Phase Transitions in Ising Systems with Long but Finite Range Interactions

M. Cassandro, E. Presutti

1996, v.2, Issue 2, 241-262

ABSTRACT

We consider an Ising system with spin-spin interaction $-J_\gammma(x,y)\sigma(x)\sigma(y)$, $x$ and $y\in Z^d$, $d\ge 2$, where $J_\gamma(x,y)\ge 0$ is a Kac potential, $\gamma>0$ the scaling parameter and $\gamma^{-1}$ the interaction range, which is strictly finite for any $\gamma>0$. The interaction is normalised so that $\beta=1$ is the mean field inverse critical temperature [J. Lebowitz and O. Penrose, Rigorous treatment of the Van der Waals Maxwell theory of the liquid vapour transition, J. Math. Phys., 1996, v.7, 98] Calling $\beta_cr(\gamma)$ the inverse critical temperature for the scaling parameter $\gamma>0$, we prove that for any $\beta>1$ there exists $\gamma_\beta>0$ such that $\beta_cr(\gamma)<\beta$ for all $\gamma\le \gamma_\beta$. Since $\beta_cr(\gamma) \ge 1$ [M. Cassandro, E. Olivieri, A. Pellegrinotti and E. Presutti, Existence and uniqueness of DLR measures for unbounded spin systems, Z. Wahr. verw. Geb., 1987, v. 41, 313-334], this shows that in dimension $d\ge 2$ the van der Waals theory correctly predicts the critical temperature in the ``Lebowitz-Penrose limit'' $\gamma\to 0^+$.

Keywords: Ising systems,phase transitions,Kac potentials

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