STATISTICAL MECHANICS

(Physics 214B)

The course is intended as an advanced introduction to the methods of Statistical Mechanics. It will be assumed that the student has already some familiarity with statistics, probability theory and elementary complex variable theory and contour integration. First-year graduate courses in Electromagnetism and Classical and Quantum Mechanics are prerequisites. Topics to be covered during the quarter will include:

Ideal (non-interacting) Fermi Systems. Thermodynamics of an ideal Fermi gas. Magnetic behavior of an ideal Fermi gas (Pauli paramagnetism, Landau diamagnetism). Electron gas in metals. Thermoionic and photoelectric emission. White dwarf stars. Thomas-Fermi statistical model of the atom.

Method of Cluster Expansions. Cluster expansion for a classical gas. Virial expansion for the equation of state. Virial coefficients. Exact results. Inhomogeneous liquids, capillary waves. Cluster expansion for quantum systems.

Method of Quantized Fields. Introduction to second quantization. Low temperature behavior of an interacting Bose gas. Characterization of low lying states, energy spectrum. Superfluidity and quantized vortex rings. Low lying states of an interacting Fermi gas. Energy spectrum of a Fermi liquid. Superconductors, Cooper pairs, BCS gap equation and quasiparticle excitation spectrum. Landau phenomenological theory.

Phase Transitions: Criticality, Universality and Scaling. Dynamical models for phase transitions. Lattice gas and binary alloys. Ising model. Mean field theory. Corrections to mean field behavior. Critical exponents. Thermodynamic inequalities. Landau theory. Scaling hypothesis for thermodynamic functions. Correlations and fluctuations. Leading thermal and magnetic exponents. Anomalous dimensions.

Models for Phase Transitions. Ising model in one dimension. Heisenberg ferromagnet and n-vector model. Ising model in two dimensions, exact solution. Statistical mechanics models in arbitrary dimensions. High and low temperature series expansions. Kosterlitz-Thouless transition. Polymers and membranes. Random systems and spin glasses. Replica trick.

Renormalization Group Approach to Phase Transitions. Scale transformations and scaling. Fixed points. The one-dimensional and two-dimensional Ising models as examples. Decimation, bond-shifting and block-spin transformations. General formulation of the renormalization group. The epsilon expansion, Wilson-Fisher fixed point. Upper and lower critical dimension. The 1/n expansion. Finite size scaling. Simulations and Monte Carlo methods.

Fluctuations and Non-Equilibrium Phenomena. Brownian motion. Langevin equation. Approach to equilibrium, Fokker-Planck equation. Wiener-Khintchine relations. Fluctuation-dissipation theorem. Onsager relations.

Recommended Books:

Statistical Mechanics, by R.K. Pathria (Butterworth & Heinemann, 1996);

Statistical Physics, by L.E. Reichl (Wiley, 1998);

Equilibrium Statistical Physics, by M.Plischke and B. Bergersen (World Scientific, 1994).

Herbert W. Hamber, FRH 3172 (x5596).

hhamber@uci.edu