A first glimpse at phase transitions: Ferromagnetism

Abstract

The Curie-Weiss model is a simplification of the Ising model to show the existence of a phase transition for ferromagnetism. In this thesis, we study the behaviour of sums of these dependent variables. We prove in general that under the appropriate assumptions, we can still conclude a version of the Law of Large Numbers. We also find that if there exists a certain m∈R, λ>0 and integer k≥1, we have that (S_n-nm)/n^1/2k converges to exp(-λs^2k/(2k)!) in distribution.
For the Curie-Weiss model this means that for β, which is a constant proportion to inverse temperature, we find that if β∈(0,1) we have S_n/n→δ(s) and S_n/√n→ N(0,σ²) in distribution where σ²=(1-β)^-1-1. At β=1 there occurs a phase transition, we still have that S_n/n→δ(s), but now S_n/n^3/4→\exp(-s⁴/12). When β>1 we can find an m>0 such that S_n/n→½[δ(s-m)+δ(s+m)].
We also study the Curie-Weiss model where we assume that it is under the influence of a magnetic field. We prove that we do not find a phase transition, and we always have S_n/n→δ(s-m) in distribution for some m∈R. Next to this we find that (S_n-nm)/√n always converges to a normal distribution.