# Algorithmic identification of thermodynamic phases

Assume that the spatiotemporal behaviour of a system $z(t,x,y)$ is given by a model $M(n,m)$. We know that this system can be in different phase i.e. $z$ can exhibit qualitatively different macroscopic properties for different $m,n$ - but we can't delineate the boundaries analytically.

Now, assume we are given a large number of simulation outputs over a range of $m$ and $n$ values. Our task is to classify the simulation outputs in phases.

My question is, how would one approach this problem generally? Statistical tests? Clustering? Addressing this question would be a very pedagogical example of how to identify a phase quantitatively.

• Phase change implies rapid change in one or more properties of a system. Isn't this idea captured by a simple derivative?
– Deep
Sep 21 '17 at 6:29
• What are $n$ and $m$ and how do they relate to $z$? Sep 22 '17 at 10:21
• $m,n$ are two parameters of the model that change its behaviour. For example, they could be temperature and external magnetisation in the Ising model. In this case, I we could look at the correlation function of the spins and distinguish whether we are super or sub critical by its value for long ranges. But this is an ad hoc solution, how would you for example distinguish between two subcritical phases? Sep 25 '17 at 9:23