A Hierarchy of Diffusive Higher-Order Moment Equations for Semiconductors

Abstract

Abstract. A hierarchy of diffusive partial differential equations is derived by a moment method and a Chapman-Enskog expansion from the semiconductor Boltzmann equation assuming dominant elastic collisions. The moment equations are closed by employing the entropy maximization principle of Levermore. The new hierarchy contains the well-known drift-diffusion model, the energy-transport equations, and the six-moments model of Grasser et al. It is shown that the diffusive models are of parabolic type. Two different formulations of the models are derived: a drift-diffusion formulation, allowing for a numerical decoupling, and a symmetric formulation in generalized dual entropy variables, inspired by nonequilibrium thermodynamics. An entropy inequality (or H-theorem) follows from the latter formulation.

A realizability-preserving discontinuous Galerkin method for the M1 model of radiative transfer

“…We solve (32) with c = 1 and T = 0 in a purely absorbing medium with r a = 0.5, r s = 0. We represent the delta function in the initial condition:…”

“…Here, solutions are computed allowing a non-equilibrium between the fields. The radiative transfer is modeled as a gray process in (32) and is coupled to the material energy balance equation in (34). We solve (32) and (34) Due to a different scale, the numerical stability parameter for the realizability limiter is set to T 0 ðxÞ ¼ 4ðsinð2pxÞ þ 2Þ;…”

“…Entropy-based models have been studied extensively in the areas of extended thermodynamics [21,46], gas dynamics [27,30,33,34,36,38,54], semiconductors [1][2][3][4]28,32,35,37,52], quantum fluids [20,22], radiation transport [9,10,14,15,[23][24][25]29,31,44,45,55,57], and phonon transport in solids [22]. Since we are interested in photons, the Eddington factor in (2) is derived using the Bose-Einstein entropy.…”

“…Again, the corresponding equation is(32) where c = 1. As a comparison, a reference solution for the transport equation is calculated by a discrete ordinates method with 10 5 discretization points in space and 256 in the angular variable l. Values on the left boundary for each ordinate l are taken from an extremely sharp Gaussian distribution, in order to mimic the forward-peaked incoming beam of photons.Fig.…”

High-Order Entropy-Based Closures for Linear Transport in Slab Geometry II: A Computational Study of the Optimization Problem

“…In transport and kinetic theory, entropy-based closures are used to derive moment models which retain fundamental properties of the underlying kinetic equations such as hyperbolicity, entropy dissipation, and positivity. The resulting models have been studied extensively in the areas of extended thermodynamics [18,48], gas dynamics [24,29,32,33,40,42,57], semiconductors [2][3][4][5]26,31,34,41,56], quantum fluids [16,19], radiative transport [9-12, 20-22, 28, 30, 46, 47, 59, 62], and phonon transport in solids [19]. In spite of their attractive mathematical properties and wide application, entropy methods still suffer from several short-comings.…”