Showing papers on "Boltzmann constant published in 1973"

Free‐volume model for ionic conductivity in polymers

Abstract: Electrical conduction in polymers under a relatively low applied electric field is considered to be ionic and is affected strongly by the structural factors of the polymers. The following equation for the electrical conductivity σ was derived in which free volume Vf, jump energy Ej, and ionic dissociation energy W were taken into consideration: σ=σ0exp{− [γ Vi*/Vf+(Ej+W/2e)(kT)−1]}, where σo is a constant, γ the numerical factor to correct the overlap of free volume, Vi* the critical volume required for transport of an ion, e the dielectric constant, k Boltzmann's constant, and T the absolute temperature. This equation describes well the conduction phenomena in polymethylmethacrylate, polystyrene, and an unsaturated polyester. Relationships between electrical conduction and free volume are discussed.

Quantum transport theory of high-field conduction in semiconductors

Abstract: Resolvent super-operator techniques are used to establish a formal expression for the exact nonlinear response of nondegenerate semiconductor electrons to an arbitrary high dc electric field in the presence of arbitrary phonon and impurity scattering The theory is based on the Liouville equation and is exact in the thermodynamic limit Infinite-order perturbation expansions of the field-dependent resolvent lead to a simple kinetic equation for the one-electron density matrix The steady-state Boltzmann high-field transport equation is derived exactly under the assumptions of weak scattering and point collisions, that is for zero collision broadening and neglecting the influence of the field within the collisions Collision broadening corrections are derived for arbitrary scattering processes

The Development of Boltzmann’s Statistical Ideas

Abstract: In 1866 Boltzmann began his scientific career with an attempt to give a purely mechanical explanation of the second law of thermodynamics. He gradually recognized the need to introduce statistical concepts in order to understand irreversibility and the second law. This paper follows the development of Boltzmann’s views, tracing the increasingly important role that he assigned to statistical concepts. Boltzmann’s critics played an important part in forcing him to clarify his own thinking about statistical mechanics, and the successive criticisms made over a period of a quarter of, a century are described and analyzed. There is also an attempt to estimate the degree to which Boltzmann’s ideas were generally understood and accepted by his contemporaries.

The Boltzmann equation: theory and applications

Abstract: Welcome Speech- Life and Personality of Ludwig Boltzmann- Philosophical Biography of Ludwig Boltzmann- The Development of Boltzmann's Statistical Ideas- The Validity and the Limitations of the Boltzmann Equation- Comment to Professor Uhlenbeck's Paper- On Higher Order Hydrodynamic Theories of Shock Structure- Sound From the Boltzmann Equation- The Generalization of the Boltzmann Equation to Higher Densities- The Three-Particle Collision Term in the Generalized Boltzmann Equation- Velocity Correlation Functions for Moderately Dense Gases- On Kinetic Equations for Particles with Internal Degrees of Freedom- Flow-Birefringence in Gases, An Example of the Kinetic Theory Based on the Boltzmann Equation for Rotating Particles- Non-Equilibrium Angular Momentum Polarization in Rotating Molecules- The Boltzmann Equation in Solid State Physics- Experimental and Theoretical Investigations in Semiconductors Concerning the Boltzmann Equation- Some Probabilistic Aspects of the Boltzmann Equation- The Statistical Interpretation of Non-Equilibrium Entropy- A Review of Computer Studies in the Kinetic Theory of Fluids- A Survey of Neutron Transport Theory- Relativistic Boltzmann Theory- Kinetic Equation for Elementary Excitations in Quantum Systems- Ergodic Theory- Ergodic Theory- Ergodic Theory and Approach to Equilibrium for Finite and Infinite Systems- Erinnerungen an Boltzmanns Vorlesungen

Stefan‐Maxwell relations for multicomponent diffusion and the Chapman‐Enskog solution of the Boltzmann equations

Abstract: It is established that corrections factors for higher order Chapman‐Enskog approximations to multicomponent diffusion coefficients are equal for a given N‐component gas mixture. This enables us to invert the mass flux equations for the diffusion forces and obtain the Stefan‐Maxwell equations in any Chapman‐Enskog approximation. The impedance matrix for these equations is symmetric regardless of whether or not the multicomponent diffusion coefficients are symmetric. Further, the impedance matrix is the same whether one chooses to invert mass flux equations involving symmetric or asymmetric diffusion coefficients.

Kinetic Approach to Neural Systems

Abstract: A kinetic model of neural systems is introduced and discussed with statistical mechanics techniques. It is assumed that, for a macroscopic description of the model, it suffices to consider only the distribution for the velocity and position of the impulses, and the distribution for the excitation and position of the neurons, at any timet. Making use of Boltzmann's method for the study of a dilute gas, coupled differential equations for the rate of change with time of the distributions have been constructed.

Instantaneous electron energy distribution function in ion waves

Abstract: The assumptions involved in the use of Boltzmann's law to describe the electron density variations in ion acoustic waves are discussed. The need for electron collisions to fill the regions of maximum potential with low‐energy electrons is pointed out. A direct method for measuring the electron energy distribution at any time during the period of an ion wave is presented. Experimental results show the existence of electrons with energy small enough for their motion to be constrained to the wave potential well. For higher energies, the distribution stays approximately the same. The method provides a direct measure of plasma potential, which is related to density. The Boltzmann law is still seen to be accurate for relative fluctuations of the order of up to 20%.

An Experiment to Measure Boltzmann's Constant

Abstract: We describe a simplification of Jean Perrin's classic experiment to determine Boltzmann's constant from the sedimentation equilibrium of colloidal suspensions. Perrin's complicated procedure for preparing suitable colloidal particles is avoided by using commercially available plastic spheres of specified diameter and density. The experiment is suitable for either an introductory or advanced laboratory.

The Statistical Interpretation of Non-Equilibrium Entropy

Abstract: Boltzmann’s original scheme leading to the statistical interpretation of non-equilibrium entropy may be summarized as follows: Dynamics → Stochastic Process (kinetic equation) → Entropy. Recent computer experiments as well as spin echo experiments in dipolar coupled systems illustrate clearly the difficulties in Boltzmann’s derivation. Indeed, they display situations for which a Boltzmann type of a kinetic equation is not valid. The main purpose of this communication will be to show that we can now construct a more general microscopic model of entropy which shows the expected monotoneous approach to equilibrium even in non-Boltzmannian situations such as experiments involving “negative time evolution”.

Fundamental equations of a mixture of gas and small spherical solid particles from simple kinetic theory.

Abstract: The fundamental equations of a mixture of a gas and pseudofluid of small spherical solid particles are derived from the Boltzmann equation of two-fluid theory. The distribution function of the gas molecules is defined in the same manner as in the ordinary kinetic theory of gases, but the distribution function for the solid particles is different from that of the gas molecules, because it is necessary to take into account the different size and physical properties of solid particles. In the proposed simple kinetic theory, two additional parameters are introduced: one is the radius of the spheres and the other is the instantaneous temperature of the solid particles in the distribution of the solid particles. The Boltzmann equation for each species of the mixture is formally written, and the transfer equations of these Boltzmann equations are derived and compared to the well-known fundamental equations of the mixture of a gas and small solid particles from continuum theory. The equations obtained reveal some insight into various terms in the fundamental equations. For instance, the partial pressure of the pseudofluid of solid particles is not negligible if the volume fraction of solid particles is not negligible as in the case of lunar ash flow.

A Comment on Boltzmann's H-Theorem and Time Reversal

Abstract: The standard textbook discussion of Boltzmann's H-theorem is analyzed and found unsatisfactory. An improved presentation is suggested.

High Field Transport in n-InSb at 77 °K and the Solution of Boltzmann's Equation by Iteration

Abstract: The solution of Boltzmann's equation for n-InSb at high fields and 77 °K is obtained by Rees' iterative method. The time independent solution obtained is in agreement with a recent Monte-Carlo calculation by Fawcett and Ruch and experimental results by Neukerman and Kino. A comparison between the Boltzmann solution and the drifted Maxwellian approach previously discussed by the authors is also presented. On a calcule la solution de la Beuation de Boltzmann pour n-InSb dans les hauts champs electriques par la methode iterative de Rees. La solution qui est independant du temps s'accorde avec un calcul Monte-Carlo par Fawcett et Ruch et avec les resultats experimentaux par Neukman et Kino. Une comparison entre la solution Boltzmann et la fonction de distribution de Maxwell emporte dans l'energie est donnee.

On Kinetic Equations for Particles with Internal Degress of Freedom

Abstract: Already in his famous 1872-paper Boltzmann even included polyatomic dilute gases. More recently this question has been reconsidered from the quantum theoretical point of view. The simplest model in this context is a gas of particles with spin, which has to be described by a one particle distribution matrix with 2 S + 1 rows and columns (S = value of spin). The H-theorem can be traced back to the unitarity of the scattering matrix for the binary collision, suitably occurringin the kinetic equation. A special spin-dependent scattering amplitude is considered for which the classical limit can easily be effectuated. In conclusion some remarks on the classical Boltzmann equation for rigid dumb-bells are made.

The Measurement of e/k in the Introductory Physics Laboratory

Abstract: The ratio of electronic charge to Boltzmann's constant can be easily determined by measuring the short-circuit collector current versus the emitter-base voltage characteristics of a silicon transistor connected in the common-base mode. The incorporation of this experiment in the introductory physics laboratory is recommended.

The numerical solution of Boltzmann's equation

Abstract: FOR the numerical solution of Boltzmann's kinetic equation it is proposed to approximate the collision frequency and the integral of reverse collisions in the cells of velocity space. The equation with the approximated collision integral is solved by an iterative method. The differential equations obtained are integrated along the characteristics in such a way that it is not necessary to remember large blocks of distribution functions in the course of the calculation. The results of a calculation of Couette's problem are given. The data obtained are compared with the results of other authors. Recently several different approaches to the calculation of rarefied gas flows have been proposed: statistical simulation [1–3], and direct solution of Boltzmann's equation [4, 5]. The kinetic simulated equations have become especially widely used [6–9]. A detailed collection of the existing numerical and analytic methods of solving Boltzmann's equation is given in the monograph [10]. Methods for the direct solution of Boltzmann's equation existing at the present time [4, 5] enable results of low accuracy to be obtained, since they require a large expenditure of computer time and a large operative computer memory. This also hinders an experimental check of the accuracy of the solution by varying the different constants characterising the accuracy of the method, since the calculations are usually performed to the limit of the possibilities of the computer. As a rule only a comparison of the results obtained by different methods can describe the accuracy of the solutions. The greatest difficulties in the solution of Boltzmann's equation are connected with evaluating the fivefold collision integral and with the necessity to store in the operative memory large blocks of distribution functions, which for one-dimensional problems depend on four variables. The use of all the prior information about the solution in the design of the method of calculation could help to obtain it with fewer difficulties. To evaluate the collision integral in this paper it is proposed to divide the velocity space into cells and in each of them to approximate the collision frequency and the integral of the reverse collisions. If a good enough approximate form can be chosen from physical considerations, the size of the cell in the velocity space can be taken to be fairly large. In the approximation of the collision integral only the coefficients of the approximate expressions have to be stored in the operative memory, therefore with the small number of cells the demands on the operative memory are considerably reduced. It is proposed to calculate the coefficients of the approximating expressions in such a way that the first n moments of the approximating expressions will be identical with the moments of Boltzmann's collision integral ( n is the number of coefficients in the approximating expressions). The collision integral can be evaluated in such a way that the conservation laws will be satisfied for any number of cells in velocity space. It may be hoped that the solution obtained will differ little from the exact solution of Boltzmann's equation for a sufficiently small number of cells, if in the subdivision of the velocity space into cells the structure of the flow has been correctly taken into account in both the free-molecular and the continuous limits. An iterative method is used to solve the equation with the approximated collision integral. It is proposed to solve the differential equations obtained along characteristics in such a way that finding the solution would not require large blocks of distribution functions to be remembered. As a test of the method the problem of the Couette flow was solved. This flow has been previously simulated by the Monte Carlo method in a linear approximation [11] ; the non-linear case for Boltzmann's equation was considered in [5], where the profile of the gas velocity was given for a relative velocity of the plates equal to 2 M for an extremely rarefied gas ( K = 4), calculated with low accuracy. Calculations carried out show that already with the subdivision of the velocity space into two half-spaces (two cells in the velocity space) with the chosen method of approximation the macroparameters obtained give a fairly accurate solution of Boltzmann's equation. Increasing the number of cells to 4 and 32 had a weak effect on the results of the calculation. The proposed method enables the computing time and the size of the operative memory of the computer required to be considerably reduced as compared with the methods described in [4, 5], and at the same time the accuracy of the calculations is increased.

Verification of the nature of the approximation of the Boltzmann integral by Krook's kinetic relaxation model

Abstract: Up to now a significant number of aerodynamic problems have been solved with the aid of Krook's kinetic relaxation model. However, because of the absence of reliable solutions of boundary problems for the Boltzmann equation, the correctness of the assumed model of the collision integral remains unclarified. In the present paper, in order to verify the nature of the approximation of the collision operator by the given model, a machine experiment is undertaken. The Boltzmann collision operator is computed for a variety of test functions characteristic of the motion of a rarefied gas and the values obtained according to it are compared to the Krook model. Some physical hypotheses embedded in the relaxation model are also examined.

Distribution of diffusing particles near an absorbing wall

Abstract: On the basis of Boltzmann's kinetic equation we obtain and solve numerically the exact diffusion integral equation for the distribution of particles which is valid throughout the whole region of transition from macroscopic to kinetic description. It is shown that in spite of strong distortion of the angular part of the distribution function, the behavior of the density at the wall varies little by comparison with the diffusion distribution.

Relativistic Boltzmann Theory

Abstract: In the last thirty years a relativistic kinetic theory of transport processes has been developed, which is based on an appropriate generalization of Boltzmann’s celebrated equation. The relevant macroscopic laws could be derived in the framework of this theory. Moreover transport coefficients could be calculated, according to various procedures: (a) the eigenvalue method, (b) the simplified collision description, (c) the moment method and (d) the orthogonal functions method.

The solution of transfer theory problems in spherical geometry

Abstract: THE method of integral transforms is applied to the solution of problems of the one-speed theory of neutron transfer with isotropic scattering in spherically symmetrical inhomogeneous media. Functions are obtained which have the meaning of elementary solutions of Boltzmann's equation in spherical geometry.

Kinetic Approach to Non-Equilibrium Phenomena

Abstract: The kinetic approach of Boltzmann to non-equilibrium phenomena in dilute gases is briefly discussed. Bogolubov’s generalization of Boltzmann’s method to dense gases is presented. The essential difficulties encountered in this attempt are exhibited and the impossibility of a straight-forward generalization of the Boltzmann equation to higher densities is concluded. Some hydrodynamical consequences of this are mentioned.