Elastic scattering of acoustic phonons on point mass defects

The problem of eigenvalues of the collision operator for a gas of acoustic phonons scattered by point mass defects of small concentration embedded in transversely isotropic media is considered. For this purpose the properties of solution of the Boltzmann-Peierls kinetic equation for spatially homogeneous states are studied. An analytic formula for the Laplace transform of the distribution function is obtained. The singularities of this Laplace transform and the initial distribution function determine the dependence of this distribution function on time. For several hexagonal materials characteristics of the singularity set are calculated. Usually the singularity set consists of a continuous part and four discrete points. However, there exist elastic hexagonal materials (⁴He, Cd, Ta, Zn) for which some of discrete points are absent. For some materials (e.g. Zr) the continuous part is very narrow.     

Exact kinetic transport equation solutions in the particle propagation theory in the scattering medium

Charged particle kinetics in an inhomogeneous medium (stochastic magnetic field) is investigated. Exact analytic expressions for Green function of kinetic equation in relaxation-time approximation are derived in one and three dimensions with arbitrary particle absorption. We separately consider the case of isotropic particle injection as well as the case of unidirectional instantaneous particle injection. The new way of solution makes it possible to get off any Cauchy-Principal Value integrals in some solutions which arise in the inverse Fourier-Laplace transform. Weak scattering regime and diffusion approximation is considered, and particle density is derived in three dimensions and arbitrary particle source.     

The Einstein-Smoluchowski promeasure versus the Boltzmann(-Peierls) equation

Beginning from the specific model of the Boltzmann-Peierls equation for the distribution function f(k, t), the time-dependent theory of fluctuations is developed from Einstein's inversion of Boltzmann's relation. The distribution function f(k, t) that satisfies the Boltzmann-Peierls equation is to be viewed as a random variable characterizing the macro-state of a gas of quasiparticles (phonons, magnons, rotons, etc.). The crucial assumption entering into the present approach is simply that the randomness in the statistical distribution of the initial values of f(k, t) can be characterized by the Einstein-Smoluchowski promeasure μ_ε. It is convenient to think of μ_ε as being the infinite-dimensional analog of Einstein's distribution law in the Gaussian approximation. The equilibrium promeasure μ_ε is used to study the correlations in time of fluctuations in the moments of f(k, t). Some problems associated with a kinetic version of the Green-Kubo approach to transport processes are carefully studied. The exact formula for the thermal conductivity coefficient _T, which is derived in this paper, emerges as a portion of the general formalism.     

Scattering of long-wavelength acoustic phonons by isotopic impurities

We consider the problem of the relaxation of an arbitrary initial distribution function of a gas of long-wave acoustic phonons scattered by isotopic impurities embedded in a crystalline medium with cubic symmetry. The spectral decomposition of the collision integral of the suitable Boltzmann-Peierls equation is obtained. The spectrum of the collision operator is purely discrete and in addition to the eigenvalue 0 consists of three other eigenvalues. Explicit analytic expressions for these eigenvalues are obtained. Within the Chapman-Enskog approximation we derive the diffusion equation for the density of phonons and obtain the explicit expression for the diffusion coefficient. The dependency of the eigenvalues of the collision operator and the diffusion coefficient on the elastic constants of the medium is studied.     

Anomalous transport in fluid field with random waiting time depending on the preceding jump length

Anomalous (or non-Fickian) transport behaviors of particles have been widely observed in complex porous media. To capture the energy-dependent characteristics of non-Fickian transport of a particle in flow fields, in the present paper a generalized continuous time random walk model whose waiting time probability distribution depends on the preceding jump length is introduced, and the corresponding master equation in Fourier-Laplace space for the distribution of particles is derived. As examples, two generalized advection-dispersion equations for Gaussian distribution and lévy flight with the probability density function of waiting time being quadratic dependent on the preceding jump length are obtained by applying the derived master equation.     

Elastic scattering of acoustic phonons on point mass defects

The relaxation of homogeneous states of long-wave acoustic phonon gas scattered by point mass defects in transversely—isotropic media is studied. The spectrum of the suitable collision operator of the Boltzmann-Peierls equation is investigated. It consists of a continuous part and several discrete eigenvalues. Both continuous and discrete part of the spectrum depend on the values of components of the elastic constant tensor. For some values of elastic constants the continuous part splits up into two separate intervals and some of the discrete eigenvalues appear in the gap. The number of discrete eigenvalues and their arrangement are also affected by elastic properties of medium.     

Anomalous transport in fluid fleld with random waiting time depending on the preceding jump length

Anomalous(or non-Fickian) transport behaviors of particles have been widely observed in complex porous media.To capture the energy-dependent characteristics of non-Fickian transport of a particle in flow fields,in the present paper a generalized continuous time random walk model whose waiting time probability distribution depends on the preceding jump length is introduced,and the corresponding master equation in Fourier-Laplace space for the distribution of particles is derived.As examples,two generalized advection-dispersion equations for Gaussian distribution and levy flight with the probability density function of waiting time being quadratic dependent on the preceding jump length are obtained by applying the derived master equation.     

The fundamental solution to equations of linear magnetic hydrodynamics in a moving medium

The fundamental solution to a system of linear differential equations of magnetic hydrodynamics in a moving medium is obtained. Using the Fourier-Laplace transform, the Green tensor function is calculated as a sum of dyadics. In this way, the integral equations of magnetic hydrodynamics can easily be derived. Particular forms of the fundamental solution that are important in applications are analyzed.     

Axisymmetric wave propagation of thermoelastic transversely isotropic half-space under buried loading using potential functions

By virtue of a new scalar potential function and Hankel integral transforms, the wave propagation analysis of a thermoelastic transversely isotropic half-space is presented under buried loading and heat flux. The governing equations of the problem are the differential equations of motion and the energy equation of the coupled thermoelasticity theory. Using a scalar potential function, these coupled equations have been uncoupled and a six-order partial differential equation governing the potential function is received. The displacements, temperature, and stress components are obtained in terms of this potential function in cylindrical coordinate system. Applying the Hankel integral transform to suppress the radial variable, the governing equation for potential function is reduced to a six-order ordinary differential equation with respect to z. Solving that equation, the potential function and therefore displacements, temperature, and stresses are derived in the Hankel transformed domain for two regions. Using inversion of Hankel transform, these functions can be obtained in the real domain. The integrals of inversion Hankel transform are calculated numerically via Mathematica software. Our numerical results for displacement and temperature are calculated for surface excitations and compared with the results reported in the literature and a very good agreement is achieved.     

On the Interaction of an Alternating Electrical Field with an Inhomogeneous Magnetized Plasma

It is found out the form of the vectorial and isotropic cartesian component for the distribution function for electrons in an inhomogeneous plasma interacting with an alternative electric field, without being done any previous supposition on the temporal dependence of these quantities. For this purpose it is applied a method of succesive approximations for solving an integro-differential equation for the isotropic component of the distribution function. The scalar and vector components of the plasma spherical harmonic expansion are calculated including transient and secular terms. Some particular quantities as time-independent part of vectorial component of distribution function and of nonlinear conductivity were also obtained. Also corrected expressions for the transport coefficients in a electrical field, are obtained.     

Adomian decomposition method for solving the telegraph equation in charged particle transport

In this paper, the analysis for the telegraph equation in case of isotropic small angle scattering from the Boltzmann transport equation for charged particle is presented. The Adomian decomposition is used to solve the telegraph equation. By means of MAPLE the Adomian polynomials of obtained series (ADM) solution have been calculated. The behaviour of the distribution function are shown graphically. The results reported in this article provide further evidence of the usefulness of Adomain decomposition for obtaining solution of linear and nonlinear problems.     

Dispersion Relations in Binary Gas Mixtures From a Kinetic Theory

Sound propagation and the initial value problems in gas mixtures of two components are investigated. By using the eigen theory of linearized Boltzmann equations, a model equations is formed, with the use of the Fourier-Laplace transform for model equations derived, the dispersion relations for both components are obtained.     

Electron Energy Distribution and Optical Characteristics of a Hollow Cathode Discharge

A model is proposed for the formation of the electron energy distribution in a hollow cathode discharge. On the basis of this model, an integral equation has been derived to calculate the electron distribution function over the entire cathode cavity volume. The equation holds true for both the isotropic distribution, and the case when the local distribution function is anisotropic. Solutions of the kinetic equation obtained are presented, for electron energies over 2–3 eV and up to the cathode fall potential. It is shown that the electron energy distribution function in this interval determines the optical characteristics of the hollow cathode discharge. A comparison is given of the calculated and measured radiation powers for the cases of the hollow cathode discharge in xenon and carbon dioxide. The discrepancy between the theoretical and experimental data does not exceed 20%.     

A solution of a non-homogeneous orthotropic cylindrical shell for axisymmetric plane strain dynamic thermoelastic problems

A solution of a non-homogeneous orthotropic elastic cylindrical shell for axisymmetric plane strain dynamic thermoelastic problems is developed. Firstly, a new dependent variable is introduced to rewrite the governing equation, the boundary conditions as well as the initial conditions. Secondly, a special function is introduced to transform the inhomogeneous boundary conditions to the homogeneous ones. Then by virtue of the orthogonal expansion technique, the equation with respect to the time variable is derived, of which the solution can be obtained. The displacement solution is finally presented, which can degenerate in a rather straightforward way to the solution for a homogeneous orthotropic cylindrical shell and isotropic solid cylinder as well as that for a non-homogeneous isotropic cylindrical shell. Using the present method, integral transform can be avoided. It is fit for a cylindrical shell with arbitrary thickness subjected to arbitrary thermal loads. It is also very convenient to deal with dynamic thermoelastic problems for different boundary conditions. Besides, the numerical calculation involved is very easy to be performed. Several examples are presented.     

Statistical description of an ensemble of ultrarelativistic particles in a spatially flat universe

The kinetic equation for massless particles in an unperturbed Friedman universe with gravitational interactions taken into account is obtained by averaging the collisionless kinetic equations over local background gravitational field fluctuations. Uniformly distributed spherically symmetric fluctuations of an ultrarelativistic fluid serve as local inhomogeneities in a spatially flat universe. It is shown that the collision integral of our kinetic equation does not vanish when the distribution function is homogeneous and isotropic.     

Application of the singularity-subtraction technique to isotropic scattering in a planar layer

A numerical procedure for solving a singular Fredholm integral equation, which describes radiative transfer in an absorbing and isotropically scattering layer exposed to collimated radiation, is studied. The integral equation is solved by subtracting out the singularity and then approximating the integral term by a Gaussian quadrature. Bidirectional and hemispherical properties are found from the source function. Any arbitrary directional distribution of incident radiation can be handled by superimposing the collimated radiation case. In particular, the case of isotropic incident radiation is presented. The method gives accurate results, even for the extreme conditions of large optical thickness and large angle of incidence.     

Hot electrons in indium nitride: a new numerical approach to solving the electron transport problem

The electron transport in indium nitride in a heating electric field has been calculated. The results obtained are in good agreement with the known experimental data. The Boltzmann transport equation in the isotropic spatially homogeneous case has been solved using a new numerical approach based on the approximation of the distribution function from the points of a relatively thin grid in momentum space and the iterative search for a stationary solution.     

微粒特征分布函数测量理论和方法的进一步研究

运用微粒散射形体微扰理论对任意形体微粒的研究结果,根据光敏区光强的Gauss分布特征和微粒穿越位置的随机性,建立了相关模型和数学方程并求得了任意位置的微粒形体影响因子分布函数的数值解,讨论了大变化区的非线性影响.其次,根据微粒数分布离散性的特征,建立了微粒粒度分布函数的反演矩阵及其实验方法.最后进行了实验验证.