No abstract

The Chapman-Enskog perturbation method for a phonon gas is investigated with the use of Callaway's model for the Boltzmann-Peierls equation. Assuming that the effective relaxation time for normal processes is small and the effective relaxation time for resistive processes is large, this perturbation method proposes to expand the phase density about a displaced Planck distribution and to include the above two relaxation times in the expansion. The main advantage of using the displaced Planck distribution is that the drift velocity of a phonon gas is incorporated into the model in a non-perturbative manner. The result is a system of nonlinear second-order parabolic equations for the energy density and the drift velocity which, unlike the usual set of hydrodynamic equations, does not restrict the magnitude of the individual components of the drift velocity and the heat flux in any way. This system is linearly stable at all wavelengths and is also fully consistent with the second law of thermodynamics in the sense that there exists a macroscopic entropy density which depends locally on the hydrodynamic variables and satisfies the balance equation with a non-negative entropy production due to both resistive and normal processes.

Beginning from the relativistic Boltzmann equation in a curved space-time, and assuming that there exists a fiducial congruence of timelike world lines with four-velocity vector field u, it is the aim of this paper to present a systematic derivation of a hierarchy of closed systems of moment equations. These systems are found by using the closure by entropy maximization. Our concepts are primarily applied to the formalism of central moments because if an alternative and more familiar theory of covariant moments is taken into account, then the method of maximum entropy is ill-defined in a neighborhood of equilibrium states. The central moments are not covariant in the following sense: two observers looking at the same relativistic gas will, in general, extract two different sets of central moments, not related to each other by a tensorial linear transformation. After a brief review of the formalism of trace-free symmetric spacelike tensors, the differential equations for irreducible central moments are obtained and compared with those of Ellis et al. [Ann. Phys. (NY)150 (1983) 455]. We derive some auxiliary algebraic identities which involve the set of central moments and the corresponding set of Lagrange multipliers; these identities enable us to show that there is an additional balance law interpreted as the equation of balance of entropy. The above results are valid for an arbitrary choice of the Lorentzian metric g and the four-velocity vector field u. Later, the definition of u as in the well-known theory of Arnowitt, Deser, and Misner is proposed in order to construct a hierarchy of symmetric hyperbolic systems of field equations. Also, the Eckart and Landau–Lifshitz definitions of u are discussed. Specifically, it is demonstrated that they lead, in general, to the systems of nonconservative equations.