# To what extent can phase transitions be made rigorous?

It seems a lot of physical intuition in statistical mechanics, for example phase transitions, critical temperature, scaling hypothesis, renormalization group methods etc. should have a purely mathematical formulation; my question is: to what extent can this be done? Can we prove statements from a standard textbook (say Statistical mechanics by Huang) in a mathematically rigorous way?

A more specific example: it is well know that the 2D ising model with no external magnetic field has a 2nd order phase transition, can this be proven rigorously?

• Are you looking for exactly solvable systems with phase transitions? The free bose gas is solvable in terms of polylogarithms and has a phase transition. May 27 at 6:21

As to your question about the order of the phase transition in the Ising model, I don't think any of the books I mention above proves the fact that the latter is of second order. In dimension $$2$$, this follows, for instance, from Onsager's exact expression for the free energy. In higher dimensions, continuity of the magnetization at $$\beta_{\rm c}$$ is proved in this paper.