Exact solution of the van der Waals model in the critical region

Adriano Barra, Antonio Moro

Published 2014-12-05Version 1

Inspired by the theory of nonlinear conservation laws, we propose a novel approach, in the framework of statistical mechanics, that naturally extends the van der Waals model to the critical region. Starting from an effective microscopic description, we derive the general functional form of its mean field partition function under the assumption named Isochoric Weights Thermodynamic ansatz. The condition that outside the critical region the model reproduces, in the thermodynamic limit, the classical van der Waals equation of state allows to fix uniquely the partition function. We show that isothermal curves develop a classical viscous shock which provides the exact analytical description of the first order gas-liquid transition of simple fluids. The solution obtained holds for finite number of particles and, in the thermodynamic limit, automatically encodes the Maxwell equal areas rule.