Derivation of the Enskog--Enskog System from a Particle System

We formally derive a coupled Enskog system for a gas–particle mixture from the hard-sphere BBGKY hierarchy. The derivation retains the finite separation between colliding centres, includes collisions within and between species and explicitly states the direction of spatial shifts and the mass-dependent prefactors of the four collision operators. The resulting system provides a kinetic starting point for two-phase flows for which the volume effect cannot be ignored. For unit contact factors, we derive the corresponding gas and particle balance laws, and establish an entropy inequality under suitable regularity assumptions.