The course will cover traditional areas of statistical mechanics with a mathematical flavor. It will describe exact results where available and heuristic physical arguments where applicable. A rough outline is given below:
- Overview: microscopic vs. macroscopic descriptions; microscopic dynamics and thermodynamics.
- Energy surface; microcanonical ensemble; ideal gases; Boltzmann’s entropy, typicality.
- Alternate equilibrium ensembles; canonical, grand-canonical, pressure, etc. Partition functions and thermodynamics.
- Thermodynamic limit; existence; equivalence of ensembles; Gibbs measures.
- Cooperative phenomena: phase diagrams and phase transitions; probabilities, correlations and partition functions. Law of large numbers, fluctuations, large deviations.
- Ising model, exact solutions. Griffith’s, FKG and other inequalities; Peierle’s argument; Lee-Yang theorems.
- High temperature; low temperature expansions; Pirogov-Sinai theory.
- Fugacity and density expansions.
- Mean field theory and long range potentials.
- Approximate theories: integral equations, Percus-Yevick, hypernetted chain. Debye-Hückel theory.
- Critical phenomena: universality, renormalization group.
- Percolation and stochastic Loewner evolution.