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.     

Anti-diffusive radiation flow in the cooling layer of a radiating shock

This paper shows that for systems with optically thin, hot layers, such as those that occur in radiating shocks, radiation will flow uphill: radiation will flow from low to high radiation energy density. These are systems in which the angular distribution of the radiation intensity changes rapidly in space, and in which the radiation in some region has a pancaked structure, whose effect on the mean intensity will be much larger than the effect on the scalar radiation pressure. The salient feature of the solution to the radiative transfer equation in these circumstances is that the gradient of the radiation energy density is in the same direction as the radiation flux, i.e. radiation energy is flowing uphill. Such an anti-diffusive flow of energy cannot be captured by a model where the spatial variation of the Eddington factor is not accounted for, as in flux-limited diffusion models or the P₁ equations. The qualitative difference between the two models leads to a monotonic mean intensity for the diffusion model whereas the transport mean intensity has a global maximum in the hot layer. Mathematical analysis shows that the discrepancy between the diffusion model and the transport solution is due to an approximation of exponential integrals using a simple exponential.     

Two alternative methods for solving ion transport equation with anisotropic approximation

By means of two alternative methods namely, the maximum entropy and Chapman-Enskog, flux limited approach the ion transport equation for slowing-down problem of low-energy light ions in solid has been solved explicitly. Maximum entropy technique yields approximate solutions in the form of locally Maxwellian distribution function, based on moments expansion truncated upon entropy maximization.The behavior of the approximate maximum entropy and the flux limited solutions have the same tendency. Knowing the distribution function obtained by flux limited approach, allows us to calculate directly the path length distributions of backscattered ions, and compared with that found by other theories such as Laplace-transform and double Legendre polynomial approximation. One can see that the flux limited approach is better than the previous method namely, (DPN) Laplace-transform.The results reported in this article provide further evidence of the usefulness of both maximum entropy and flux limited for obtaining the solution of ion transport equation in compact form.     

Spatial and angular distribution of the heavy ion charge under channeling in crystals

A kinetic theory of passage of multiply charged heavy ions through crystals is developed that allows for diffusion in the transverse momentum space and ion-crystal charge exchange. The theory provides an adequate explanation for the observed angular distributions of heavy ions passing through oriented crystals, makes it possible to calculate the partial angular distributions of different charge states, and treats the discovered effects of “cooling” and “heating” of channeled ion beams in physical terms. The angular and spatial distribution of channeled ions with different energies is calculated. Whether a channeled beam of multiply charged heavy ions will be cooled or heated is related to the dependence of the electron capture and loss probabilities on the impact parameter when the ions interact with atomic chains. This interaction governs the run of the angular and spatial distribution of the channeled ion charge.     

On the solution of time-dependent transport equation with time-varying cross sections

The time-dependent neutron transport equation in an infinite medium with time-varying cross sections has been solved by means of two techniques namely, flux-limited approach and maximum entropy method. The behaviour of the distribution function are shown graphically. Knowing the distribution function allows us to calculate directly some physical parametres of special interest such as the reflection function. The results reported in this article provide further evidence of the usefulness of both maximum entropy and flux limited methods for obtaining time-dependent solution of neutron transport in compact form.     

Molecular Dynamics Simulations of Liquid Phosphorus at High Temperature and Pressure

By performing ab initio molecular dynamics simulations, we have investigated the microstructure, dynamical and electronic properties of liquid phosphorus （P） under high temperature and pressure. In our simulations, the calculated coordination number （CN） changes discontinuously with density, and seems to increase rapidly after liquid P is compressed to 2.5 g/cm3. Under compression, liquid P shows the first-order liquid-liquid phase transition from the molecular liquid composed of the tetrahedral P4 molecules to complex polymeric form with three-dimensional network structure, accompanied by the nonmetal to metal transition of the electronic structure. The order parameters Q6 and Q4 are sensitive to the microstructural change of liquid P. By calculating diffusion coefficients, we show the dynamical anomaly of liquid P by compression. At lower temperatures, a maximum exists at the diffusion coefficients as a function of density; at higher temperatures, the anomalous behavior is weakened. The excess entropy shows the same phenomena as the diffusion coefficients. By analysis of the angle distribution functions and angular limited triplet correlation functions, we can clearly find that the Peierls distortion in polymeric form of liquid P is reduced by further compression.     

Minimum entropy production closure of the photo-hydrodynamic equations for radiative heat transfer

In the framework of a two-moment photo-hydrodynamic modelling of radiation transport, we introduce a concept for the determination of effective radiation transport coefficients based on the minimization of the local entropy production rate of radiation and (generally nongrey) matter. The method provides the nonequilibrium photon distribution from which the effective (variable) absorption coefficients and the variable Eddington factor (VEF) can be calculated. For a single band model, the photon distribution depends explicitly on the frequency dependence of the absorption coefficient. Without introducing artificial fit parameters, multi-group or multi-band concepts, our approach reproduces the exact results in both limits of optically thick (Rosseland mean) and optically thin (Planck mean) media, in contrast to the maximum entropy method. Also the results for general nonequilibrium radiation between the limits of diffusive and ballistic photons are reasonable. We conjecture that the reason for the success of our approach lies in the linearity of the underlying Boltzmann equation of the photon gas. The method is illustrated and discussed for grey matter and for a simple example of nongrey matter with a two-band absorption spectrum. The method is also briefly compared with the maximum entropy concept.     

Radiation transport and internal reflection in a two region, turbid sphere

We construct an integral equation for the flux intensity in a scattering and absorbing two-region turbid spherical medium using the integro-differential form of the radiative transfer equation. The sphere is uniformly irradiated by an external source of arbitrary angular distribution and contains a distributed volume source. Anisotropic scattering is accounted for by the transport approximation. The Fresnel boundary conditions, which incorporate reflection and refraction, are used at the outer surface and at the interface between the two regions. In this respect, some new interfacial boundary conditions are introduced. For the special case of a non-scattering medium, we obtain exact solutions for specular reflection. Some numerical examples are given which show qualitative agreement with some recent work of other authors. Of particular interest are the emergent angular distribution and the albedo of the surface as a function of the refractive index and the radii of the two regions. We also draw attention to the fact that the boundary conditions at the interface differ according to the relative values of the refractive indices in the two regions. The interfacial boundary conditions for use in diffusion theory are derived and compared with those of Aronson [Boundary conditions for diffusion of light. J opt Soc Am 1995;12:2532]. In appendix B, we show how diffusion theory may be used to include scattering into the problem in a simple way.     

Saturation of thermoacoustic mixture separation

The theory for thermoacoustic mixture separation is extended to include the effects of a nonzero concentration gradient. New data are presented, which are in excellent agreement with this theory. The maximum concentration gradient which may be achieved in a binary mixture of gases through this separation process is intrinsically limited by the fractional pressure amplitude, by the tidal displacement, and by the size of the thermal diffusion ratio. Ordinary diffusion further detracts from the attainable final concentration gradient and can become the dominant remixing process as the cross section of the duct is increased. Rayleigh streaming also works against thermoacoustic separation, and an estimate of the molar flux from streaming is given.     

Entropy, entropy flux and entropy rate of granular materials

The aim of this work is to analyze the entropy, entropy flux and entropy rate of granular materials within the frameworks of the Boltzmann equation and continuum thermodynamics. It is shown that the entropy inequality for a granular gas that follows from the Boltzmann equation differs from the one of a simple fluid due to the presence of a term which can be identified as the entropy density rate. From the knowledge of a non-equilibrium distribution function-valid for processes closed to equilibrium-it is obtained that the entropy density rate is proportional to the internal energy density rate divided by the temperature, while the entropy flux is equal to the heat flux vector divided by the temperature. A thermodynamic theory of a granular material is also developed whose objective is the determination of the basic fields of mass density, momentum density and internal energy density. The constitutive laws are restricted by the principle of material frame indifference and by the entropy principle. Through the exploitation of the entropy principle with Lagrange multipliers, it is shown that the results obtained from the kinetic theory for granular gases concerning the entropy density rate and entropy flux are valid in general for processes close to equilibrium of granular materials, where linearized constitutive equations hold.     

高空核爆炸火球的二维辐射流体力学计算

本文介绍了高空核爆炸火球的二维辐射流体力学计算的主要计算方法和部分计算结果,用柱对称最大熵变Eddington因子P-1近似方程描述了高空核爆炸的辐射输运过程,讨论了P-1近似的边界条件,对能量方程和P-1近似方程的耦合,给出了一个新的差分格式,提高了耦合计算的精度,显著地减少了计算量,同时有效地克服了辐射输运计算中出现的非物理振荡现象。     

The distribution function for a subsystem experiencing temperature fluctuations

A nonlinear generalization of the Landau-Lifshitz theory of hydrodynamic fluctuations for the simplest case in which only energy flux and temperature fluctuations are observed is used to derive the distribution function for a subsystem with a fluctuating temperature, which coincides with the Levy distribution taken to be one of the main results of the so-called Tsallis’s nonextensive statistics. It is demonstrated that the same distribution function is obtained from the principle of maximum of information entropy if the latter is provided by Renyi’s entropy, which is an extensive quantity. The obtained distribution function is to be used instead of the Gibbs distribution in constructing the thermodynamics of systems with significant temperature fluctuations.     

Dust Self‐Organized Structures II. Solutions of Master Equations for Small Diffusion

Master equations for spherical dust structures are solved numerically using the asymptotic solutions at the center of the structures for the case of absence of external ionization and small diffusions. The structures are determined by a single parameter, the external plasma flux at the surface of the structure. The equilibrium states that are possible in a limited range of this parameter are investigated numerically. It is demonstrated that in the range of existence of equilibria the structures are changing their shapes and type of distributions inside the structures. For large external fluxes the ion and dust distributions can have peaks inside the structures while for low external fluxes the dust distribution has a single maximum at the structure center. The lower is the external flux supporting the structure the larger is its size. An increase of the external flux decreases the accumulation of dust and ions at the center. The total number of dust confined by the structure is larger for larger size structures. Estimates of dust crystallization inside structures are given. The role of diffusion is calculated by perturbations and is shown to be small in all structure regions except the structure edges. In the perturbation theory we use the exact expressions of the diffusion coefficients calculated previously numerically. The regions with dust density peaks inside the structures have been calculated with two order of magnitude larger precision that allows to resolve the structure parameter dependencies inside the peaks. It is shown that although in peaks the gradients of all parameters are increased the diffusion flux is still small and that the continuity and hydrodynamic approach are applicable within an accuracy about several %‐s (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Time-dependent radiative transfer using Chapman-Enskog maximum entropy method

The time-dependent problems of radiative transfer involve a coupling between radiation and material energy fields and are nonlinear because of proposed temperature dependence of the medium characteristics in semi-infinite medium with Rayleigh anisotropic scattering. By means of the limited flux, Chapman-Enskog and maximum entropy technique the time-dependent radiative transfer equation has been solved explicitly. The maximum entropy method is used to solve the resulting differential equation for radiative energy density. The calculations are carried out for temperature (normalized dimensionless) Θ(x,τ), radiative energy density and net flux with Rayleigh and anisotropic scattering for different space at different times.     

Relating Eddington factors to flux limiters

Variable Eddington factors and flux-limiters have been introduced in the P-1 and diffusion equations, respectively, to handle situations when the underlying intensity is too anisotropic for the unmodified theories to remain valid. We present a derivation of a relation between the two for which a new approach to the diffusion approximation is used. Algebraic expressions for Eddington factors satisfying the moment conditions are not satisfactory for closing the P-1 equations but, by using the derived relation, yield acceptable flux-limited diffusion theories.     

Interior radiances in optically deep absorbing media—III. Scattering from haze L

The interior radiances are calculated within an optically deep absorbing medium scattering according to the Haze L phase function. The dependence on the solar zenith angle, the single scattering albedo, and the optical depth within the medium is calculated by the matrix operator method. The development of the asymptotic angular distribution of the radiance in the diffusion region is illustrated through a number of examples; it depends only on the single scattering albedo and on the phase function for single scattering. The exact values of the radiance in the diffusion region are compared with values calculated from the approximate equations proposed by Van de Hulst. The variation of the radiance near the lower boundary of an optically thick medium is illustrated with examples. The attenuation length is calculated for various single scattering albedos and compared with the corresponding values for Rayleigh scattering. The ratio of the upward to the downward flux is found to be remarkably constant within the medium. The heating rate is calculated and found to have a maximum value at an optical depth of two within a Haze L layer when the sun is at the zenith. The location of this maximum moves toward the top of the haze layer as the solar zenith angle increases and also as the single scattering albedo decreases. When the single scattering albedo is less than 0·8, the downward flux is so small within the diffusion region that experimental measurements are probably not possible.     

Maximum entropy principle for rarefied polyatomic gases

The aim of this paper is to show that the procedure of maximum entropy principle for the closure of the moments equations for rarefied monatomic gases can be extended also to polyatomic gases. The main difference with respect to the usual procedure is the existence of two hierarchies of macroscopic equations for moments of suitable distribution function, in which the internal energy of a molecule is taken into account. The field equations for 14 moments of the distribution function, which include dynamic pressure, are derived. The entropy and the entropy flux are shown to be a generalization of the ones for classical Grad’s distribution. The results are in perfect agreement with the recent macroscopic approach of extended thermodynamics for real gases.     

高效偶阶离散纵标法研究及初步验证

采用TAKEDA基准题验证高效偶阶离散纵标法(HEPSN)的临界计算功能,数值结果表明:有效增殖因数和各区域平均中子通量密度值与基准值吻合良好,堆芯区域的平均中子通量密度误差在0.7%以内,其他区域较扩散理论误差显著减小。相较于传统离散纵标法,HEPSN计算效率更高,需要的存储更小;相较于扩散理论,HEPSN的计算精度更高。因而,高效偶阶离散纵标法在大规模输运计算中具有很好的应用前景。     

Maximum entropy image reconstruction from projections

The maximum entropy method is applied to image reconstruction from projections, of which angular view is restricted. The relaxation parameters are introduced to the maximum entropy reconstruction and after iteration the median filtering is implemented. These procedures improve the quality of the reconstructed image from noisy projections.