Spin systems and phase transitions

Supervisor: Tyler Helmuth

Research area: probability.

Description:

Phase transitions are familiar: water turns from liquid to gas at the boiling point. Knowing that all matter is made of atoms, and that atoms follow Newton’s equations, how could we predict this behaviour? The field of statistical mechanics is devoted to this question in all of its many guises. It turns out to be fruitful to set Newton’s equations aside, and instead turn this into a probability problem. This project concerns a special case, so-called spin systems. These serve as models for various physical phenomena (liquid-vapour transition; magnetism; crystallization) and have the appeal of being comparatively simple to define with mathematical precision. Despite this simplicity, spin systems display rich and varied behaviour, and their mathematical analysis is the subject of much ongoing research.

Group Project:

The group project will revolve around learning basic concepts and results about spin systems through the lens of the Ising model. By the end of the group project, you will have collectively learned:

  • The definition of the Ising model and the associated objects of interest.
  • Mathematical definitions of phase coexistence, uniqueness, and phase transitions.
  • Correlation inequalities for the Ising model
  • The phase diagram of the Ising model, and methods of proof of the various phenomena that occur.

Mode of operation and evidence of learning for the group project

The project will revolve around learning by reading, with a focus on theory, mathematical rigour, and the development of conceptual understanding. Students will demonstrate their understanding by: solving relevant problems; exploring examples and theoretical applications of the material; clearly communicating in both written and oral formats.

Individual Project:

The individual project will revolve around your interests. Some possibilities include:

  • Applying the techniques developed in the group project to (discrete) spin systems other than the Ising model, e.g., the $q$-state Potts model or the hard-core model
  • Investigating the basic phenomena of spin systems with continuous symmetries, e.g., the Mermin-Wagner theorem
  • Investigating continuous particle systems, e.g., via the cluster expansion for the hard-sphere gas
  • Exploring the critical behaviour of spin systems, e.g., the solution of the two-dimensional Ising model
  • Investigating dynamical (Markov chain) aspects of spin systems, e.g., fast and slow mixing phenomena
  • Related topics in the theoretical computer science and combinatorics, e.g., approximate counting and sampling.

Mode of Operation and Evidence of Learning for the individual project

The project will revolve around learning by reading, with a focus on theory, mathematical rigour, and the development of conceptual understanding. Students will demonstrate their understanding by: solving relevant problems; exploring examples and theoretical applications of the material; clearly communicating in both written and oral formats.

Prerequisites and Co-requisites

Probability II

Resources (in progress)