ISSN:
1089-7658
Source:
AIP Digital Archive
Topics:
Mathematics
,
Physics
Notes:
The purpose of this paper is to examine the relationship between the entropy balance equation, the Gibbs formula, and the Boltzmann equation. Consider a system of a mixture of gases contained in an arbitrary region Ω with volume V, where no chemical reactions take place. Let fi be the one-particle distribution function of species i. First, suppose there exist some fi, such that the entropy density ρS, the entropy flux Js, the entropy production σ, and the Boltzmann H-function, H(t), satisfy, respectively, the entropy balance equation and the Boltzmann H-equation under appropriate boundary conditions on the surface ∂Ω of Ω. Then fi can be shown to satisfy the Boltzmann equation. Under the functional hypothesis, where fi depends on time t and the spatial coordinates r only in terms of the thermodynamic variables—particle density of species i, ρi, hydrodynamic velocity v, energy density E, stress tensor π¯i, heat flux Q'i, and mass flux Ji, and possibly the spatial derivatives of { ρi, v, E, π¯i, Qi, Ji}, the entropy balance equation together with the semipositive definiteness of the entropy production, σ≥0, then provides an alternative method of solving the Boltzmann equation. The thermodynamic variables, in turn, are governed by their corresponding evolution equations with appropriate boundary conditions.Second, to the linear order of the spatial gradients of the temperature T, the hydrodynamic velocity v, and the ratio of the particle number ni /n, the entropy balance equation then yields a generalized Gibbs formula and a nonlinear solution of fi in terms of the thermodynamic variables, such that σ≥0. The generalized Gibbs formula is an exact one-form of the thermodynamic variables that contains the equilibrium Gibbs formula. Furthermore, if fi is linearized, it is identical to the expression given by Grad's 13-moment method. Finally, we consider the stability problem of the evolution equations for π¯i, Q'i, and Ji.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1063/1.527213
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