Mathematical physics and ergodic theory of lattice systems - Ising model, quasicrystals 1000-1M22MIK

The lecture will be devoted to the study of mathematical models of systems of interacting particles located on the nodes of regular lattices. As an example illustrating the existence of magnets, the Ising model of interacting spins will be presented. We will prove spontaneous symmetry breaking - the existence of a phase transition.

We will discuss Hilbert's 18-th problem and its relation to quasicrystals - microscopic models of interacting particles for which the energy functional minimum is reached only at non-periodic configurations. Non-periodic tiling planes and their connections with the ergodic theory of symbolic dynamical systems will be presented. We will also deal with one-dimensional systems - Thue-Morse and Fibonacci sequences and Sturm systems in general.

Fundamental open problems will be presented: the existence of non-periodic Gibbs measures and the existence of one-dimensional non-ergodic cellular automata.

We do not assume knowledge of physics or mathematics beyond courses in the first two years of study.

Lecture schedule
1. Why do magnets exist? Ising model of interacting spins
2. Spontaneous symmetry breaking in the ferromagnetic Ising model
3. Minimization of the free energy functional
4. The exact solution of the one-dimensional Ising model
5. Generalizations of the Ising model - classical lattice-gas models
6. Percolation
7. Non-periodic tilings - Hilbert's 18-th problem
8. Microscopic models of quasicrystals - systems without non-periodic ground states
9. Non-periodic Gibbs measures
10. Symbolic dynamic systems - Thue-Morse and Fibonacci sequences
11. Ergodic theory of non-periodic systems
12. Topology of non-periodic systems
13. One-dimensional systems of interacting particles without periodic ground states
14. Cellular automata