Showing papers on "Boltzmann constant published in 1970"

The mathematical theory of non-uniform gases : an account of the kinetic theory of viscosity, thermal conduction, and diffusion in gases

Abstract: Foreword Introduction 1. Vectors and tensors 2. Properties of a gas: definitions and theorems 3. The equations of Boltzmann and Maxwell 4. Boltzmann's H-theorem and the Maxwellian velocity-distribution 5. The free path, the collision-frequency and persistence of velocities 6. Elementary theories of the transport phenomena 7. The non-uniform state for a simple gas 8. The non-uniform state for a binary gas-mixture 9. Viscosity, thermal conduction, and diffusion: general expressions 10. Viscosity, thermal conduction, and diffusion: theoretical formulae for special molecular models 11. Molecules with internal energy 12 Viscosity: comparison of theory with experiment 13. Thermal conductivity: comparison of theory with experiment 14. Diffusion: comparison of theory with experiment 15. The third approximation to the velocity-distribution function 16. Dense gases 17. Quantum theory and the transport phenomena 18. Multiple gas mixtures 19. Electromagnetic phenomena in ionized gases Historical summary Name index Subject index References to numerical data for particular gases (simple and mixed).

Improved Variational Principles for Transport Coefficients

Abstract: A sequence of variational principles for converting a trial solution of a linearized Boltzmann equation into bounds on a transport coefficient is presented. For systems in which the Boltzmann collision operator has a bounded eigenvalue spectrum, we obtain an infinite sequence of lower bounds which begins with the familiar result of Ziman. For an arbitrary trial function, this sequence converges monotonically to the exact transport coefficient. Application of the first two terms has been made to the lattice thermal conductivity of a model simulating solid argon; the second bound lies considerably higher than the first.

The Linearized Boltzmann Equation. Navier-Stokes and Burnett Transport Coefficients

Abstract: The Boltzmann equation is linearized around total equilibrium and solved by means of a projection operator, which projects the single particle distribution function on the slowly varying local density, temperature, and stream velocity. For small spatial variations the exact solution approaches asymptotically to the Chapman-Enskog normal solution with a relaxation time of the order of the mean free time. The heat current and pressure tensor obtained from this solution are connected with the gradients of the local temperature and stream velocity in a nonlocal and noninstantaneous manner by means of time correlation kernels. For large times and small gradients, the transport currents are expanded in gradients of the local densities, yielding expressions for the Navier-Stokes and linear Burnett coefficients as matrix elements of the linearized Boltzmann collision operator. An Onsager relation between two linear Burnett coefficients is demonstrated.

Hard‐Square Solids at High Densities

Abstract: An asymptotic expansion for the Helmholtz free energy FN of a system of N hard squares of side σ is obtained, with the result FN / NkBT ≃ lim r→1 2 ln(λ / σ) − 2 ln(τ − 1) + C + D(τ − 1) + E(τ − 1)2 + ···, where kB is Boltzmann's constant, λ is the mean thermal de Broglie wavelength, and τ is the ratio of the system area to the close‐packed area. A formal expression is derived for the constant C, while D, E, ··· are the coefficients in the expansion of 2 ln(τ1/2 + 1) in powers of (τ − 1). The modified cell‐cluster technique is used to evaluate C, with the result C = − 0.26042240.

On the sign of successive time derivatives of Boltzmann's H function

Abstract: It has been conjectured in the literature that for the Boltzmann equation: (i) Boltzmann's H has the property of possessing successive (semi-) definite time derivatives with alternating sign, (ii) this property is suitable for selecting H from further possibly existing (Lyapunov) functions with (semi-) definite first time derivative. The authors show at first that, in contradiction to (ii), for Maxwell's model of a discrete velocity gas the corresponding variant of H is not the unique Lyapunov function having the property (i) and suitable to define a 'non-equilibrium entropy' with respect to the (spatially homogeneous) Boltzmann equation of this model. Secondly, in the case of the complete spatially inhomogeneous Boltzmann equation, H is generally not (semi-)definite in contradiction to the above-mentioned conjecture (i).

Estimates of Error Bounds of Gas Transport Properties

Abstract: Error bounds are calculated for transport properties that do not have the form − (f, L−1f), where L is the linearized semiclassical Boltzmann collision operator. The error bound generated depends on error bounds estimated for two other gas properties that do have this form. The method is illustrated for the thermal diffusion ratio of a gas mixture of mechanically similar rigid spheres and for the perturbation of the rate of change of temperature in a gas of reacting rigid sphere molecules. In the Appendix a rederivation is given of some general relations between diffusive and thermal fluxes.

Relativistic Boltzmann Theory and the Grad Method of Moments

Abstract: It has been the custom, when writing on relativistic Boltzmann theory, to justify such studies by recounting the various physical systems to which they can apply. By now one is sufficiently acquainted with relativistic plasmas, massive stellar systems and the like to make such justifications unnecessary. While applicability is of course the final justification for any physical theory, there is one other that I would like to mention briefly. It is the aesthetic appeal, so often emphasized by Dirac. The relativistic Boltzmann equation is both simple and elegant. From it one can obtain many beautiful results, such as those of Ehlers, Gerun and Sachs. It brings into play virtually the whole of relativity theory in one way or another and is amenable to analysis by such modern mathematical techniques as fiber bundle theory. It also affords a unifying view that is lacking in the classical theory, One need only compare the classical and relativistic treatments of radiative transfer theory, which is a special application of the Boltzmann equation to zero rest-mass particles to appreciate this fact. Finally I would mention that, even in the more mundane matter of finding approximate solutions, there are decided advantages to a relativistic treatment over the corresponding classical treatment.

Intermediate Operators and the Kinetic Theory of Gases

Abstract: A decreasing set of intermediate operators, L(N), are defined that are greater than the linearized Boltzmann operator in the sense that they correspond to collision models with a lower rate of entropy production With these operators upper bounds for transport properties may be calculated to supplement the lower bounds calculated by the usual variation method Two examples are given, both for the rigid‐sphere model One is the thermal conductivity, and the other, the perturbation to the reaction rate by the depletion of the faster moving molecules In the second instance, new lower bounds are calculated as well The extension to other collision models is indicated We also find that a measured transport property determines a mean relaxation frequency that must lie within the spectrum of relaxation frequencies of the fluctuation variable corresponding to the flux

Time Correlation Functions in a Binary Gas Mixture

Abstract: The low density forms of the correlation functions which yield the coefficients of mutual and thermal diffusion in a binary gas mixture are derived and examined from two points of view. One approach uses the binary collision expansion of the existing correlation function expressions, another uses the physical arguments given by Mori to extract these correlation functions from the Boltzmann equation by considering the evolution of the system from an arbitrary initial state. Both approaches indicate that the time correlation functions can be expressed in terms of single momentum averages of two dynamical functions which obey coupled integrodifferential equations of the Boltzmann type. These equations are solved by Sonine polynomial expansion and calculations are performed for a hard‐sphere gas mixture in order to illustrate the results. The correlation functions which characterize the thermal diffusion coefficient are discussed in some detail and several interesting aspects of the dynamics are pointed out.

Theoretical model for a nonequilibrium helium arc plasma

Abstract: A semiempirical model for a nonequilibrium helium arc plasma is used to interpret systematic deviations in existing spectroscopic data and to aid in computing nonequilibrium plasma transport properties. Two groups of bound electronic states are assumed to exist which follow Boltzmann distributions at Te (upper states) and TR ≠ Te (lower states). Under certain conditions, TR can be interpreted as a radiation temperature characterizing the optically thick part of the spectrum and can be calculated using a restricted Rosseland approximation. The nonequilibrium plasma is characterized by electron densities as much as several orders of magnitude greater than the local thermodynamic equilibrium value. The model reproduces the experimental data in detail once the electron temperature profile is given.

On the Stability Behaviour of Boltzmann's Equation

Abstract: The Boltzmann equation for an “isolated” gas system is assumed to form a “dynamical system” in a compact subset of the space of continuous distribution functions (existence assumption). Then the asymptotic stability in the sense of Lyapunov of the total Maxwell distribution is investigated (“approach to equilibrium”). Further the influence of persistent perturbations on the stability behaviour of Boltzmann's equation is considered (“structural stability”).

Fundamental relations in the theory of linear viscoelasticity under transient test conditions

Abstract: It is well known [8] that the viscoelastic parameters of polymer materials can be expressed uniquely by various functions of time and frequency. These functions are subjected to definite laws interrelating them as a consequence of the Boltzmann superposition principle, so that the applicability of viscoelastic characteristics is determined only by the type of tests performed on real objects. Conventionally, the test conditions have been steady-state, kinetic (relaxation or creep), and dynamic (periodically applied load or strain). The functions determined experfmentally here are, then, respectively the relaxation (or creep) function and the complex elasticity (or compliance) function, ff at least one of these functions is determined for the entire time or frequency range, then the viscoelastic spectra can be deduced from it completely.

Some Case of the Non-Uniform and Non-Steady State of Gas-Mixture,

Abstract: : In Part I, basing on the Boltzmann equations for a gas-mixture and the equations of conservation (mainly on the equation of momentum), the author obtains fundamental relations and properties defining the mean molecular quantities of an arbitrary constituent (i-gas) of some non-uniform and non-steady gas-mixture occupying the whole physical space. One assumes that every i-gas is monatomic, and its distribution-function is Maxwellian. Molecules are mass-points or perfectly rigid elastic smooth spheres. The purpose of Part 2 is the definition of the fundamental quantities and properties connected with the state of the whole gas-mixture, and a more detailed consideration of the equations.

On the classical approximation in the quantum statistics of equivalent particles

Abstract: It is shown here that the microcanonical ensemble for a system of noninteracting bosons and fermions contains a subensemble of state vectors for which all particles of the system are distinguishable. This “IQC” (inner quantum-classical) subensemble is therefore fully classical, except for a rather extreme quantization of particle momentum and position, which appears as the natural price that must be paid for distinguishability. The contribution of the IQC subensemble to the entropy is readily calculated, and the criterion for this to be a good approximation to the exact entropy is a logarithmically strengthened form of the usual criterion for the validity of classical statistics in terms of the thermal de Broglie wavelength and the average volume per particle. Thus, it becomes possible to derive the Maxwell-Boltzmann distribution directly from the ensemble in the classical limit, using fully classical reasoning about the distinguishability of particles. The entropy is additive—theN! factor of the Boltzmann count cancels out in the course of the calculation, and the “N! paradox” is thereby resolved. The method of “correct Boltzmann counting” and the lowest term of the Wigner-Kirkwood series for the partition function are seen to be partly based on the IQC subensemble, and their partly nonclassical nature is clarified. The clear separation in the full ensemble of classical and nonclassical components makes it possible to derive the classical statistics of indistinguishable particles from their quantum statistics in a controlled, explicit way. This is particularly important for nonequilibrium theory. The treatment of molecular collisions along too-literally classical lines turns out to require exorbitantly high temperatures, although there are suggestions of indirect ways in which classical nonequilibrium theory might be justified at ordinary temperatures. The applicability of exact classical ergodic and mixing theory to systems at ordinary temperatures is called into question, although the general idea of coarse-graining is confirmed. The concepts on which the IQC idea is based are shown to give rise to a series development of thermostatistical quantities, starting with the distinguishable-particle approximation.

Analysis of the Velocity Field in Plasmas from the Doppler Broadening of Spectral Emission Lines

Abstract: Publisher Summary An emission line broadened by the Doppler Effect represents the envelope of the unresolved displacements of the line caused by the motion of the radiating atoms or ions in the direction of observation. The motion may not be purely thermal but can, in general, be combined with mass motion that in itself can be of two forms—systematic and random. In principle, Doppler broadening provides a simple and fairly accurate means of measuring the temperature of an inaccessible radiating medium. Specifically, when the temperature of hot plasma is to be measured, the contribution to the broadening by mass motion may be considerable in view of the ability of plasma to sustain collective forms of motion. Apart from its general applicability, the kinetic temperature carries with it two other advantages. Although plasma does not behave as a perfect gas, the kinetic temperature scale requires no correction to bring it in line with the thermodynamic scale, provided k is selected as the Boltzmann constant.

Solution of boltzmann's equation for inelastic scattering

Abstract: Boltzmann's equation for electrons in a metal is solved, with inelastic processes included by the use of the Van Hove scattering function. Known results for phonon and impurity-limited conductivities are derived in a briefer and simpler way. A generalized formula is given which includes all possible resistivity effects in the weak-scattering limit. A closed-form expression for the resistivity due to isobaric impurities is derived as a special case, and reduces, in the Debye approximation, to a known result.

Conditions of applicability of the Boltzmann equation

Abstract: The existence of certain characteristic times, introduced by Bogolyubov [1], is of fundamental importance for the derivation of the Boltzmann equation from the Liouville equations. In the present paper characteristic spatial scales are also introduced, which permit a more detailed study of the influence of spatial gradients and boundary conditions. A convenient formalism, which is a generalization of the formalism of [2], is used in this study. The following has been shown for a Boltzmann gas (compare [1–4]): In this study we assume the asymptotic convergence of all expansions in a small parameter and the uniqueness of the solutions of all the equations encountered.

A Numerical Scheme For Free-surface Flow Modeling Based On Kinetic Theory

Abstract: Although most of numerical schemes based on shallow water equations work well for continuous surface flows, they are not suitable for flows containing vertical motions, such as the dam break problem, expansion and shock waves problems because they are based on the governing equations which have no effective mechanism to account for the vertical motion. To overcome this shortcoming, the authors of this paper apply the Boltzmann equation of gas kinetic theory to the surface water flow modeling and develop a set of equations governing surface water flows which incorporate the energy losses at the vertical motion. The test examples show that the Boltzmann equation based scheme has high resolution and high accuracy in modeling the discontinuities flows. In addition, Boltzmann based scheme is highly accurate in modeling the smooth surface flows.

A non-equilibrium kinetic description of shock-wave structure

Abstract: A formulation for the shock-wave structure is devised by the approximation of Boltzmann's equation by a simpler kinetic model. Initially, the distribution function in Boltzmann's collision integral is expressed in terms of a function of deviation from local equilibrium, the magnitude of which is unrestricted, and the analysis is specialized to hard sphere molecules. A model is then derived by assigning to the deviation function the first-order term of Chapman-Enskog's sequence which leads to Navier-Stokes equations. The model equation is shown to possess a description of a gas in a non-equilibrium state and to imply a Prandtl number value of $\frac{2}{3}$, the formulation also containing the Bhatnagar-Gross-Krook model as a special case. In applying the kinetic model to the shock problem, the collision frequency of the loss term is replaced by a set of mean frequencies (independent of the molecular velocity) each of which characterizes a specific macroscopic quantity. The shock equations are evaluated numerically for argon employing an interation scheme that is initiated by the Navier-Stokes solution. One iteration only to the flow variables is performed. For weak shocks the iteration proves to be in very close agreement with the Navier-Stokes solution for a Prandtl number of $\frac{2}{3}$; at higher Mach numbers, the iteration predicts a progressively larger deviation, especially in the temperature profile. In addition, the density and velocity profiles exhibit a 'kink' at Mach numbers 5 and 10. Unlike the Navier-Stokes predictions, the results also show that for high Mach numbers the total enthalpy within the shock no longer remains sensibly constant.

The Velocity Constants of Free Ions and Ion Pairs. I

Abstract: The specific velocity constants of free ion-molecule reactions and ion pair-molecule reactions are discussed. Assuming an equilibrium between solvated and desolvated reactive centers, the specific velocity constant of ion pair with a given charge-charge distance is expressed by where κ is the transmission coefficient, k the Boltzmann constant, T the absolute temperature, h the Planck constant, [Si]a the concentration of the solvent i, K‡ the equilibrium constant of activation, and KM and ki the equilibrium constants for solvation by the monomer M and by the solvent i, respectively. The KM and Ki are derived from eletrostatic free energies for solvation. The apparent specific velocity constant is obtained by averaging the k's with respect to the charge-charge distance. The ratio of the reactivity of free ions and that of ion pairs is further calculated. Agreement with experiment is satisfactory.