Local Maxwellian Research Articles

Some remarks about continuity properties of local Maxwellians and an existence theorem for the BGK model of the Boltzmann equation

Continuity of local Maxwellians in various topologies of L/sup 1/ is studied. The existence and convergence of approximate solutions of the nonlinear BGK model of the Boltzmann equation are proved.

Lagrangian formulation of neoclassical transport theory

Neoclassical transport theory is developed in a Lagrangian formulation in contrast to the usual Eulerian development. The Lagrangian formulation is constructed from the three actions: magnetic moment, parallel invariant, and bounce-averaged poloidal flux. By averaging over the fast orbital time scales, an equation in the actions alone of the Fokker–Planck type is obtained. The coefficients give the rates of the elementary neoclassical scattering processes. This action space form of the kinetic equation contains in explicit form processes like banana diffusion and the kinematic Ware pinch. The associated fluxes can be computed by simple moments without having deviations from the local Maxwellian. All the trapped-particle contributions are of this explicit type. Another class of fluxes arise from perturbations to the Maxwellian and are termed implicit. The decomposition of the fluxes into explicit and implicit parts is a key feature of the Lagrangian formulation. These contributions correspond to distinct physical processes and have separate Onsager symmetry theorems (explicit and implicit) for their respective transport matrices. The theory does not depend on the details of the Fokker–Planck coefficients but only on some very general properties and is thus applicable—without modification of the formalism—to nonaxisymmetric and turbulent systems. This general formulation is the primary purpose of the work. To benchmark the theory, in the present paper, the tokamak transport coefficients (for the Lorentz gas) are computed and compared to the known Eulerian results, demonstrating the equivalence of the two formulations. The elementary processes responsible for the neoclassical pinch and bootstrap effects (somewhat obscured in the Eulerian picture) are identified and the physical basis for their Onsager symmetry relationship is clarified.

Expansion About a Local Maxwellian for Evaluating the Spatially Dependent Neutron Spectrum

The problem of the spatial variation of thermal-neutron fluxes is investigated in the framework of energy-dependent diffusion theory. Neutron flux is expanded here in terms of Laguerre polynomials in energy, weighted with a local Maxwellian distribution in which the neutron temperature depends on the space variable. The method proposed is applied to the diffusion of thermal neutrons in moderating systems with a nonuniform temperature profile. 3 figures.

Generalization of the Moment Method of Maxwell-Grad for Multi-Temperature Gas Mixtures and Plasmas

Abstract A coupled system of balance equations is derived for the coefficients of orthogonal expansions of the velocity distribution functions. The orthogonal functions are not specified, but the initial func­ tions must be local Maxwellians with different temperatures for different species of particles. Closed expressions for the matrix elements of the non-linearized Boltzmann operator are given, whose dominant terms are determined and compared.

Analysis of pressure distribution for free molecule flow in circular tube.

Pressure distribution in a circular tube under a free molecule regime was analyzed by utilizing local Maxwellian and Monte Carlo methods. It is recognized that pressure slip at the entrance and exit of the tube exist and the distribution is linearly changed along the tube axis. The magnitudes of the slips increase with the decrease of the ratio of length to radius of tube and the ratio of pressure at the exit to that at the entrance.

Variational Solution of the Linearized Boltzmann Equation with Applications to Helium

High accuracy variational solutions for the Navier-Stokes coefficients of thermal conductivity and viscosity are obtained from a linearized Boltzmann equation for a dilute one-component gas with spherically symmetric interactions. Results are presented for quantum-mechanical hard spheres and for three helium potential models at 1–20°K. The deviation of the one-particle distribution function from a local Maxwellian is also obtained from the variational solutions and is compared with an approximate treatment of the Hilbert-Enskog equation.

Asymptotic solutions for the distribution function in non-equilibrium flows. Part 1. The weak shock

This paper is the first part of an investigation of the molecular velocity distribution function in non-equilibrium flows. In this part, the general features of the distribution are discussed and illustrated by a detailed study of its asymptotic expansions in different velocity domains for a weak shock, employing a simple relaxation model for the collisions. Using the strength of the shock as a small parameter, the Chapman–Enskog distribution is derived, in a less restrictive way than in previous analyses, as the first two terms in a suitable ‘inner’ asymptotic expansion of the distribution in velocity space, valid only for not very high velocities. It is shown that the consideration of two further limits, called the intermediate and outer, is necessary for a complete description of the distribution in velocity space. The uniformly valid composite expansion demonstrates the slow approach to equilibrium of fast molecules. The outer solution depends on integrals over the flow and is in general ‘global’, in contrast to the inner solution which is essentially local; this introduces certain asymmetries on a fine scale even in a weak shock. It is shown, for example, that fast molecules moving towards the hot side accumulate by collisionless streaming, whereas those moving towards the cold side attenuate like a molecular beam and represent essentially a ‘precursor’ of the hot side. A simple approximation for the distribution in the precursor is derived, and found to contain, in the outer limit, a large perturbation on the local Maxwellian; this results in an approach to equilibrium like $\exp (-|x|^{\frac{2}{3}})$ on the cold side.A heuristic extension of the argument to the true Boltzmann equation leads to the result that for molecules with an interparticle potential varying as the inverse m-power of the distance, the approach to equilibrium through the precursor is like $\exp (-|x|^l)$, where l = m/(m + 2).

Propagation of Density Disturbances in a Dense Hard-Sphere Gas

Approximate calculations of S(K, ω)—the double Fourier transform of the density—density correlation function—for the linearized Enskog equation for hard spheres are presented. The interaction terms are the usual Boltzmann collision integral and the collisional transfer terms. The former is replaced by a single relaxation-time kinetic model, while the collisional transfer terms are evaluated, as a first approximation, with a local Maxwellian for the distribution function. The numerical results at small K exhibit the expected peak at ω=cK due to sound propagation, in agreement with the predictions of the linearized hydrodynamic calculations. A peak is found to persist at large wavenumbers corresponding to a modified form of collective excitation. However, this peak disappears when first-order corrections to the local Maxwellian approximation in the collisional transfer terms are made. The feasibility of light- and neutron-scattering experiments to investigate the regions of interest is discussed.

Comments on the Solution to the Boltzmann Equation for a Weakly Ionized Plasma

Hydrodynamic equations describing the motion of electrons in a weakly ionized plasma are derived formally from the Boltzmann equation by means of the Chapman-Enskog procedure. Two cases are considered, each with a stationary Maxwellian distribution ascribed to the atoms. In the first, electron-electron collisions are ignored, and the electron distribution function is determined by a balance between the electron-atom collisions and the electric field. There is only one conservation law, a hydrodynamic equation for the density. In the second case, electron-electron collisions are dominant, the distribution function is a local Maxwellian, and there are five conservation laws---equations for the density, drift velocity, and temperature. The equations used previously by the author to describe low-frequency oscillations are obtained in either case if the electron-atom collision frequency is independent of velocity. Otherwise, the zeroth-order equations are still exact, but first-order corrections are required, as illustrated by the example of the velocity-independent mean free path. Our results are somewhat different from those of Davidov, who made different assumptions about the dominant collision mechanisms.

Radio Emission by Electrons in the Fine-Structure Electric Fields of Shock Waves

The structure of shock waves in fully ionized hydrogen in the absence of a magnetic field has been investigated by Greenberg and Trève using the Mott-Smith distribution for the protons and a local Maxwellian for the electrons. They find that large charge-separation effects occur inside the shock due to the electron-proton mass difference, thus giving rise to strong stationary electric-field oscillations in the shock transition. This behavior is typical of other theories of plasma shock structure. The radio noise emitted by electrons in the plasma accelerated in these electric fields is now calculated. In traversing the electric oscillations inside the shock, those electrons with a few times the thermal energy radiate like an assembly of oscillators at frequencies capable of propagating ahead of the shock. Although the particular shock-wave model of Greenberg and Trève has been adopted for the purposes of the numerical discussion, the parameters of the radiation calculation are readily adjusted for possible changes in the shock-wave model which more detailed treatments of shock structure may bring. The application of this process to Type II solar radio bursts is discussed. Assuming that such shock waves are produced by a solar flare expanding into the corona, a sufficiently high intensity of radio emission is found to provide an alternate possible interpretation for this source of solar noise.