Maximum Entropy Closure Research Articles

Consistency of the phenomenological theories of wave-type heat transport with the hydrodynamics of a phonon gas

Various phenomenological theories of wave-type heat transport, which can be interpreted as the models of an isotropic rigid heat conductor with an internal vector state variable, have been proposed in the literature with the objective to describe the second sound propagation in dielectric crystals. The aim of this paper is to analyze the relation between these phenomenological approaches and the phonon gas hydrodynamics. The four-moment phonon gas hydrodynamics based on the maximum entropy closure of the moment equations with nonlinear isotropic phonon dispersion relation is considered for this purpose. We reformulate the equations of this hydrodynamics in terms of energy and quasi-momentum as the primitive fields and subsequently demonstrate that, from the macroscopic point of view, they can be understood as describing the reference model of an isotropic rigid heat conductor with quasi-momentum playing the role of the internal vector state variable. This model is determined by the entropy function and the additional scalar potential, but if the finite domain of phonon wave vectors is approximated by the whole space, the additional potential can be expressed in terms of the entropy function and its first derivatives. Then the transformation of primitive fields and the expansion of thermodynamic potentials in powers of the square of quasi-momentum enable us to compare the reference model with the models proposed earlier in the literature. It is shown that the previous models require some subtle modifications in order to achieve full consistency with phonon gas hydrodynamics.

A DISCUSSION ON THE STOCHASTIC MODELING OF SPRAY FLOWS

Most theoretical and numerical models of multiphase flows can be placed into one of three categories: Eulerian-Eulerian, Lagrangian-Eulerian, or stochastic methods. Stochastic methods generally involve the development and evolution of a probability density function (pdf) to describe random variables representing particle or droplet properties. This article presents a survey of stochastic spray modeling frameworks that have appeared in the literature. A review of the spray equation and the sectional approach provides a background to a more extensive discussion on specific modeling methods, including assumed pdfs, maximum-entropy moment closure, and particle methods. Several examples are provided to highlight each modeling technique as well as an examination of the advantages and disadvantages of each method.

Exact Maximum Entropy Closure of the Hydrodynamical Model for Si Semiconductors: The 8-Moment Case

An exact closure is obtained of the 8-moment model for semiconductors based on the maximum entropy principle in the case of silicon semiconductors. The validity of the model is assessed, and comparisons with an approximate closure are presented and discussed.

A 3-D multiband closure for radiation and neutron transfer moment models

We derive a 3D multi-band moment model and its associated closure for radiation and neutron transfer. The new closure is analytical and nonlinear but very simple. Its derivation is based on the maximum entropy closure and assumes a Wien shape for the intensity when used in the Eddington tensor. In the multi-band approach, the opacity is re-arranged (binned) according to the opacity value. The multi-band model propagates identically all photons/neutrons having the same opacity. This has been found to be a good approximation on average since the transport is mostly determined by the opacities and less by the frequencies. This same concept is used to derive the closure. We prove on two complex test atmospheres (the solar atmosphere and an artificial atmosphere) that the closure we have derived has good accuracy. All approximations made in deriving the model have been carefully numerically checked and quantified.

Maximum-Entropy Closures for Kinetic Theories of Neuronal Network Dynamics

We analyze (1 + 1)D kinetic equations for neuronal network dynamics, which are derived via an intuitive closure from a Boltzmann-like equation governing the evolution of a one-particle (i.e., one-neuron) probability density function. We demonstrate that this intuitive closure is a generalization of moment closures based on the maximum-entropy principle. By invoking maximum-entropy closures, we show how to systematically extend this kinetic theory to obtain higher-order, kinetic equations and to include coupled networks of both excitatory and inhibitory neurons.

Nine-moment phonon hydrodynamics based on the maximum-entropy closure: one-dimensional flow

The maximum-entropy distribution functions applied to Callaway's model for phonon gas dynamics lead to a hierarchy of closed systems of moment equations. The system of equations for the energy density and the heat flux is the first member of this hierarchy of closures. Here emphasis is placed on analysing the next member, the 9-moment maximum-entropy system that involves the flux of the heat flux as an extra gas-state variable. After presenting a study of the one-dimensional, rotationally symmetric reduction of this system, we explicitly calculate a single generating function of three Lagrange multipliers in terms of which the reduced system of three evolution equations for these multipliers can be cast into a symmetric hyperbolic form. In the context of determining the Lagrange multipliers as explicit functions of the moment densities, we discuss new aspects of the expansion of various non-equilibrium quantities about quasi-equilibrium states. This expansion is fundamentally a non-equilibrium expansion that includes the heat flux in a non-perturbative manner, i.e., there are no unphysical limitations on the magnitude of the nonvanishing component of the heat flux to maintain a theory. Results are presented both at the first order in the expansion and at the second order. This enables us to verify the internal consistency of our approach and to justify the non-equilibrium generalization of the method of Grad.

A half-moment model for radiative transfer in a 3D gray medium and its reduction to a moment model for hot, opaque sources

We present in this paper a new 3D half-moment model for radiative transfer in a gray medium, called the M 1 1 / 2 model, which uses maximum entropy closure. This model is a generalization to 3D of the 1D version recently proposed in (J. Comp. Phys. 180 (2002) 584). The direction space Ω is divided into two pieces, Ω + and Ω - , in a dynamical way by the plane perpendicular to the total radiative flux, and the half moments are defined from these subspaces. The model closure and the integrations of the radiative transfer equation performed on the moving Ω ± spaces are detailed. 1D planar results, which have motivated the extension of the model of (J. Comp. Phys. 180 (2002) 584) to multi-dimensions, are shown. These results are very good. The model is thereafter derived for 3D spherically symmetric geometry, where the correctness of the non-trivial border terms can be checked. Two 3D spherically symmetric problems are numerically solved in order to show the accuracy of the closure and the role of the border terms. Once again, compared to the solution obtained with a ray tracing solver, results are very good. From the 3D half-moment model, a new moment model, called M 1 + , is derived for the particular case of a 3D hot and opaque source radiating into a cold medium, for applications such as simulations of stellar atmospheres and fires. Two-dimensional numerical results are presented and compared to those obtained solving the RTE and with other moment models. They demonstrate the very good accuracy of the M 1 + model, its good convergence properties, and better prediction compared to all other existing moment models in its domain of applicability.

When Maximizing Entropy gives the Rational Closure

We introduce a generalization of epsilon-probability functions and a new correspondence with rational consequence relations. The rational consequence relations corresponding to the generalized epsilon-probability functions of maximum entropy are investigated and it is shown that in this case the analogous notion of `maximum entropy closure' equals the rational closure, thus reconfirming the rational closure of a conditional knowledge base as the simplest, least prejudiced, rational consequence relation satisfying that knowledge base.

On the approximation of the Fokker-Planck equation by momentsystems

The aim of this paper is to show that moment approximations of kineticequations based on a maximum-entropy approach can suffer from severedrawbacks if the kinetic velocity space is unbounded. As example, westudy the Fokker-Planck equation where explicit expressions for themoments of solutions to Riemann problems can be derived. The quality ofthe closure relation obtained from the maximum-entropy approach as wellas the Hermite/Grad approach is studied in the case of five moments. Itturns out that the maximum-entropy closure is even singular inequilibrium states while the Hermite/Grad closure behaves reasonably. Inparticular, the admissible moments may lead to arbitrarily large speeds ofpropagation, even for initial data arbitrary close to global eqilibrium.

One-dimensional Riemann solvers and the maximum entropy closure

The maximum entropy closure for the two-moment approximation of the neutron transport equation is presented. We use a robust Roe-type Riemann solver to solve the resulting moment equations. We also present three boundary conditions to use with this method. The ghost cell method effectively implements the Mark boundary condition by placing phantom cells just outside the physical system. This method is extremely simple to implement and gives reasonable results. The boundary Eddington factor method implements the Marshak boundary condition. While it yields good results at boundaries with incoming neutrons, it does not do so well at vacuum boundaries. The partial numerical flux method is an extension of the Marshak boundary condition, allowing us to specify extra angular information about the incoming neutron distribution. The neutron flux calculations with this method are generally the best out of the three boundary conditions presented here. Several simple steady-state and time-dependent problems illustrate the qualities, both good and bad, of the maximum entropy closure.

Assessment of a High Resolution Centered Scheme for the Solution of Hydrodynamical Semiconductor Equations

Hydrodynamical models are suitable to describe carrier transport in submicron semiconductor devices. These models have the form of nonlinear systems of hyperbolic conservation laws with source terms, coupled with Poisson's equation. In this article we examine the suitability of a high resolution centered numerical scheme for the solution of the hyperbolic part of these extended models, in one space dimension. Because of the lack of physically significant exact analytical solutions, the method is assessed against a benchmark for the system of compressible, unsteady Euler equations with source terms, which has an exact solution; the latter is shown to be nearly identical to the numerical one. The method is then used to solve the extended hydrodynamical model (EM) based on the maximum entropy closure recently introduced by Anile, Romano, and Russo, simulating a ballistic diode n+ -n-n+ , which models a metal oxide semiconductor field effect transistor (MOSFET) channel. Results are presented for the reduced- and full-equation EM formulation at steady state, for an initially discontinuous electron density at the junctions. Transient results show the evolution of highly nonlinear waves emanating from the neighborhood of the junctions.

Moment Equations with Maximum Entropy Closure for Carrier Transport in Semiconductor Devices: Validation in Bulk Silicon

Balance equations based on the moment method for the transport of electrons in silicon semiconductors are presented. The energy band is assumed to be described by the Kane dispersion relation. The closure relations have been obtained by employing the maximum entropy principle.The validity of the constitutive equations for fluxes and production terms of the balance equations has been checked with a comparison to detailed Monte Carlo simulations in the case of bulk silicon.

Stress distribution in sandpiles – A variational approach

Two variational closure models are proposed in order to describe stress distribution and arch formation in sandpiles located on a rigid base. The maximum strength and the maximum entropy closures are shown to yield the normal stress minimum below the pile apex. Stress distributions in sandpiles are determined numerically. The force fluctuations in a model regular granular assembly are studied by Monte Carlo simulation method. A dependence of the level of the fluctuations vs. the macroscopic stress state is determined.

Stationary neutrino radiation transport by maximum entropy closure

We obtain the angular distributions that maximize the entropy functional for Maxwell-Boltzmann (classical) Bose-Einstein, and Fermi-Dirac radiation. In the low and high occupancy limits, the maximum entropy closure is bounded by previously known variable Eddington factors that depend only on the flux. For intermediate occupancy, the maximum entropy closure depends on both the occupation density and the flux. The Fermi-Dirac maximum entropy variable Eddington factor shows a scale invariance, which leads to a simple, exact analytic closure for fermions. This two-dimensional variable Eddington factor gives results that agree well with exact (Monte Carlo) neutrino transport calculations out of a collapse residue during early phases of hydrostatic neutron star formation.

Maximum entropy distribution and closure for Bose-Einstein and Fermi-Dirac radiation transport

view Abstract Citations (46) References (11) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Maximum Entropy Distribution and Closure for Bose-Einstein and Fermi-Dirac Radiation Transport Cernohorsky, J. ; Bludman, S. A. Abstract We derive and study the variable Eddington factors following without approximation from the maximum entropy distribution. A unified formalism is developed for classical Maxwell-Boltzmann, Bose-Einstein, and Fermi-Dirac radiation. In the classical and high occupancy limits the maximum entropy closure is bounded by previously known variable Eddington factors that depend only on the flux: the Levermore-Pomraning closure for Bose radiation at high occupation density and Minerbo's flux-limiter for the Boltzmann limit of either quantum statistics. For immediate occupancy, the maximum entropy closure depends on both the occupation density and the flux. The Fermi-Dirac maximum entropy variable Eddington factor exhibits scale invariance, which leads to a simple analytic closure for fermion radiation. This Fermi-Dirac variable Eddington factor agrees well with Monte Carlo neutrino transport closures during early stages of neutron star formation. Publication: The Astrophysical Journal Pub Date: September 1994 DOI: 10.1086/174640 Bibcode: 1994ApJ...433..250C Keywords: Eddington Approximation; Fermi-Dirac Statistics; Maximum Entropy Method; Neutron Stars; Quantum Statistics; Radiation Transport; Star Formation; Stellar Radiation; Closure Law; Formalism; Maxwell-Boltzmann Density Function; Neutrinos; Transport Properties; Astrophysics; RADIATION MECHANISMS: NONTHERMAL; STARS: NEUTRON full text sources ADS |