Statistical Mechanics

Fall 2002

Course Overview



Instructors: Susan McKay and Dave Feldman.

Meeting Times: TBA

Goals

In general, this course is designed to serve as a bridge between graduate course-work and your research in classical statistical mechanics. We hope that you leave this course with an expanded knowledge base and a set of skills that will help you succeed at and enjoy research. To achieve this overall goal:
  1. We want you to gain a firm understanding of the big ideas and central themes of statistical mechanics and critical phenomena.
  2. We want you to learn some essential "tools of the trade" --- techniques that everyone working in stat mech needs to know about.
  3. We want you to gain experience presenting, teaching, and talking about statistical mechanics.
  4. We want you to gain experience learning semi-independently from graduate texts, monographs, and research articles.


Topics Covered

We anticipate covering all of the following topics:
  1. Starting points for statistical mechanics: microcanonical ensemble, canonical ensemble, grand canonical ensemble, maximum entropy formulation, minimum free energy formulation.
  2. Canonical models of classical statistical mechanics: Ising, Heisenberg, Potts, XY, Edwards-Anderson, etc.
  3. Critical Phenomena. Critical exponents, universality classes.
  4. Mean Field Theory.
  5. One-dimensional systems. Transfer matrix techniques.
  6. Monte Carlo Methods
  7. Renormalization Group methods
  8. Introduction to disordered systems


In addition, we expect to cover some of the following. The exact topics we cover will depend on time and student interest.

  1. Introduction to Dynamical Systems: Iterated maps, Lyapunov exponents, bifurcations, definition of chaos.
  2. Spatially extended dynamical systems: Cellular Automata. Coupled map lattices.
  3. Information theoretic analyses of statistical mechanical and dynamical systems: Entropy rate, metric entropy, excess entropy.
  4. Networks: random graph theory, small world networks, scale-free networks, statistics of networks, statistical models on networks.
  5. High and low temperature expansions.
  6. Systems far from equilibrium.

Evaluation

Your evaluation will be roughly based on the following:


Other Details

  1. There is no required text for the course. We have put together an annotated bibliography of some stat mech texts. We suggest having some of these books at your disposal, but there's probably no need to buy them.
  2. There will be around eight problem sets.
  3. Over the course of the term, you will give around five short, in-class presentations. These presentations will be informal discussions of a subtopic of statistical mechanics. These presentations will require some preparation, and may necessitate tracking down outside references.
  4. You are strongly encouraged to work together on homework. You can also consult us, other faculty, friends, and family. However, the homework you hand in should represent your own understanding. This means that if your friends get a homework problem and you don't understand how they did it, you shouldn't photocopy their solution and turn it in. But it's fine to talk to them, figure out the solution, and then write up the solution based on your understanding.
  5. There will not be a final exam.


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