# A Simple Kinetic Model for the Phase Transition of the van der Waals Fluid

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## 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.

## Keywords

Boltzmann equation Kinetic theory for non-ideal gases Phase transitions Nonlinear dynamics## Notes

### Acknowledgements

The present work was supported in part by JSPS KAKENHI Grant Number 17K18840 and by JSPS and MAEDI under the Japan-France Integrated Action Program (SAKURA).

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