Phase transitions in the neighbourhood of mean field theory

• Main Author: Timmers, Dennis
• Language: English
• Published: University of British Columbia 2012
• Online Access: http://hdl.handle.net/2429/43716

We study systems consisting of interacting spin particles which can have a positive or negative spin. We consider an Ising model and a type of Widom-Rowlinson (WR) model. The interactions between spin particles are regulated by Kac potentials which carry a parameter ɣ. It is known that in the Kac limit, as ɣ tends to zero, models with Kac potentials become mean field theory. Mean field theory is known to exhibit phase transitions. The focus of this work is to prove phase transitions not only in the Kac limit but also near the Kac limit i.e., for ɣ small but strictly positive. Placing the Ising and WR model in a rectangular box with side-length L and periodic boundary conditions defines finite volume Gibbs measures. The infinite volume Gibbs state ν is the limit of the finite volume Gibbs measures as L tends to infinity. A particle system exhibits a phase transition if ν is a mixture of ergodic states. The main achievement of this thesis is the development of a new method to prove phase transitions. We first apply the Kac-Siegert transformation which reformulates the particle system by introducing an auxiliary field. The spin-spin interactions are replaced by interactions of the spin particles with the auxiliary field. The main idea of this dissertation is to study the mean auxiliary field. In principle it should be easier to work with the mean field because, as we will show, it is approximately Gaussian. By a new expansion around mean field theory we prove that for ɣ strictly positive but small the infinite volume Gibbs state for the auxiliary field, for both the Ising and the WR model, is a mixture of two ergodic states. It is shown that this implies that the infinite volume Gibbs state for both the Ising and WR model is a mixture of two ergodic states. One Gibbs state predominantly has positive spin particles, the other Gibbs state predominantly has negative spin particles. The new expansion is related to the Glimm Jaffe Spencer expansion around mean field theory.