k-Particle kinetic equations: in search of the nonequilibrium entropy

Abstract

Systematic development of various Liapunov functionals (generalizations of the H-functions) in the kinetic theory is studied. The functional are monotone functions of time, whose stationary points determine the equilibria of the system governed by the corresponding kinetic equation. The mathematical structure is general enough to embrace kinetic equations for the N-particle distribution functions (the fully hierarchy equation) as well as the kinetic equations of the reduced description, i.e., the equations for the k-particle distribution functions. In the case of the hierarchy of N equations (including the exact hierarchy) the stationary points of the functionals are of the same functional form as the k-particle distribution function in the equilibrium statistical mechanics. For k=1 and the closure relation as in the revised Enskog equation, the first member of the family becomes the H-function found by Resibois [10]. As an application of the explicit form of the Liapunov functionals various existence and stability results for the corresponding kinetic equations are presented.