On the nonequilibrium statistical mechanics of a binary mixture. I. The distribution functions

Kinetic derivation of Cahn-Hilliard fluid models

“…The corresponding matrices, however, may be singular in the situation of thermodynamic instabilities. The resulting macroscopic equations are found to be that of nonideal Navier-Stokes-Fourier-Fick fluids with density dependent transport coefficients [49,50,51], generalizing the results obtained for single species fluids [52,56,57,58,54].…”

“…The triplet particle distribution functions f ijk similarly depend on (x i , x j , x k , t) but triple collision effects are not taken into account in this work. For a single gas, the equations of the BBGKY hierarchy have been derived independently by Bogoliubov [66], Born and Green [67], Kirkwood [68], and introduced by Yvon [69], and we refer to the literature [50,51,56,57,70] for more details. The aim of this work being the derivation of macroscopic equations governing Cahn-Hilliard type fluids, we only use a simplified kinetic theory of moderately dense gas mixtures.…”

“…The kinetic theory of dense gas mixtures involves generalized Boltzmann equations governing the species one-particle distribution functions f i , i ∈ S. These equations are obtained by expanding multiple distribution functions with clusters of independant particles and using Bogoliubov's method [49,50,51], in a similar way as for single species fluids [52,56,57,58,54].…”

“…The kinetic theory of dense gas mixtures is based on generalized Boltzmann equations for the species one-particle distribution functions [49,50,51], generalizing the situation of single species fluids [52,53,54,55,56,57,58]. Such Boltzmann equations are obtained from the two first BBGKY hierarchy equations and involve cluster expansions as well as Bogoliubov's functional property.…”

“…Assuming that repulsive potentials are hard, some collision integrals for dense gas mixtures are also simplified in such a way that the diffusive fluxes are directly expressed in terms of chemical potential gradients in agreement with thermodynamics. In comparison, linear system inversions have been used in previous work in order to transform fluxes expressed in terms of density gradients into fluxes expressed in terms of chemical potential gradients [50,51]. The corresponding matrices, however, may be singular in the situation of thermodynamic instabilities.…”