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

## 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).

## References

- 1.Chapman, S., Cowling, T.G.: The Mathematical Theory of Non-uniform Gases, 3rd edn. Cambridge University Press, Cambridge (1995)zbMATHGoogle Scholar
- 2.Hirschfelder, J.O., Curtiss, C.F., Bird, R.B.: The Molecular Theory of Gases and Liquids. Wiley, New York (1964)zbMATHGoogle Scholar
- 3.Grmela, M.: Kinetic equation approach to phase transitions. J. Stat. Phys.
**3**, 347 (1971)ADSMathSciNetCrossRefGoogle Scholar - 4.Frezzotti, A., Gibelli, L., Lorenzani, S.: Mean field kinetic theory description of evaporation of a fluid into vacuum. Phys. Fluids
**17**, 012102 (2005)ADSCrossRefzbMATHGoogle Scholar - 5.Kobayashi, K., Ohashi, K., Watanabe, M.: Numerical analysis of vapor–liquid two-phase system based on the Enskog–Vlasov equation. AIP Conf. Proc.
**1501**, 1145 (2012)ADSCrossRefGoogle Scholar - 6.Frezzotti, A., Barbante, P.: Kinetic theory aspects of non-equilibrium liquid–vapor flows. Mech. Eng. Rev.
**4**, 16–00540 (2017)CrossRefGoogle Scholar - 7.Swift, M.R., Orlandini, E., Osborn, W.R., Yeomans, J.M.: Lattice Boltzmann simulations of liquid–gas and binary fluid systems. Phys. Rev. E
**54**, 5041 (1996)ADSCrossRefGoogle Scholar - 8.Gonnella, G., Lamura, A., Sofonea, V.: Lattice Boltzmann simulation of thermal nonideal fluids. Phys. Rev. E
**76**, 036703 (2007)ADSCrossRefGoogle Scholar - 9.Rowlinson, J.S., Widom, B.: Molecular Theory of Capillarity. Dover, New York (2002)Google Scholar
- 10.Carlen, E.A., Carvalho, M.C., Esposito, R., Lebowitz, J.L., Marra, R.: Phase transitions in equilibrium systems: microscopic models and mesoscopic free energies. Mol. Phys.
**103**, 3141 (2005)ADSCrossRefGoogle Scholar - 11.van Kampen, N.G.: Condensation of a classical gas with long-range attraction. Phys. Rev.
**135**, A362 (1964)MathSciNetCrossRefGoogle Scholar - 12.Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I. Springer-Verlag, Berlin (1987)CrossRefzbMATHGoogle Scholar