Non-Analyticity and the van der Waals Limit

TL;DR

This paper investigates the non-analytic behavior of the free energy in the Kac model during first order phase transitions in the van der Waals limit, revealing non-analyticity persists near the transition point.

Contribution

It demonstrates that the free energy lacks analytic continuation at the phase transition point in the van der Waals limit for certain temperature ranges.

Findings

Existence of a temperature threshold $eta_0$ for non-analyticity.

Non-analyticity of $f_eta(m)$ along the spontaneous magnetization path.

Persistence of non-analyticity as $eta$ and $ heta$ vary within specified bounds.

Abstract

We study the analyticity properties of the free energy $f_\ga(m)$ of the Kac model at points of first order phase transition, in the van der Waals limit $\ga\searrow 0$. We show that there exists an inverse temperature $\beta_0$ and $\ga_0>0$ such that for all $\beta\geq \beta_0$ and for all $\ga\in(0,\ga_0)$, $f_\ga(m)$ has no analytic continuation along the path $m\searrow m^*$ ($m^*$ denotes spontaneous magnetization). The proof consists in studying high order derivatives of the pressure $p_\ga(h)$, which is related to the free energy $f_\ga(m)$ by a Legendre transform.