Project IV (MATH4072) 2010-11


The Ising and other solvable models in statistical mechanics

Ed Corrigan

Description

The Ising model grew out of attempts to understand spontaneous magnetisation - the property that some materials (such as a piece of iron) have to remain magnetised when an external magnetic field is removed. Spontaneous magnetisation is associated with a phase transition, meaning it disappears as the temperature is raised above a certain critical point. The one-dimensional Ising model does not exhibit this behaviour but the two-dimensional Ising model does. Moreover many features of it may be determined exactly (L Onsager was the first to do this in 1944). The three dimensional Ising model is more tricky, with its properties (so far) being determined via numerical simulations. Since the early work on the Ising model, and particularly after Baxter's work in the 1960s and 1970s, there has been huge interest in discovering and determining the properties of other exactly solvable models of a similar type. This effort continues to the present.

There are many directions this project could take, ranging from a review of Onsager's work and its further developments by CN Yang and others, to exploring generalisations of the Ising model leading to interesting mathematics surrounding the Yang-Baxter equations, or to creating your own simulations.

Prerequisites

  • Mathematical Physics II   MATH2071
  • Statistical Mechanics MATH4231  (or MATH3351)  would be an advantage but not essential.
  • Experience with programming  (for example in Matlab, Maple, C, C++ ...) and a desire to use a computer could be useful but not essential.

Resources

email: Ed Corrigan