Enskog and Enskog-Vlasov equations with a slightly modified correlation factor and their H theorem
Shigeru Takata, Aoto Takahashi
TL;DR
This work addresses extending kinetic theory to non-ideal dense gases by modifying the Enskog collision factor with a simple density-functional form. It introduces the Enskog equation with a slight modification (EESM) using the correlation factor $g(X,Y) = S(R(X)) + S(R(Y))$ with $R(X) = \dfrac{1}{m}\int_D \rho(\mathbf{Y})\theta(\sigma-|\mathbf{Y}-\mathbf{X}|) d\mathbf{Y}$, and proves an H-theorem for both Enskog and Enskog–Vlasov equations under three boundary paradigms. By selecting explicit forms of the scalar function $S$, the model recovers classical equations of state (e.g., van der Waals and Carnahan–Starling) and yields a corresponding pressure relation $p = \rho RT(1+2b\rho S(2b\rho))$; the framework is designed to be numerically implementable and extensible to multi-component systems. The results provide thermodynamic consistency, a practical path to dense-gas simulations, and a basis for future work on boundary conditions, Vlasov coupling, and fluid-dynamic limits.
Abstract
A novel modification of the original Enskog equation is proposed. The modification is much simpler than that is made in the modified (or revised) Enskog equation proposed by van Beijeren \& Ernst in 1973 and does not require a consideration of many-body configuration. The proposed modification is general enough to be adapted to various equations of states for non-ideal gases. It is shown that the H-theorem can be established for the Enskog and the Enskog--Vlasov equation with the proposed modification.