Decomposition of entropy and the choice of nonequilibrium variables for relativistic classical and quantum gases

In this paper, beginning from the fundamental principles of general-relativistic kinetic theory of monatomic ideal gases – both classical and quantum, we investigate an expression for the Boltzmann entropy s (per unit volume), with a view to a deeper understanding of the generalized Gibbs relation and other postulates of extended (irreversible) thermodynamics. The main upshot of our analysis is that if M is any finite set of the moments of the distribution function f (by definition, M consists of conserved and nonconserved variables), then there exists a phase-space function z such that s can exactly be written as a sum of two structurally different parts: the first part is a function of M and thus represents the entropy of extended (irreversible) thermodynamics, while the second part gives the functional contribution to s independent of M and vanishing at z=0; this part depends only on z. Originally, the method we develop here resulted from a detailed consideration of the case where M consists of conserved variables (internal energy, density, etc.). This paper extends Banach's previous work (Physica A 275 (2000) 405) in that M contains also nonconserved variables such as, e.g., number flux vector, heat flux, and stress deviator. Our general ideas are illustrated for three important systems: classical ultrarelativistic gases and quantum photon and neutrino gases.