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Prerequisite

Statistical Mechanics (8.333)

Outline

1. Collective Modes: Hydrodynamic Limit; Importance of Symmetries and Dimensionality; Introduction to Phase Transitions and Critical Phenomena.
   
   
2. The Landau-Ginzburg Model: Mean-Field Theory; Critical Exponents; Goldstone Modes and the Lower Critical Dimension; Fluctuations and the Upper Critical Dimension.
   
   
3. Universality: Self-similarity; the Scaling Hypothesis; Kadanoff's Heuristic Renormalization Group (RG), and Exponent Identities.
   
   
4. Perturbation Theory: Diagrammatic Expansions; Wilson's Momentum Space RG, and the Taming of Divergent Perturbation Series by Epsilon-expansions.
   
   
5. Lattice Models: Ising, Potts, etc.; Position-space RGs (Cumulant, Migdal-Kadanoff); Monte-Carlo Simulations; Finite-size Scaling.
   
   
6. Series Expansions: Low Temperatures and High Temperatures; Duality; Random Walk Generating Functions; Exact Solution of the Two-dimensional Ising Model.
   
   
7. Two-dimensional Films: Algebraic Order; Topological Defects; Melting and the Hexatic Phase; the Non-linear Sigma Model.
   
   (If time permits, one of the following topics:)
   
   
8. Dynamics: Langevin Equations; Conservation Laws; Dynamic Universality Classes.
   
   
9. Random Systems: Annealed versus Quenched Impurities; Harris' Criterion; Random Bonds; Random Fields; Spin-glasses.
   
   
10. Scaling Theories of Polymers, and other Networks.

Textbooks

This course does not follow a particular text. The following are useful reference books:

Ma, Shang-keng. Modern Theory of Critical Phenomena. Reading, MA: W. A. Benjamin, Advanced Book Program, 1976. ISBN: 9780805366709.

Stanley, H. Eugene. Introduction to Phase Transitions and Critical Phenomena. New York, NY: Oxford University Press, 1993. ISBN: 9780195014587.

Amit, Daniel J. Field Theory, the Renormalization Group, and Critical Phenomena. Rev. 2nd ed. Singapore: World Scientific, c1984. ISBN: 9789971966102, and 971966115 (pbk).

Huang, Kerson. Statistical Mechanics. 2nd ed. New York, NY: Wiley, c1987. ISBN: 9780471815181.

Negele, John W., and Henri Orland. Quantum Many-particle Systems. Redwood City, CA: Addison-Wesley Pub. Co., c1988. ISBN: 9780201125931.

Feynman, Richard Phillips. Statistical Mechanics. Reading, MA: Addison-Wesley, 1998. ISBN: 9780201360769.

Parisi, Giorgio. Statistical Field Theory. Redwood City, CA: Addison-Wesley Pub. Co., 1988. ISBN: 9780201059854.

Problem Sets

The 12 homework assignments are an important part of this course, and the final average homework score will count for 50% of the final grade. You may consult with classmates in "study groups", as long as you write out your own answers, and do not use solution-sets from previous years.

No problem sets will be accepted after the solutions have become available. Problem sets handed in after the due date but before the solutions have been posted are subject to a 50% grade penalty.

Exams

There will be a midterm exam and a final exam. Each exam score will count for 25% of the final grade. A missed midterm will be averaged into the final grade as zero, unless an excuse is obtained in advance. Excuses are granted only for very serious circumstances attested to by the Dean or a medical doctor. A student who has been excused may be required to take a makeup exam.

Grading

Final grades will be determined from:

Your final letter grade will reflect our best attempt to evaluate objectively your performance in the course:

A: Exceptionally good performance, demonstrating a superior understanding of the subject matter, a foundation of extensive knowledge, and a skillful use of concepts and/or materials.

B: Good performance, demonstrating capacity to use the appropriate concepts, a good understanding of the subject matter, and an ability to handle the problems and materials encountered in the subject.

C: Adequate performance, demonstrating an adequate understanding of the subject matter, an ability to handle relatively simple problems, and adequate preparation for moving on to more advanced work in the field.

D: Minimally acceptable performance, demonstrating at least partial familiarity with the subject matter and some capacity to deal with relatively simple problems, but also demonstrating deficiencies serious enough to make it inadvisable to proceed further in the field without additional work.

F: Failed. This grade also signifies that the student must repeat the subject to receive credit.