Basic concepts of statistical physics:
Density matrix, partition function, thermodynamic averages. Entropy and microcanonical ensemble: reversibility, egodicity, typicality. Legendre transforms and thermodynamic potentials. Canonical and grandcanonical distribution. Quantum statistics: fermions, bosons, anyons. Fermi gas, Bose-Einstein condensation.
Theory of phase transitions:
Classification (continous and discontinous transitions), analyticity of thermodynamic potentials, order parameter, symmetries. Quantum phase transitions, topological phases (Kitaev model). Existence of phase transitions: Mermin-Wagner theorem, Perron theorem. Solvable models: 1D and 2D Ising. Critical exponents and universality. Mean field approximation, Landau theory,
Ginzburg criterium. Renormalization group theory in real space: examples of 1D and 2D Ising.
Non-equilibrium statistical physics:
Linear response theory: response function, fluctuation-dissipation theorem; calculation of transport coefficients – an example of electrical conductivity. Symmetries of transport coefficients, Onsager relations. Stochastic processes: Langevin equation, Brownian motion, gaussian processes, Fokker-Planck equation. Nonequilibrium fluctuation theorems: Tasaki - Crooks theorem , Jarzynski equality, generalization of the II. Law of thermodynamics. Systems far from equilibrium: master equations, principle of detailed balance, stationary states, exclusion models.