Syllabus

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Description

This is the second term in a two-semester course on statistical mechanics. Basic principles are examined in 8.334, such as the laws of thermodynamics and the concepts of temperature, work, heat, and entropy. Topics from modern statistical mechanics are also explored including the hydrodynamic limit and classical field theories.

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:)

  1. Dynamics: Langevin equations; conservation laws; dynamic universality classes.
  2. Random systems: annealed versus quenched impurities; Harris' criterion; random bonds; random fields; spin-glasses.
  3. Scaling theories of polymers, and other networks.

Textbooks

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

Amazon logo Kardar, Mehran. Statistical Physics of Fields. New York, NY: Cambridge University Press, 2007. ISBN 9780521873413.

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

Amazon logo 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: Amazon logo 9789971966102, and 971966115 (pbk).

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

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

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

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

Problem Sets

The homework assignments are an important part of this course, and the final average homework score will count for 30% 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.

There are 6 problem sets. Occasionally, there are problems marked as optional in the problem sets. If attempted, these problems will be graded as other problems, and their score added to the total. The overall grade for the course has a 30% contribution from the (required) problem sets. Thus, perfect scores on all the non-optional problems leads to the maximal grade of 30 from the problem sets. The optional problems provide a chance to reach the 30%-score for the problem sets, even when some of the required problems are not correct.

Exams

There are 3 in-class tests; in-class tests each count for 15% of the final grade.

Final Project

Students choose one of the following two for a final project:

  1. Research project: Write a brief paper (two to four pages in Physical Review format) on a subject of your choice, relevant to the topics of this course.
  2. Teaching site: Design a Web site that can be used to teach a topic related to Collective Behaviors to non-specialists.

The final project counts for 25% of the course grade.

Grading

Final grades will be determined from:


ACTIVITIES PERCENTAGES
Homework 30%
In-class tests (3x15%) 45%
Final project 25%