Nonlinear moment method for the isotropic Boltzmann equation and invariance of the collision integral

A new approach is proposed for the development of a nonlinear moment method of solving the Boltzmann equation. This approach is based on the principle of invariance of the collision integral with respect to the choice of basis functions. Sonine polynomials with a Maxwellian weighting function are taken as these basis functions for the velocity-isotropic Boltzmann equation. It is shown that for arbitrary interaction cross sections the matrix elements corresponding to the moments of the nonlinear collision integral are not independent but are coupled by simple recurrence formulas by means of which all the nonlinear matrix elements are expressed in terms of linear ones. As a result, a highly efficient numerical scheme is constructed for calculating the nonlinear matrix elements. The proposed approach opens up prospects for calculating relaxation processes at high velocities and also for solving more complex kinetic problems. Zh. Tekh. Fiz. 69, 22–29 (June 1999)     

Generalization of the Hecke theorem for the nonlinear Boltzmann collision integral in the axisymmetric case

The properties of the nonlinear collision integral in the Boltzmann equation are studied. Expansions in spherical Hermitean polynomials are used. It was shown [1] that the nonlinear matrix elements of the collision operator are related to each other by simple expressions, which are valid for arbitrary cross sections of particle interaction. The structure of the collision operator and the properties of the matrix elements are studied for the case when the interaction potential is spherically symmetric. In this case, the linear Boltzmann operator satisfies the Hecke theorem. The generalized Hecke theorem, from which it follows that many nonlinear matrix elements vanish, is proved with recurrence relations derived. It is shown that the generalized Hecke theorem is a consequence of the ordinary Hecke theorem.     

Asymptotics of collision integral matrix elements in the isotropic case

The method of nonlinear moments, when used to solve the Boltzmann equation, necessitates the calculation of collision integral matrix elements. The matrix elements are hard to calculate numerically, especially at large indices. The asymptotics of the matrix elements are constructed. In terms of the model of pseudopower particle interaction, a formula free of summation is derived. This makes it possible to find the asymptotic behavior of linear and nonlinear elements when two indices are large. For an arbitrary interaction cross section, asymptotic expansions of linear and nonlinear matrix elements in one index are obtained. For Maxwellian molecules, asymptotic formulas are derived for three large indices.     

Symmetry of the nonlinear collision operator matrix and new prospects in the moment method for solving the Boltzmann equation

Traditionally, the moment method has been used in kinetic theory to calculate transport coefficients. Its application to the solution of more complicated problems runs into enormous difficulties associated with calculating the matrix elements of the collision operator. The corresponding formulas for large values of the indices are either lacking or are very cumbersome. In this paper relations between matrix elements are derived from very general principles, and these can be employed as simple recurrence relations for calculating all the nonlinear and linear anisotropic matrix elements from assigned linear isotropic matrix elements. Efficient programs which implement this algorithm are developed. The possibility of calculating the distribution function out to 8–10 thermal velocities is demonstrated. The results obtained open up prospects for solving many topical problems in kinetic theory. Zh. Tekh. Fiz. 69, 6–9 (September 1999)     

Isotropic matrix elements of the collision integral for the Boltzmann equation

We have proposed an algorithm for constructing matrix elements of the collision integral for the nonlinear Boltzmann equation isotropic in velocities. These matrix elements have been used to start the recurrent procedure for calculating matrix elements of the velocity-nonisotropic collision integral described in our previous publication. In addition, isotropic matrix elements are of independent interest for calculating isotropic relaxation in a number of physical kinetics problems. It has been shown that the coefficients of expansion of isotropic matrix elements in Ω integrals are connected by the recurrent relations that make it possible to construct the procedure of their sequential determination.     

On the Velocity Distribution Function of Light Ions in Heavy Gases in an Electric Field

The special properties presented by the Boltzmann collision operator in the case of an induced-dipole interaction (Maxwellian interaction) between ions and neutrals are exploited to obtain in such case a proper solution of the Boltzmann equation for light ions in heavy gases in an electric field. Meanwhile it is proved that consideration of more than two terms of the spherical harmonic expansion of the ion velocity distribution requires to improve the usual accuracy of the terms deriving from the collision integral; vice versa, the improvement of the accuracy of the collision terms requires to retain more than two terms of the spherical harmonic expansion. The consistency of our procedure and results in the approximation neglecting the square of the ion-neutral mass ratio with respect to unity is discussed. Finally, the most significant velocity averages are calculated on the basis of the obtained ion distribution. In the limit of the above approximation they are shown to agree with Wannier's results.     

COLLISIONAL EFFECTS IN CLOUD-IN-CELL SCHEME

In this paper, a physical model and numerical method in cloud-in-cell(C1C) scheme for simulation of weak collisional effects are proposed.Collisions are introduced at reconstruction time in terms of a one-dimensional Fokker-Planck operator., The friction term of the collision,operator is equivalent to the introduction of a velocity-dependent acceleration a(u)=-uy(u) into the equation of motion of the particles. The diffusion term is represented by finite differences over the nonuniform velocity grid. Col lisional relaxation of two Maxwellian beams is studied.The numerical results are in good agreement. with the analgtical values.     

Implementation of an inelastic collision operator into KIPP‐SOLPS coupling and its effects on electron parallel transport in the scrape‐off layer plasmas

This paper develops an inelastic collision operator for the Kinetic Code for Plasma Periphery (KIPP) code to investigate the kinetic effects of electron cooling due to inelastic collisions. It is fully tested based on the self‐consistent KIPP‐SOLPS coupling algorithm by being compared to the ADAS database. The collisional radiative rate coefficients from the ADAS database for deuterium atomic physics can be recovered using the inelastic collision operator with assuming Maxwellian electrons, which shows that the inelastic collision operator works well for various plasma conditions. Across a wide range of plasma conditions in the scrape‐off layer, KIPP‐SOLPS coupling simulation results with the implementation of an inelastic collision operator are not significantly different from results using a simpler uniform cooling scheme. The uniform scheme is thus recommended rather than including computationally intensive inelastic collision physics.     

Approach to equilibrium in a one-dimensional,two-component gas of Maxwellian molecules

The coupled Boltzmann equations describing the evolution of the velocity distributions of a one-dimensional, two-component gas of Maxwellian molecules are analyzed. When the two species have different masses, the system approaches equilibrium. The complete eigenvalue spectrum of the linearized collision operator is obtained, and is found to exhibit an interesting dependence on the mass ratio. The response of one species to an external field, when the other species is regarded as a host fluid, is also examined.     

The relation between the velocity and mass distributions. The role of collisionless relaxation processes

We consider steady-state mass distributions (mass functions) attained at the nonlinear stage of fragmentation as a result of fragment coalescence. The influence of the fragment velocity distribution on the mass function is discussed. The kinetic equations governing the fragments quasiparticles are solved using the group symmetry properties of the collision integral. We have calculated power indices and locations of the breaking points for the mss spectrum (luminosity function) associated with the transition from the collisionless Lynden-Bell type to the Maxwellian distribution, as well as with the predominance of either purely geometric or Newtonian collision cross sections. The power indices found are in a reasonable agreement with the values observed for star or galaxy clusters.     

Line strengths,collision strengths and excitation rates for multiply-charged silicon ions

Line strengths, collision strengths and excitation rates have been calculated for a variety of transitions in multicharged silicon ions from Si(Vi) to Si(XIV). The collision strengths were evaluated in an LS coupling scheme in the distorted wave approximation neglecting exchange except for the helium-like transitions. Excitation rates were then obtained by integrating the collision strength over a Maxwellian velocity distribution function. These results are then described by a simple two-parameter fit for the rates.     

Computer simulation and analysis of the Holweck pump in the transient regime

The results of simulating the Holweck pump by numerically solving the Boltzmann kinetic equation are reported. The nonlinear collision integral is calculated using the conservative projection method. The translation operator is approximated with tetrahedral grids. The ratio of pressures in pumped-in and pumpedout containers is studied as a function of the gas density (rarefaction) and the radius and rotation velocity of the rotor at Knudsen numbers close to unity.     

Calculation of the collision integral linear kernel for Maxwellian molecules in the isotropic case

Application of the method of nonlinear moments to solve the Boltzmann equation generates the need to sum a series that is the expansion of the distribution function in basis functions. This series converged only if the Grad test is fulfilled. Such a limitation can be removed if the expansion of the distribution function is summed over the index related to only the expansion in velocity magnitude. In this case, the distribution function and the collision integral become expanded in only spherical harmonics and the expansion coefficients satisfy integro-differential equations. The kernels of these equations are the sums of the Sonine polynomials in the velocities of colliding and outgoing particles multiplied by matrix elements of the collision integral. For a number of arguments, the direct calculation of the kernels requires that a very large number of terms in the sum be taken into consideration. In this respect, an approach seems to be promising in which the asymptotics of the matrix elements and Sonine polynomials at large indices are used and summation over index is replaced by integration. In this paper, we apply this approach to calculate the linear kernel in the isotropic case, assuming that interaction between particles is described by a pseudopower law. With this approach, the collision integral kernel can be calculated with a high accuracy using as little as a few tens of series terms and the asymptotic estimate of the residue.     

Global Existence and Full Regularity of the Boltzmann Equation Without Angular Cutoff

We prove the global existence and uniqueness of classical solutions around an equilibrium to the Boltzmann equation without angular cutoff in some Sobolev spaces. In addition, the solutions thus obtained are shown to be non-negative and C ^∞ in all variables for any positive time. In this paper, we study the Maxwellian molecule type collision operator with mild singularity. One of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator. This norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated H-theorem.     

Stable Equilibrium Based on Lévy Statistics:A Linear Boltzmann Equation Approach

To obtain further insight on possible power law generalizations of Boltzmann equilibrium concepts, we consider stochastic collision models. The models are a generalization of the Rayleigh collision model, for a heavy one dimensional particle M interacting with ideal gas particles with a mass m<<M. Similar to previous approaches we assume elastic, uncorrelated, and impulsive collisions. We let the bath particle velocity distribution function to be of general form, namely we do not postulate a specific form of power-law equilibrium. We show, under certain conditions, that the velocity distribution function of the heavy particle is Lévy stable, the Maxwellian distribution being a special case. We demonstrate our results with numerical examples. The relation of the power law equilibrium obtained here to thermodynamics is discussed. In particular we compare between two models: a thermodynamic and an energy scaling approaches. These models yield insight into questions like the meaning of temperature for power law equilibrium, and into the issue of the universality of the equilibrium (i.e., is the width of the generalized Maxwellian distribution functions obtained here, independent of coupling constant to the bath).     

Zum Einfluß der Energieabhängigkeit der Elektronenstoßfrequenz auf die elektrische Leitfähigkeit schwach ionisierter Plasmen

With regard to experimental applications in plasma diagnostics numerical approximations are given for the Gurevic-type correction functions, which in the kinetic theory of weakly ionized plasmas describe the deviations from the Lorentzian electrical conductivity. The approximations are based either upon the dependance of the collision frequency on the power of the electron velocity, or on a first order Taylor expansion around the most probable electron velocity of a Maxwellian distribution. For all assumptions regarding the velocity dependance of the collision frequency the influence of temperature (and pressure) on the effective collision frequency is indicated.     

Boundary symmetries in linear differential and integral equation problems applied to the self-consistent Green’s function formalism of acoustic and electromagnetic scattering

Explicit symmetry relations for the Green’s function subject to homogeneous boundary conditions are derived for arbitrary linear differential or integral equation problems in which the boundary surface has a set of symmetry elements. For corresponding homogeneous problems subject to inhomogeneous boundary conditions implicit symmetry relations involving the Green’s function are obtained. The usefulness of these symmetry relations is illustrated by means of a recently developed self-consistent Green’s function formalism of electromagnetic and acoustic scattering problems applied to the exterior scattering problem. One obtains explicit symmetry relations for the volume Green’s function, the surface Green’s function, and the interaction operator, and the respective symmetry relations are shown to be equivalent. This allows us to treat boundary symmetries of volume-integral equation methods, boundary-integral equation methods, and the T matrix formulation of acoustic and electromagnetic scattering under a common theoretical framework. By specifying a specific expansion basis the coordinate-free symmetry relations of, e.g., the surface Green’s function can be brought into the form of explicit symmetry relations of its expansion coefficient matrix. For the specific choice of radiating spherical wave functions the approach is illustrated by deriving unitary reducible representations of non-cubic finite point groups in this basis, and by deriving the corresponding explicit symmetry relations of the coefficient matrix. The reducible representations can be reduced by group-theoretical techniques, thus bringing the coefficient matrix into block-diagonal form, which can greatly reduce ill-conditioning problems in numerical applications.     

The collision frequency of electron-neutral-particle in weakly ionized plasmas with non-Maxwellian velocity distributions

The collision frequencies of electron-neutral-particle in weakly ionized complex plasmas with the non-Maxwellian velocity distributions are studied. The average collision frequencies of electron-neutral-particle in plasmas are accurately derived. We find that these collision frequencies are significantly dependent on the power-law spectral indices of non-Maxwellian distribution functions and so they are generally different from the collision frequencies in plasmas with a Maxwellian velocity distribution, which will affect the transport properties of the charged particles in plasmas. Numerically analyses are made to show the roles of the spectral indices in the average collision frequencies respectively.     

A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources

In this paper, we propose a general time-discrete framework to design asymptotic-preserving schemes for initial value problem of the Boltzmann kinetic and related equations. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We propose to penalize the nonlinear collision term by a BGK-type relaxation term, which can be solved explicitly even if discretized implicitly in time. Moreover, the BGK-type relaxation operator helps to drive the density distribution toward the local Maxwellian, thus naturally imposes an asymptotic-preserving scheme in the Euler limit. The scheme so designed does not need any nonlinear iterative solver or the use of Wild Sum. It is uniformly stable in terms of the (possibly small) Knudsen number, and can capture the macroscopic fluid dynamic (Euler) limit even if the small scale determined by the Knudsen number is not numerically resolved. It is also consistent to the compressible Navier–Stokes equations if the viscosity and heat conductivity are numerically resolved. The method is applicable to many other related problems, such as hyperbolic systems with stiff relaxation, and high order parabolic equations.