New properties of a class of generalized kinetic equations

We give some properties of a new class of hard-sphere kinetic equations of great generality, introduced earlier by Polewczak. The assumptions used to obtain the general class are very weak, and the equations include not only the standard and revised Enskog equations, but also generalizations thereof that can be expected to yield essentially exact transport coefficients. In particular, there is a natural two-particle realization that is obtained from maximizing the information entropy subject to prescribed two-particle and one-particle probability distribution functions;k-particle analogs fork > 2 also naturally follow. We obtain Liapunov functionals for the whole class of equations under consideration and discuss the question of which of these functionals can be expected to play the role ofH-functions. We also obtain several more special results that include new lower bounds on the potential part of theH-function for the revised Enskog equation. The bounds are instrumental in obtaining global existence theorems and also imply that the necessary condition for invertibility of the nonequilibrium extension of local activity as a functional of local density is satisfied.