Statistical Mechanics and Lattice Models
Instructor: Professor Allon Percus
Contact information: CGU Math South Building, tel. 909-607-0744
Office hours: Mondays 11-12, or by appointment.
Course description: An intermediate-level graduate course emphasizing fundamental techniques in mathematical physics. Topics include: random walks, lattice models, asymptotic/thermodynamic limit, critical phenomena, transfer matrix, duality, polymer model, mean field, variational method, renormalization group, finite-size scaling.
Background and prerequisites: Math 251 or equivalent course in probability. The course will assume basic familiarity with thermodynamics, as well as undergraduate-level real analysis and linear algebra.
Textbook: All required materials will be posted on sakai, under the “Resources” section.
Assignments and grading: Homework will generally be assigned weekly, and will be due in class the following week. An automatic one-day extension will be granted to those who submit it by e-mail. There will be a take-home final exam. Grading will be approximately as follows:
- 50% Homework
- 40% Final exam
- 10% Class participation
You may work together on homework assignments, but all your answers must be explained in your own words. Your homework submission must also state the names of all those with whom you collaborated. Please familiarize yourselves with CGU’s academic honesty policies.
- Jan 21: Probabilistic background. Central limit theorem; large deviations.
- Jan 27: Random walks. Master equation; first-passage probability; critical dimension.
- Feb 3: Statistical mechanics models. Ising and related models; equilibrium properties; thermodynamic limit.
- Feb 10: Phase transitions. First-order transitions; Second-order transitions; universality.
- Feb 17: PROBLEM SESSION
- Feb 24: Exact solutions. Transfer matrix; Onsager solution; spherical model.
- Mar 3: Rigorous approaches. Griffiths’ inequalities; Peierls argument.
- Mar 10: Duality. High-temperature expansion; low-temperature expansion; duality.
- Mar 17: SPRING BREAK
- Mar 24/Mar 31: Mean field methods. Variational approach; complete graph; trees.
- Mar 31/Apr 7: Fundamentals of renormalization group. One-parameter renormalization; two-parameter renormalization.
- Apr 14: Guest lecture: the Hopfield associative memory network
- Apr 21: Real-space renormalization group. Majority rule; decimation.
- Apr 28: Monte Carlo methods. Random number generation; importance sampling; Metropolis method; Finite-size scaling.
- May 5: Statistical mechanics of computation. Satisfiability; phase transitions; algorithmic consequences.