Linear alternative to the Boltzmann equation without linearization: II. Electron-electron scattering

A new form of projector suggested in the previous paper of the series is applied to Time-Convolutionless Generalized Master Equations (TCL-GME) and the problem of spinless interacting electron gas with finite concentration. As a result, a closedlinear kinetic equation for the single-electron density matrix (Wigner distribution function) is obtained. In the second order in the short range interaction and after a short period of forgetting details of initial conditions, a new linear equation with time-independent coefficients results providing a linear alternative to the famous Boltzmann-Uehling-Uhlenbeck equation for quantum gases.The author is really grateful to the International Centre for Theoretical Physics, Trieste for invitation to the Research Workshop on Condensed Matter Physics 1993 where a greater part of this work was completed.     

Two linearization procedures for the Boltzmann equation in ak=0 Robertson-Walker space-time

It is the aim of this paper to describe two different linearization procedures for the Boltzmann equation in ak=0 Robertson-Walker space-time. These procedures are discussed with a view to obtaining an asymptotic form of the Boltzmann equation for the late stages of cosmic expansion where the behavior appropriate to a nonrelativistic gas is encountered. Using the asymptotic kinetic equations, a necessary and sufficient condition is formulated under which every small perturbation of the equilibrium distribution function, either classical or relativistic, decays to zero as time goes on. The same condition can be extracted from each of two linearization procedures, and in this sense a comparison is made of these approaches which reveals mutual agreement. Also, applying an asymptotic theory of the Einstein-Boltzmann system, we show that the final state of a gas is dust (i.e., a fluid with zero temperature and pressure). Comparison with the predictions of the Eckart fluid model is briefly presented.     

Existence of the scattering matrix for the linearized Boltzmann equation

Following Hejtmanek, we consider neutrons in infinite space obeying a linearized Boltzmann equation describing their interaction with matter in some compact setD. We prove existence of theS-matrix and subcriticality of the dynamics in the (weak-coupling) case where the mean free path is larger than the diameter ofD uniform in the velocity. We prove existence of theS-matrix also for the case whereD is convex and filled with uniformly absorbent material. In an appendix, we present an explicit example where the dynamics is not invertible onL ₊ ¹ , the cone of positive elements inL ¹.A. Sloan fellow; research partially supported by the U.S. NSF under Grant GP 39048     

并行时域电场积分方程方法在动态海面二维电磁散射中的应用

采用并行时域电场积分方程方法对动态海面的二维瞬态散射特性进行研究。为了保证该方法的后期稳定性,时间基函数和空间基函数采用二阶B样条基函数和三角基函数,矩阵元素采用时间维度精确解析、空间奇异部分精确解析进行计算;为了减少对无限海面进行截断带来的边缘效应,入射波采用锥形调制高斯脉冲;结合信息传递接口（MPI）技术和稀疏矩阵压缩存储技术,对不同时刻的海面进行瞬态散射分析。大量的数值算例证明了该方法在计算动态海面的二维瞬态散射问题时的正确性,还可以保证后期的稳定性,提高计算效率,减少对计算机内存需求。     

The homogeneous Boltzmann hierarchy and statistical solutions to the homogeneous Boltzmann equation

An existence and uniqueness result for the homogeneous Boltzmann hierarchy is proven, by exploiting the statistical solutions to the homogeneous Boltzmann equation.     

Linear magnetoresistance and magnetic field dependent scattering probability

Within the framework of the Boltzmann equation it is possible to find a linear behaviour for the Magnetoresistance. Instead of using a purely random impurity scattering model, one makes this model slightly magnetic field dependent. Keeping only the linear term of the power expansion in the magnetic field, one finds a linear dependence of the Magnetoresistance.     

The bobylev approach to the nonlinear Boltzmann equation

The Bobylev approach to the nonlinear Boltzmann equation is reviewed. The linearized problem is discussed and it is shown that eigenfunctions decaying like a negative power of the velocity are possible with Maxwell molecules only. The relaxation to equilibrium according to the nonlinear equation is discussed and the Krook-Wu conjecture on the status of the BKW mode is shown to be false in general. The buildup of the high-energy tails is considered and a phenomenon observed by Tjon is given a simple explanation. Finally, the method is illustrated with numerical calculations performed for two sets of initial conditions.     

An alternative to the Teukolsky equation

General Relativity and Gravitation - We disclose the effects of Lifshitz dynamical exponent z on the properties of holographic paramagnetic-ferromagnetic phase transition in the background of...     

The reduced basis method for the electric field integral equation

We introduce the reduced basis method (RBM) as an efficient tool for parametrized scattering problems in computational electromagnetics for problems where field solutions are computed using a standard Boundary Element Method (BEM) for the parametrized electric field integral equation (EFIE). This combination enables an algorithmic cooperation which results in a two step procedure. The first step consists of a computationally intense assembling of the reduced basis, that needs to be effected only once. In the second step, we compute output functionals of the solution, such as the Radar Cross Section (RCS), independently of the dimension of the discretization space, for many different parameter values in a many-query context at very little cost. Parameters include the wavenumber, the angle of the incident plane wave and its polarization.     

Haldane exclusion statistics and the Boltzmann equation

We generalize the collision term in the one-dimensional Boltzmann-Nordheim transport equation for quasiparticles that obey the Haldane exclusion statistics. For the equilibrium situation, this leads to the golden rule factor for quantum transitions. As an application of this, we calculate the density response function of a one-dimensional electron gas in a periodic potential, assuming that the particle-hole excitations are quasiparticles obeying the new statistics. We also calculate the relaxation time of a nuclear spin in a metal using the modified golden rule.     

Bistatic electromagnetic scattering from a three-dimensional perfect electric conducting object above a Gaussian rough surface based on the Kirchhoff–Helmholtz and electric field integral equation

A hybrid integral equation is developed to solve the problem of electromagnetic (EM) scattering from a three-dimensional (3D) perfect electric conducting (PEC) object above a two-dimensional (2D) PEC or dielectric Gaussian rough surface. Firstly, the Kirchhoff–Helmholtz (KH) equation is adopted to describe the wave reflection on the rough surface; only one integral operation on the rough surface is needed, and the scattering from the object can be described by solving the electric field integral equation (EFIE) on the surface of the object. Moreover, according to scattering theory, the KH equation and the EFIE are coupled together (KH-EFIE) to describe wave propagation between the object and the rough surface. Then method of moments (MoM) is adopted to solve the KH-EFIE, and the current is obtained to calculate the scattering field. Finally, compared with other methods, the accuracy of the proposed approach is validated, and its efficiency is proved to be much higher than numerical solutions. Furthermore, by calculating the statistic composite radar cross-section (RCS) of the object/surface and the difference radar cross-section (DRCS) of the object, the influence of the rough surface root mean square (rms) height, the correlation length, the medium permittivity, the shape of the object, and the altitude of the object on the scattering characteristic is investigated.     

Description of the approach to equilibrium in the Boltzmann equation

An integral transform of the Boltzmann equation with a clear physical interpretation is introduced. It is applied to different interaction models and initial conditions, obtaining relevant information about the way the equilibrium is reached. This method leads quite naturally to the introduction of an N-pole approximant of the distribution function which seems to be a rather useful technique not only in view of its simplicity but also because of its capability to keep track of the temporal evolution features of the chosen interaction model.     

On the fluid-dynamical approximation to the Boltzmann equation at the level of the Navier-Stokes equation

The compressible and heat-conductive Navier-Stokes equation obtained as the second approximation of the formal Chapman-Enskog expansion is investigated on its relations to the original nonlinear Boltzmann equation and also to the incompressible Navier-Stokes equation. The solutions of the Boltzmann equation and the incompressible Navier-Stokes equation for small initial data are proved to be asymptotically equivalent (mod decay ratet ^–5/4) ast+ to that of the compressible Navier-Stokes equation for the corresponding initial data.     

An experimental test of the Boltzmann equation: Argon

Recent theoretical work has shown that the kinetic theory of dilute gases, as expressed in the Boltzmann equation and the Chapman-Enskog method of solution, is not rigorously based in the sense that the Boltzmann equation has not been shown rigorously to follow from the laws of classical or quantum mechanics. The widely-held view that the theory is justified by agreement with experiment is not firmly based, since no stringent test has ever been made. This paper describes such a test, in which an independently and precisely determined potential function is used to calculate transport properties for argon, which are found to be in excellent agreement with the best modern measurements.     

Nonlocal corrections to the Boltzmann equation for dense Fermi systems

A kinetic equation which combines the quasiparticle drift of Landau's equation with a dissipation governed by a nonlocal and noninstant scattering integral in the spirit of Snider's equation for gases is derived. Consequent balance equations for the density, momentum and energy include quasiparticle contributions and the second-order quantum virial corrections. The medium effects on binary collisions are shown to mediate the latent heat, i.e. an energy conversion between correlation and thermal energy. An implementation to heavy ion collisions is discussed.