Showing papers on "Boltzmann constant published in 1972"

On the Boltzmann equation for rigid spheres

Abstract: A formal deduction of the Boltzmann equation from the Liouville equation is presented for the case of rigid spheres. The main result is that, in the Boltzmann limit (number of molecules tending to infinity, diameter tending to zero, finite mean free path), the Boltzmann equation follows under the assumptions of a sufficiently smooth N-particle distribution function for a smooth limit to exist and of an initial datum which satisfies (at least for N → ∞) the chaos assumption. Possible extensions to molecules interacting with central forces and to dense gases are briefly discussed.

Shock‐Wave Structure using Nonlinear Model Boltzmann Equations

Abstract: The structure of strong plane shock waves in a perfect monatomic gas was studied using four nonlinear models of the Boltzmann equation. The models involved the use of a simplified collision operator with velocity‐independent collision frequency, in place of the complicated Boltzmann collision operator. The models employed were the BGK and ellipsoidal models developed by earlier authors, and the polynomial and trimodal gain function models developed during the work. An exact set of moment equations was derived for the density, velocity, temperature, viscous stress, and heat flux within the shock. This set was reduced to a pair of coupled nonlinear integral equations and solved using specially adapted numerical techniques. A new and simple Gauss‐Seidel iteration was developed during the work and found to be as efficient as the best earlier iteration methods. Extensive comparisons were made of the model results with Monte Carlo solutions, and significant aspects of the comparisons are discussed.

Distribution function of electrons and kinetic coefficients in cesium-argon mixture discharge

Abstract: The distribution function of electrons in a cesium-argon mixture discharge is found in the present paper by solving Boltzmann's equation. The elastic and inelastic electron-atom collisions and electron-electron collisions are taken into consideration.

Non-Ohmic Electron Transport on Liquid Helium

Abstract: The transport coefficients in high electric fields are obtained for electrons bound in image-potential-induced surface states on a dielectric liquid surface. A two-dimensional Boltzmann equation is solved in the diffusion approximation, assuming the principle electron scatterers are gas atoms and surface waves. At high temperatures, where gas-atom scattering dominates both energy and momentum relaxation, the transport coefficients are field independent even though the average electron energy is much higher than the product of Boltzmann's constant and the liquid temperature. At low temperatures, where surface-wave scattering dominates the momentum relaxation, the conductivity and Hall mobility increase rapidly with increasing electric field. For $^{4}\mathrm{He}$ this non-Ohmic transport should occur below 1 K at fields below 0.1 V/cm.

Solution of Boltzmann's equation for semiconductors using a spherical harmonic expansion. (Acoustic phonon scattering)

Abstract: A method of solution of the steady state Boltzmann equation for small fields using a spherical harmonic expansion of the nonequilibrium part of the distribution function has been derived for acoustic phonon scattering in nondegenerate semiconductors. The method is easy to apply for arbitrarily complicated scattering functions and has some other advantages over previous treatment. It is possible to extend the method to include other scattering mechanisms and the effects of a magnetic field. Applications to silicon and germanium are discussed. As a by-product, a new sum rule for Clebsch-Gordan coefficients is derived.

Quantum Mechanical Derivation of the Correction to the Boltzmann Equation Due to the Duration of Collisions

Abstract: Expressions are studied for the rate of change of momentum distribution that result from using a recent formulation of a quantum mechanical derivation of the Boltzmann equation obtained by Snider and Sanctuary. The gas is supposed to consist of spherical and structureless particles. If the distribution function is restricted so as initially to be spatially uniform, it will remain spatially uniform and the rate of change of momentum distribution is as given by the Boltzmann collision term. In case the distribution function initially is spatially nonuniform, the nonuniformity persists in the Boltzmann equation, and in addition to drift terms, a term appears containing the duration of the collisions. This can be related to the Enskog modification of dense gases in the case of hard sphere particles. A new quantum mechanical expression of this correction is presented.

The Grad’s solution of the boltzmann equation in terms of irreducible spherical tensors and talmi coefficients

Abstract: The algebra associated with Grad’s method of solving the Boltzmann equation has been simplified by expanding the distribution function into a polynomial based on spherical polar co-ordinates and using the Talmi coefficients for the collision part. This simplified reformulation has been applied for calculating the components of the stress tensor for a fully ionized plasma in a strong magnetic field from both the Vlasov and the simplified Boltzmann equations.

Calculation of Electron Density in Nonparabolic Bands

Abstract: It is shown that the integral F(e,η)=2π−1/2 ∫ 0∞y1/2(1+y/e)1/2(1+2y/e)dyexp(y−η)+1, which is useful for determining the electron density in a nonparabolic energy band of Kane type, may be evaluated in terms of a Bessel function in the Boltzmann limit and that in the narrow‐gap (small e) limit, an approximation suggested by Bebb and Ratliff is quite accurate.

Degeneracy of the Spectra of the Linearized Boltzmann Collision Operator for the Maxwellian Molecules

Abstract: The degeneracy between the spectra corresponding to scalar and vector solutions to the linearized Boltzmann collision operator for the Maxwellian molecules is shown to be a consequence of the Galilean translational invariance of the equations and the spherical symmetric property of the scalar solution.

Solution of the Boltzmann kinetic equation for the relaxation of a gas mixture

Abstract: The temporal behavior is considered of a quiescent mixture of gases of different temperatures with spatially uniform distribution. The process of heating a cold gas by a hot gas is treated on the basis of the Boltzmann kinetic equation. The mixture is assumed to be composed of absolutely hard smooth spheres, and the initial distribution functions for each gas is taken to the Maxwellian. With such a choice of initial distribution functions, it is shown that the solution of the Boltzmann kinetic equation depends only on the velocity modulus and the time.

Calculation of the transport coefficients for dilute gases using an interactive solution of Boltzmann's equation.

Abstract: The second approximation to Boltzmann's equation for dilute gases, in the Chapman-Enskog form, is solved for several different potential models by using an iterative procedure. This method is presented as an alternative to the standard one of Chapman and Cowling. The first order perturbation function, p, to the local Maxwellian distribution, f[(o)], obtained as an infinite series [mathematical equation] where the p[(i)]'s are obtained by repeated iterations of the Boltzmann equation. The transport coefficients, such as thermal conductivity, lambda, and viscosity, eta[s], are obtained from p as infinite series [mathematical equation] In the case of the Maxwellian and Pseudo-MaxweIlian potential models, for a single component gas, both the p[(i)]'s and lambda[(i)]'s and eta[s][(i)]'s are found to follow exact geometrical progression which can s be summed. The transport coefficient obtained by performing these sums are found to correspond to the analytical values obtained by the Chapman-Cowling method. For the rigid sphere model the lambda[(i)]'s still seem to approximate closely to a geometrical progression. On assuming an exact geometrical progression for the lambda[(i)]'s, one obtains a value for the thermal conductivity which is in close agreement with that obtained by Chapman-Cowling. The iterative method is extended to deal with binary mixtures for the case of Maxwellian and Pseudo-Maxwellian potential models where exact geometrical progressions are again exhibited, and values for the transport coefficients are found to correspond to those for the Chapman-Cowling method. The significance of the form of the results obtained by the iterative procedure is discussed particularly with regard to the possibility of extending the method to deal with more general repulsive potentials and higher order mixtures.

Experimental validation of modified boltzmann type of model and shift model for multilane traffic flow

Abstract: THE BOLTZMANN TYPE OF STATISTICAL MODELS PROPOSED BY PRIGOGINE ET AL. FOR TIME-INDEPENDENT, SPACE- HOMOGENEOUS, MULTILANE TRAFFIC ASSUMES THAT TRAFFIC FLOW IS DESCRIBED BY 3 PROCESSES-INTERACTION, RELAXATION, AND ADJUSTMENT-AND HAS BEEN SHOWN EARLIER TO PRODUCE POOR VALIDATION RESULTS. IN THIS PAPER WE PROPOSE A NEW MODEL THAT USES A SHIFTING PROCESS, A CONCENTRATION-DEPENDENT PARAMETER REPLACING THE DESIRED SPEED-DENSITY IN THE RELAXATION PROCESS. THE RESULTING MODIFIED BOLZMANN TYPE OF MODEL IS SHOWN TO HAVE MANY DESIRABLE PROPERTIES AND FITS EXPERIMENTAL SPEED DATA WITHIN THE DATA ERROR MARGIN. SECOND MODEL, CALLED THE SHIFT MODEL, USES ONLY THE SHIFTING PROCESS AND NEGLECTS THE INTERACTION, RELAXATION, AND ADJUSTMENT PROCESSES OF THE BOLZMANN TYPE OF MODEL, GIVES RESULTS ONLY SLIGHTLY LESS OPTIMAL THAN THE RESULT OF THE BEST POSSIBLE FIT OBTAINED BY THE MODIFIED BOLTZMANN TYPE OF MODEL, AND IS FAR BETTER THAN THE ORIGINAL BOLTZMANN TYPE OF MODELS. SOME APPLICATIONS OF THE SHIFT MODEL ARE DISCUSSED.

Radiating gas around a body with consideration of temperature difference between electrons and ions

Abstract: The flow around a blunt body at hypersonic speed by a current of nonequilibrium ionized monatomic nonviscous radiating gas is studied, with consideration of temperature difference between the electron gas and the ion-atom gas. Atomic excitation due to collisions with electrons and subsequent ionization, as well as photoionization, are taken into consideration. Since the value of the shock wave separation is small in comparison with the characteristic dimension of the body, the radiation transfer equation is written in the local onedimensional planar layer approximation. The influence of incident flow parameters upon the flow field across the shock wave and the distribution of radiation thermal flux are studied. Terminology: r, radius vector.~ calculated from the center of curvature of the body about which flow occurs; rT, rb, radius vectors of the body surface and shock wave; ~, separation of the shock wave e = r b - rT; L, characteristic dimension of the body; V, W, vector and modulus of total gas velocity, respectively; Win, maximum gas velocity (velocity of escape into a vacuum); u, component of gas velocity along the radius vector; p, gas pressure; p, gas density; Pi, ion density; c~, degree of nonequilibrium gas ionization, cz =pi/p;O~E, degree of equilibrium gas ionization; Ta, temperature of atoms and ions, Te, electron temperature; Te, , excitation temperature; Tin, body surface temperature; A, atom in the fundamental state; A* atom in an excited state; u, frequency; u., T., ionization frequency and temperature; m a , mass of an atom; me, e, electronic mass ] ] and charge; CE, electron-atom excitation cross section; nea , ionization reaction velocity for electronatom shock; he, electron density; R, specific gas constant; K, Boltzmann constant; E, internal energy per unit mixture mass; ~, spectral absorption coefficient per unit mass of monatomic gas; 6, blackness coefficient of body surface; ~-~, optical coordinate; Pi, mean cosine of angle for givenangular zone of photon propagation; I ~, radiation spectral intensity; Bu(Te) , Planck function; q u, q, spectral and total radiant o~ energy flux vectors q --- ~qvd~ ; M, Much number; ~r b, indices related to gas parameters in undisturbed flow O region and in shock wave, respectively~ 1. Fundamental System of Equations. In order to describe the nonequilibrium processes in the shock layer it is necessary to define a certain kinetic model for the gas. In this study the kinetics proposed for argon by Chaplin in [1] wilI be used. Consideration is made of reactions A--]-e~A*+e, A*+e~,~_A+~-e, A~hv~A*~e