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

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