Non-Analyticity and the van der Waals Limit

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 $\beta \ge {\beta }_0$ and for all $\ga\in(0,\ga_0)$, $f_\ga(m)$ has no analytic continuation along the path $m\searrow m^{\ast }$ ($m^{\ast }$ 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.