization of local observables. Local integrals of motion.
b. Floquet systems and the existence of a prethermal state.
- [21/1/2025]: Definition of thermodynamic states, thermodynamic equilbrium state; Zero, First and Second law of thermodynamics; microscopic reversibility VS macroscopic irreversibility: Loschmidt and Zermelo paradoxes; example of pressure as time-average. Conceptual problems of the definitions.
- [22/1/2025]: Review of Boltzmann ergodic hypothesis. Microcanonical and canonical ensemble, orthodicity and equivalence of the ensembles. [Exercise Sheet #1]
- [28/1/2025]: Ergodic theory: Birkhoff-Kinchin ergodic theorem, Equivalent characterization of ergodic systems, translation on the torus. Mixing, characterization of mixing systems, mixing and approach to equilibrium. Poincaré recurrence theorem.
- [29/1/2025]: Hamiltonian perturbation theory. Canonical transformation, homological equation, normal form, resonances. Poincaré theorem on nonexistence of integrals of motion. [Exercise Sheet #2]
- [3/2/2025]: Equipartition of energy, FPUT problem and paradox. Analysis of the numerical outcomes. KAM Theorem and appicability to FPUT (Rink-Kappeler-Henrici).
- [4/2/2025]: Comments on KAM theorem. Approximation of FPUT with Toda. Zabusky and Kruskal derivation of KdV from FPUT; integrability of KdV.
- [18/2/2025]: KdV as a resonant normal form of the FPUT problem. Hamiltonian PDEs and perturbation theory [Exercise Sheet #3].
- [19/2/2025]: Quantum mechanics on a lattice, local operators. Abanin-De Roeck-Ho-Huveneers theorem. Sketch of the proof.
- [25/2/2025]: Proof of the normal form theorem ([A], Main theorem) in the time-periodic setting [Exercise Sheet #4].
- [26/2/2025]: Physical consequences and their proofs ([A]. Physical consequences).
The students who want to do the exam are required to solve the weekly exercise sheets and to discuss them with the instructor at the end of the course (short oral examination).