Syllabi for Statistical Mechanics PCS 8302 (spring 08)

Instructor: M Howard Lee

This course being a continuation of PCS 8301, we will follow the
syllabus for the entire sequence given below. We will begin with the
renormalization solution of the Ising model. See B 6. Depending on the
interest and progress made, some topics in D may be covered.

The course will be governed by the same requirements for PCS 8301 (SMI)
except for the regular exams. The grades will be based on the homework.
See the syllabus for SMI for additional details.


A. Foundation of stat mech

1. Gibbs' entropy function and microcan. ensembles
2. LaGrange's method of multipliers; temperature and chemical potential
3. Applications to classical ideal gases
4. two-level system and negative temperature


B. Canonical ensembles and applications to quantum problems

1. Quantum partition function and ensemble averaging: free energy,
fluctuations
2. Einstein and Debye solids; phonons and specific heat
3. Planck's radiation, cosmic background, entropy of the universe
4. magnetism and lattice problems
a. intro to paramagnet, ferromagnet, anti-ferromagnet
b. phenomenology of phase transitions in magnets
c. ID Ising model: exact solution by a transfer matrix method
5. Phase transitions and critical phenomena in magnetic systems
a. mean field theory of the Heisenberg model and classical exponents
b. scaling theory, Rushbrooke's inequality for critical exponents
c. graphs and approximate methods
6. Wilson's Renormalization group method
a. Concept of a fixed point and relevant variables
b. Application to 1d Ising models
c. Real space RG: 2d Ising model


C. Grand canonical ensembles and many-body problems

1. classical problems of adsorption, site vacancy and migrations
2. many-fermion systems
a. chemical potential and Fermi energy, applications to normal metals
b. relativistic problems: degenerate stars and Chandrasekhar limit.
3. many-boson systems
a. Bose Einstein condensation and first order phase transition
b. applications to atomic Bose gases
c. Introduction to the low-T behavior of liquid He-4
4. Classical fluids of hard sphere and other potentials
a. Ursell-Mayer cluster expansions
b. Yang-Lee zeros and mechanisms of phase transitions
c. pair correlation functions, long range order, structure factors


D. Special Topics

1. chaos and 1d mapping
a. Feigenbaum numbers and scaling
b. Feigenbaus's route to chaos and other scenarios
c. attractors
2. Julia and Mandelbrot sets
relationships to Yang-Lee zeros, 1d Ising model and transformations
3. Linear response theory of Kubo
a. Van Hove's relations for the scattering cross section and the
correlation function.
b. conductivity formula and diffusion
c. sum rules, fluctuation-dissipation theorem
d. generalized Langevin eq; relaxation and memory functions
4. Off-diagonal long range order (ODLRO) of Yang, Onsager-Penrose
applications to superconductivity, superfluidity, and other
quantum systems.
5. Introduction to many-body theory
a. second quantization
b. correlation energy of an electron gas, charged Bose gas
c. Bogoliubov's theory of an interacting many Bose gas
d. Yang-Huang-Lee theory of a hard sphere Bose gas
e. BCS theory of superconductivity