A simple kinetic model for the phase transition of the van der Waals fluid

TL;DR

This paper introduces a minimal kinetic model for van der Waals fluid phase transitions, linking microscopic interactions with thermodynamic properties and deriving a Cahn--Hilliard type equation for simulations.

Contribution

It presents a simplified kinetic model that captures non-ideal gas effects and derives a continuum equation for phase transition analysis.

Findings

The model determines the non-ideal gas potential from the van der Waals equation of state.

A monotonic functional related to Helmholtz free energy is identified via the H theorem.

Numerical simulations using the derived Cahn--Hilliard equation demonstrate phase transition dynamics.

Abstract

A simple kinetic model, which is presumably minimum, for the phase transition of the van der Waals fluid is presented. In the model, intermolecular collisions for a dense gas has not been treated faithfully. Instead, the expected interactions as the non-ideal gas effect are confined in a self-consistent force term. Collision term plays just a role of thermal bath. Accordingly, it conserves neither momentum nor energy, even globally. It is demonstrated that (i) by a natural separation of the mean-field self-consistent potential, the potential for the non-ideal gas effect is determined from the equation of state for the van der Waals fluid, with the aid of the balance equation of momentum, (ii) a functional which monotonically decreases in time is identified by the H theorem and is found to have a close relation to the Helmholtz free energy in thermodynamics, and (iii) the Cahn--Hilliard type equation is obtained in the continuum limit of the present kinetic model. Numerical simulations based on the Cahn--Hilliard type equation are also performed.