Extended kinetic theory

Abstract

The domain of applicability of one-point kinetic theory is enlarged by admitting two-point distribution functions or some of their moments as extra state variables. Kinetic equations governing the time evolution in the extended state space are introduced as particular realizations of the nonlinear Onsager–Casimir equation. This abstract equation collects the common structure of equations arising in the analysis of compatibility of two levels of description (e.g., kinetic theory and equilibrium thermodynamics or kinetic theory and hydrodynamics). The structure consists of a Poisson bracket and two potentials. The Poisson bracket expresses kinematics, the first potential, called the thermodynamic potential, generates the time evolution and links the theory with thermodynamics. The second potential, called the dissipative potential, introduces dissipation into the time evolution that is compatible with the kinematics and thermodynamics. To find a particular realization of the nonlinear Onsager–Casimir equation means to identify kinematics (Poisson bracket) and the two potentials, that correspond to a particular system and situations under consideration. This way of introducing kinetic equations has two advantages. First, the kinetic equations are guaranteed to be intrinsically consistent and compatible with more macroscopic theories. Second, the clear physical meaning of kinematics and the two potentials, together with the possibility of discussing them separately, allows one to use a broad range of physical insights and considerations to specify them for particular systems and situations of interest.