Entropic derivation of the spectral Eddington factors

In general, the closure of the finite system of moment equations by the corresponding maximum entropy distribution function results in the symmetric conservative system of first-order partial differential equations for the Lagrange multipliers of the constrained Boltzmann entropy maximization problem. Then the transformation of dependent variables yields the system of conservation equations for the moments which is consistent with the additional conservation equation identified with the balance of entropy. The objective of this paper is to employ these facts for the analysis of the spectral Eddington factors obtained from the maximum entropy distribution functions. The supposition that the spectral Eddington factors should depend on the energy density and the heat flux only through the single variable representing the heat flux normalized in some way by the energy density predominates in the literature on the subject. Here, it is demonstrated that this is true only for classical Maxwell-Boltzmann radiation and, in this case, the well-known results of Minerbo are recovered. A similar single-variable dependence postulated by Cernohorsky and Bludman for fermionic radiation cannot be justified since it leads to the contradiction with the consistency conditions between the moment evolution equations and the entropy balance. For Bose-Einstein radiation, we rederive and analyze the results given in the literature for low-energy and high-energy limits. We also show that, except for those limiting cases, the Eddington factor for bosonic radiation cannot be represented as a function of a single normalized variable. In the present approach, the entropy function plays a crucial role in determining the system of evolution equations for the energy density and the heat flux. In this system, the flux of the heat flux, and hence the Eddington factor, is determined by the additional scalar potential uniquely related to the entropy function for each type of statistics. Since the Eddington factor cannot be expressed in terms of elementary functions, we propose to use the polynomial approximation. Namely, for Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein statistics, we expand the entropy function in powers of the square of the heat flux and also calculate the corresponding power series expansion of the additional potential. By truncating the latter, we obtain the Eddington factor represented as the eighth-order polynomial in the heat flux with coefficients being the elementary functions of the energy density and the parameter which determines statistics. Finally, we analyze the behavior of the scalar Eddington factors in the limiting case when the normalized heat flux tends to one.