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:
I. Overview: microscopic vs. macroscopic descriptions; microscopic dynamics and thermodynamics.
II. Energy surface; microcanonical ensemble; ideal gases; Boltzmann’s entropy, typicality.
III. Alternate equilibrium ensembles; canonical, grand-canonical, pressure, etc. Partition functions and thermodynamics.
IV. Thermodynamic limit; existence; equivalence of ensembles; Gibbs measures.
V. Cooperative phenomena: phase diagrams and phase transitions; probabilities, correlations and partition functions. Law of large numbers, fluctuations, large deviations.
VI. Ising model, exact solutions. Griffith’s, FKG and other inequalities; Peierle’s argument; Lee-Yang theorems.
VII. High temperature; low temperature expansions; Pirogov-Sinai theory.
VIII. Fugacity and density expansions.
IX. Mean field theory and long range potentials.
X. Approximate theories: integral equations, Percus-Yevick, hypernetted chain. Debye-Hückel theory.
XI. Critical phenomena: universality, renormalization group.
XII. Percolation and stochastic Loewner evolution.
If you have any questions about the course please email me: email@example.com. We can then set up a time to meet.