Pub Date : 2012-12-01DOI: 10.1080/00411450.2012.747540
B. Guo, Yongqian Han
In this article, the nonlinear transfer equations with absorbing boundary condition, which describe the spatial transport of radiation in a material medium, are considered. We first establish the well-posedness of solutions for the radiative transfer equations based on the principle of contraction mapping and the comparison principle. Then we show that the radiative transfer equations have diffusion limits as the mean free path tends to zero if the specific intensity of radiation entering the system through the boundary of the domain is uniform with respect to the incoming direction. Our proof is based on asymptotic expansions. We show that the validity of these asymptotic expansions relies only on the smoothness of initial data and boundary functions, while two hypotheses, Fredholm alternative and centering condition, are removed.
{"title":"Diffusion Limit of a Small Mean Free Path of Radiative Transfer Equations with Absorbing Boundary Condition","authors":"B. Guo, Yongqian Han","doi":"10.1080/00411450.2012.747540","DOIUrl":"https://doi.org/10.1080/00411450.2012.747540","url":null,"abstract":"In this article, the nonlinear transfer equations with absorbing boundary condition, which describe the spatial transport of radiation in a material medium, are considered. We first establish the well-posedness of solutions for the radiative transfer equations based on the principle of contraction mapping and the comparison principle. Then we show that the radiative transfer equations have diffusion limits as the mean free path tends to zero if the specific intensity of radiation entering the system through the boundary of the domain is uniform with respect to the incoming direction. Our proof is based on asymptotic expansions. We show that the validity of these asymptotic expansions relies only on the smoothness of initial data and boundary functions, while two hypotheses, Fredholm alternative and centering condition, are removed.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"41 1","pages":"552 - 582"},"PeriodicalIF":0.0,"publicationDate":"2012-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2012.747540","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58909793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2012-10-01DOI: 10.1080/00411450.2012.675217
E. Sauter, F. S. de Azevedo, M. Thompson, M. Vilhena
The critical eigenvalues of the transport equation play an important role in the description of the dynamics of transport problems both in nuclear physics as well as in radiative transport theory. This article treats the problem of calculating numerically the critical spectrum of the transport equation with semireflecting boundary conditions. The eigenvalue problem is solved using spectral methods and numerical results are presented. The scattering kernel is considered to be one of three types, namely, isotropic, linearly anisotropic, or Rayleigh scattering, even although more general kernels could be considered.
{"title":"Eigenvalues of the Anisotropic Transport Equation in a Slab","authors":"E. Sauter, F. S. de Azevedo, M. Thompson, M. Vilhena","doi":"10.1080/00411450.2012.675217","DOIUrl":"https://doi.org/10.1080/00411450.2012.675217","url":null,"abstract":"The critical eigenvalues of the transport equation play an important role in the description of the dynamics of transport problems both in nuclear physics as well as in radiative transport theory. This article treats the problem of calculating numerically the critical spectrum of the transport equation with semireflecting boundary conditions. The eigenvalue problem is solved using spectral methods and numerical results are presented. The scattering kernel is considered to be one of three types, namely, isotropic, linearly anisotropic, or Rayleigh scattering, even although more general kernels could be considered.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"41 1","pages":"448 - 472"},"PeriodicalIF":0.0,"publicationDate":"2012-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2012.675217","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58909602","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2012-10-01DOI: 10.1080/00411450.2012.671223
J. Densmore
The method of moments is a well-known technique for determining exact expressions for spatial and angular moments of radiation distributions in infinite, homogeneous media. These moments can further be used to calculate other quantities of interest. We examine how the method of moments is altered when the underlying transport problem is spatially continuous but involves a grid and moments are computed on this grid instead of through integration over the entire domain. For the problem we consider, we employ both singular-eigenfunction and Fourier-transform approaches to show that when moments are evaluated in this manner (i) the flux-weighted average of x remains equal to the source-weighted average of x, but (ii) the flux-weighted average of (x−xa )2 is greater than the source-weighted average of (x−xa )2 by an additional error term, where x is the spatial variable and xa is an arbitrary point. We also demonstrate that the two resulting expressions for this error term are equivalent.
{"title":"Spatial Moments of Continuous Transport Problems Computed on Grids","authors":"J. Densmore","doi":"10.1080/00411450.2012.671223","DOIUrl":"https://doi.org/10.1080/00411450.2012.671223","url":null,"abstract":"The method of moments is a well-known technique for determining exact expressions for spatial and angular moments of radiation distributions in infinite, homogeneous media. These moments can further be used to calculate other quantities of interest. We examine how the method of moments is altered when the underlying transport problem is spatially continuous but involves a grid and moments are computed on this grid instead of through integration over the entire domain. For the problem we consider, we employ both singular-eigenfunction and Fourier-transform approaches to show that when moments are evaluated in this manner (i) the flux-weighted average of x remains equal to the source-weighted average of x, but (ii) the flux-weighted average of (x−xa )2 is greater than the source-weighted average of (x−xa )2 by an additional error term, where x is the spatial variable and xa is an arbitrary point. We also demonstrate that the two resulting expressions for this error term are equivalent.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"41 1","pages":"389 - 405"},"PeriodicalIF":0.0,"publicationDate":"2012-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2012.671223","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58909940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2012-10-01DOI: 10.1080/00411450.2012.671213
R. Pettersson
This article considers the time and space-dependent linear Boltzmann equation for elastic or inelastic (granular) collisions. First, in the angular cut-off case or with hard sphere collisions, mild L 1-solutions are constructed as limits of iterate functions. Then, in the case of hard potentials together with diffuse boundary conditions, global boundedness in time of higher velocity moments is proved, using our old collision velocity estimates together with a Jensen inequality.
{"title":"On Boundedness of Higher Velocity Moments for the Linear Boltzmann Equation with Diffuse Boundary Conditions","authors":"R. Pettersson","doi":"10.1080/00411450.2012.671213","DOIUrl":"https://doi.org/10.1080/00411450.2012.671213","url":null,"abstract":"This article considers the time and space-dependent linear Boltzmann equation for elastic or inelastic (granular) collisions. First, in the angular cut-off case or with hard sphere collisions, mild L 1-solutions are constructed as limits of iterate functions. Then, in the case of hard potentials together with diffuse boundary conditions, global boundedness in time of higher velocity moments is proved, using our old collision velocity estimates together with a Jensen inequality.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"41 1","pages":"345 - 355"},"PeriodicalIF":0.0,"publicationDate":"2012-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2012.671213","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58909708","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2012-10-01DOI: 10.1080/00411450.2012.671217
V. Glivici-Cotruţă, B. Merk
The time-dependent, one-dimensional diffusion equation is solved for a finite slab of two layers. An external source is supplied to one of the layers. The differential equations are subject to the reflecting boundary conditions at the two outer boundary surfaces. The flux and the current density are continuous across the interface between two media. The exact analytical solution is expressed in terms of a Green’s function. The solution is developed by the application of the Laplace transformation.
{"title":"An Analytical Solution of the Time-Dependent Diffusion Equation in a Composite Slab","authors":"V. Glivici-Cotruţă, B. Merk","doi":"10.1080/00411450.2012.671217","DOIUrl":"https://doi.org/10.1080/00411450.2012.671217","url":null,"abstract":"The time-dependent, one-dimensional diffusion equation is solved for a finite slab of two layers. An external source is supplied to one of the layers. The differential equations are subject to the reflecting boundary conditions at the two outer boundary surfaces. The flux and the current density are continuous across the interface between two media. The exact analytical solution is expressed in terms of a Green’s function. The solution is developed by the application of the Laplace transformation.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"41 1","pages":"356 - 367"},"PeriodicalIF":0.0,"publicationDate":"2012-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2012.671217","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58909789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2012-10-01DOI: 10.1080/00411450.2012.671207
Tobias Gebäck, M. Asadzadeh
In this article, we derive equations approximating the Boltzmann equation for charged particle transport under the continuous slowing down assumption. The objective is to obtain analytical expressions that approximate the solution to the Boltzmann equation. The analytical expressions found are based on the Fermi-Eyges solution, but include correction factors to account for energy loss and spread. Numerical tests are also performed to investigate the validity of the approximations.
{"title":"Analytical Solutions for the Pencil-Beam Equation with Energy Loss and Straggling","authors":"Tobias Gebäck, M. Asadzadeh","doi":"10.1080/00411450.2012.671207","DOIUrl":"https://doi.org/10.1080/00411450.2012.671207","url":null,"abstract":"In this article, we derive equations approximating the Boltzmann equation for charged particle transport under the continuous slowing down assumption. The objective is to obtain analytical expressions that approximate the solution to the Boltzmann equation. The analytical expressions found are based on the Fermi-Eyges solution, but include correction factors to account for energy loss and spread. Numerical tests are also performed to investigate the validity of the approximations.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"41 1","pages":"325 - 336"},"PeriodicalIF":0.0,"publicationDate":"2012-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2012.671207","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58910014","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2012-10-01DOI: 10.1080/00411450.2012.671225
P. Zweifel
This talk was presented at the ICTT-22 meeting held in Portland, Oregon on September 12–16, 2011.
本次演讲于2011年9月12日至16日在俄勒冈州波特兰举行的ICTT-22会议上发表。
{"title":"Kenneth Case and his Singular “Eigenfunctions”","authors":"P. Zweifel","doi":"10.1080/00411450.2012.671225","DOIUrl":"https://doi.org/10.1080/00411450.2012.671225","url":null,"abstract":"This talk was presented at the ICTT-22 meeting held in Portland, Oregon on September 12–16, 2011.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"41 1","pages":"406 - 417"},"PeriodicalIF":0.0,"publicationDate":"2012-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2012.671225","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58909965","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2012-10-01DOI: 10.1080/00411450.2012.671222
R. Sospedra‐Alfonso, B. Shizgal
We study the relaxation of Li+ ions dilutely dispersed in He at equilibrium. We employ the Henyey-Greenstein phase function to model the angular dependence of the differential scattering cross section for Li+-He collisions. We solve the spatially homogeneous linear Boltzmann equation for this model cross section with the collision operator explicitly given in terms of the scattering kernel. With the Quadrature Discretization Method based on the speed polynomials, the Boltzmann equation is reduced to a set of ordinary differential equations. This numerical method provides a rapid convergence for the Li+ distribution function. We study the relaxation of the shape of the Li+ distribution function in terms of the Kullback-Leibler information relative to the steady state and local in time Maxwellians. A comparison of the relaxation times of these two functionals show that there is no formation of a local Maxwellian during the relaxation process. This was verified for several values of the g-parameter in the Henyey-Greenstein phase function model and the initial average energies investigated.
{"title":"Henyey-Greenstein Model in the Shape Relaxation of Dilute Gas Mixtures","authors":"R. Sospedra‐Alfonso, B. Shizgal","doi":"10.1080/00411450.2012.671222","DOIUrl":"https://doi.org/10.1080/00411450.2012.671222","url":null,"abstract":"We study the relaxation of Li+ ions dilutely dispersed in He at equilibrium. We employ the Henyey-Greenstein phase function to model the angular dependence of the differential scattering cross section for Li+-He collisions. We solve the spatially homogeneous linear Boltzmann equation for this model cross section with the collision operator explicitly given in terms of the scattering kernel. With the Quadrature Discretization Method based on the speed polynomials, the Boltzmann equation is reduced to a set of ordinary differential equations. This numerical method provides a rapid convergence for the Li+ distribution function. We study the relaxation of the shape of the Li+ distribution function in terms of the Kullback-Leibler information relative to the steady state and local in time Maxwellians. A comparison of the relaxation times of these two functionals show that there is no formation of a local Maxwellian during the relaxation process. This was verified for several values of the g-parameter in the Henyey-Greenstein phase function model and the initial average energies investigated.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"41 1","pages":"368 - 388"},"PeriodicalIF":0.0,"publicationDate":"2012-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2012.671222","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58909928","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2012-10-01DOI: 10.1080/00411450.2012.682618
L. Barletti, G. Frosali
We derive semiclassical diffusive equations for the local electron densities in a semiconductor characterized by a two-band k·p Hamiltonian, under the action of a strong external field. By using a spinorial formalism, we consider the quantum kinetic (Wigner) system endowed with a Bhatnagar-Gross-Krook (BGK)-like interaction term. Diffusive equations are derived by the Chapman-Enskog method. The closure of such equations is obtained by using the quantum version of the minimum entropy principle. In practice, it is unfeasible to put in an explicit form the diffusive equations in the general case, even in the semiclassical limit. Then we investigate the case in which band parameters have little influence on the dynamics at the macroscopic scale.
{"title":"Diffusive Limits for a Quantum Transport Model with a Strong Field","authors":"L. Barletti, G. Frosali","doi":"10.1080/00411450.2012.682618","DOIUrl":"https://doi.org/10.1080/00411450.2012.682618","url":null,"abstract":"We derive semiclassical diffusive equations for the local electron densities in a semiconductor characterized by a two-band k·p Hamiltonian, under the action of a strong external field. By using a spinorial formalism, we consider the quantum kinetic (Wigner) system endowed with a Bhatnagar-Gross-Krook (BGK)-like interaction term. Diffusive equations are derived by the Chapman-Enskog method. The closure of such equations is obtained by using the quantum version of the minimum entropy principle. In practice, it is unfeasible to put in an explicit form the diffusive equations in the general case, even in the semiclassical limit. Then we investigate the case in which band parameters have little influence on the dynamics at the macroscopic scale.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"41 1","pages":"473 - 493"},"PeriodicalIF":0.0,"publicationDate":"2012-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2012.682618","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58909657","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2012-10-01DOI: 10.1080/00411450.2012.672360
Richard Sanchez
We consider the artificial conversion of the discrete–ordinates (SN) equations into a system of spherical harmonic (PN) equations. This is done by adding to the SN equations an artificial source that has two components. The first component transforms the SN scattering term into PN-like scattering, while the second modifies the SN streaming operator into a lower-order PN streaming operator. Denoting by and the spaces of solutions of the SN and PN equations, respectively, we define SN-PN equivalence via a constructive Proposition based on two linear morphisms, and , such that if ψ is the solution of the SN equations with source S+π*(S), then π K ψ is solution of the PN equations with source π K S. We proceed then to prove this Proposition by constructing the two components of the artificial source. We also prove that when the morphism π* is not unique, and propose a general form for the second component of the artificial source, which is shown to comprise all artificial sources previously proposed in the literature.
我们考虑将离散坐标方程(SN)人工转换为球谐方程(PN)系统。这是通过在SN方程中加入一个有两个组成部分的人工源来实现的。第一个组件将SN散射项转换为类PN散射，第二个组件将SN流算子修改为低阶PN流算子。我们分别用和表示SN和PN方程的解的空间，通过一个基于两个线性态射的构造命题定义了SN-PN等价，并且，使得如果ψ是源为S+π*(S)的SN方程的解，则π K ψ是源为π K S的PN方程的解，然后通过构造人工源的两个分量来证明这个命题。我们还证明了当态射π*不唯一时，并给出了人工源第二分量的一般形式，它包含了以前文献中提出的所有人工源。
{"title":"On SN-PN Equivalence","authors":"Richard Sanchez","doi":"10.1080/00411450.2012.672360","DOIUrl":"https://doi.org/10.1080/00411450.2012.672360","url":null,"abstract":"We consider the artificial conversion of the discrete–ordinates (SN) equations into a system of spherical harmonic (PN) equations. This is done by adding to the SN equations an artificial source that has two components. The first component transforms the SN scattering term into PN-like scattering, while the second modifies the SN streaming operator into a lower-order PN streaming operator. Denoting by and the spaces of solutions of the SN and PN equations, respectively, we define SN-PN equivalence via a constructive Proposition based on two linear morphisms, and , such that if ψ is the solution of the SN equations with source S+π*(S), then π K ψ is solution of the PN equations with source π K S. We proceed then to prove this Proposition by constructing the two components of the artificial source. We also prove that when the morphism π* is not unique, and propose a general form for the second component of the artificial source, which is shown to comprise all artificial sources previously proposed in the literature.","PeriodicalId":49420,"journal":{"name":"Transport Theory and Statistical Physics","volume":"41 1","pages":"418 - 447"},"PeriodicalIF":0.0,"publicationDate":"2012-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1080/00411450.2012.672360","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"58910040","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：