Showing papers on "Boltzmann constant published in 1976"

Statistical mechanics of chemical equilibria and intramolecular structures of nonrigid molecules in condensed phases

Abstract: An exact classical statistical mechanical theory is developed which describes how intermolecular forces alter the average intramolecular structures of nonrigid molecules and how these forces affect the equilibrium constant of chemically reacting species. The theory lends itself to computationally convenient approximations. Three illustrative applications of the theory are given. First, the equilibrium constant for the reaction N2O2?2NO2 is studied. Second, the shift in the chemical bond lengths of N2 and O2 from their gas phase values to those in the liquid are calculated from the theory. Third, the average conformational structure of n‐butane in various dense fluid solvents is predicted. The basic methods used in the derivation of the theory are the techniques of physical cluster series (as opposed to the usual Mayer ’’mathematical’’ cluster expansions), and topological reductions. The formalism gives rise to a renormalization of chemical bonding Boltzmann factors so that the theoretical expressions are not swamped by essentially infinite and unnormalized cluster functions. Along with the results on chemical equilibria and intramolecular structures, the interaction site cluster series for intermolecular correlation functions is rederived in a more general manner than the derivation originally presented by Ladanyi and Chandler.

Duality transformation in a many‐component spin model

Abstract: It is shown that the duality transformation relates a spin model to its dual whose Boltzmann factors are the eigenvalues of the matrix formed by the Boltzmann factors of the original spin model. The duality relation valid for finite lattices is obtained, and applications are given.

Use of generalized orthogonal polynomial solutions of Boltzmann’s equation in certain aeronomy problems: Auroral ion velocity distributions

Abstract: We present a method of solution of Boltzmann's equation that is valid for conditions where the species velocity distribution function departs significantly from a Maxwellian. This method should be particularly useful for problems in aeronomy and space physics where there are large or rapidly varying forces acting on a neutral or ionized gas or for problems dealing with highly rarefied gases where the mean free path for collisions is comparable to or greater than the characteristic scale length of the problem. The method of solution, which was first introduced by Mintzer, corresponds to expanding the species distribution function in a generalized orthogonal polynomial series with an arbitrary weight factor. The specific form of the weight factor depends on the details of the problem. The method is illustrated by calculating auroral ion velocity distributions for a relaxation collision model, and the results are compared with the velocity distributions obtained from an exact solution to Boltzmann's equation.

On relativistic kinetic gas theory XV. The Ritz method in relativistic Boltzmann theory

Abstract: The transport coefficients of a multicomponent gas mixture are approximated by means of a variational technique. It is shown that the procedure is equivalent to a more heuristic method developed earlier.

On the Navier-Stokes equations with constant total temperature

Abstract: For various applications in fluid dynamics, it is assumed that the total temperature is constant. Therefore, the energy equation can be replaced by an algebraic relation. The resulting set of equations in the inviscid case is analyzed. It is shown that the system is strictly hyperbolic and well posed for the initial value problems. Boundary conditions are described such that the linearized system is well posed. The Hopscotch method is investigated and numerical results are presented.

Momentum of a fast particle in a dissipative medium with an excitation threshold

Abstract: Using a quantum mechanical non-equilibrium formalism, the momentum distribution of an electron is evaluated for various times after it has been inserted with a given momentum into a dissipative medium. The medium is assumed to have an excitation threshold, and so might represent plasmons or optical phonons. The initial momentum of the electron is taken to be large, allowing an approximation based on the neglect of recoil correlations to be used. The distributions obtained are compared with those resulting from the Boltzmann equation, and the effects of virtual processes and of recoil correlations illustrated. At short times, considerable differences from the Boltzmann predictions are seen. At longer times, the distributions approach the Boltzmann predictions if the coupling of the electron to the medium is weak.

The broadening of the calcium resonance line in a high-temperature helium atmosphere

Abstract: The paper describes experimental measurements of the shape of the Ca I resonance line at 4227 A in a high-temperature helium atmosphere. A ballistic piston compressor was used to produce hot helium in the temperature range from 3000 to 7000 K and the number-density range from 2 to 4 by 10 to the 20th power per cu cm, which conditions approximate those postulated for the atmospheres of certain cool white-dwarf stars. The Boltzmann temperature of the calcium atoms was measured by the brightness-emissivity method, the absorption line shape was measured with a twelve-channel polychromator, and the helium kinetic temperature was calculated from the equation of an ideal gas. The observed deviation from thermodynamic equilibrium is illustrated. The value of the ratio of the damping constant to the helium number density at 5000 K is found to be 45 (+ or - 4) by 10 to the -22nd power A cu cm, or 4.7 (+ or - 0.4) by 10 to the -9th power rad/s cu cm. It is noted that no existing line-broadening calculation for the calcium-helium system is consistent with the present results. Recent studies of the 4227-A line in spectra of the sun and the white dwarf van Maanen 2 are discussed.

A transformation property of the Boltzmann equation

Abstract: where t denotes the time, x< ,i = 1, 2, 3, denote the spatial coordinates, and c{ ,i = 1, 2, 3, denote the components of velocity. The usual summation convention is used, so that the repeated index i in (1.1) is summed from 1 to 3. The notation / = f(t, , cj is the phase density function whose value represents the number density of the gas molecules on the phase space at the time t. Specifically, if 5D is any domain in the phase space, then the expected number of gas molecules with their phase coordinates belonging to 3D at the time I is given by the integral

Kinetic Theory of Gases-I

Abstract: The kinetic theory of the classical perfect gas (largely due to Maxwell and Boltzmann) was the first statistical theory to be introduced into physics. The equilibrium results of kinetic theory can now all be obtained as special cases of the general results of statistical mechanics obtained in chapter 7 but the detailed nature of the kinetic theory can give added insight into processes such as the pressure exerted by a gas on the wall of a container. The kinetic theory approach can also be extended to non-equilibrium situations and, as shown in the next chapter, may be used to calculate the transport coefficients (thermal conductivity, coefficient of viscosity, etc.) of a perfect gas. The kinetic theory of the perfect gas is of great importance in applied physics because the results are directly relevant to real gases at low pressure as in vacuum systems, the upper atmosphere and space research.

An Empirical Form for the Orientation-Ordering Intermolecular Potential in Para-Azoxyanisole

Abstract: We demonstrate here that the angle-independent factor of the orientation-ordering intermolecular potential (OIP) U in para-azoxyanisole (PAA) can be well represented by an equation of the form: u = A/T - B, where A and B are positive and independent of temperature T but dependent on pressure P; u is defined as U/kT, where k is the Boltzmann constant. This form for u has been derived from experimentally observed temperature dependence of u at a constant P. In such a determination of u, use has been made of the long-range orientational order parameter Q measured by an NMR technique and a universal u-Q relation. This u-Q relation, formally similar to the one that can be derived from the Maier-Saupe theory, has been obtained from a phenomenological theory and the intermolecular interaction treated under the molecular field approximation. We apply this empirical form for u in understanding a few experimental results in PAA, and show that reasonable agreements can be obtained if certain other facts are...

Theory of kinetic equations for systems with production and absorption of particles

Abstract: This paper is concerned with systems which are not in thermal equilibrium because of production and absorption of particles. On the basis of a new Hamiltonian describing such a nonequilibrium system, we develop a method for deriving kinetic equations for singlet density, time correlations of density, etc., including all the higher order interactions necessary to describe the production and absorption of particles. The foundations of the Boltzmann and Langevin equations for neutron distributions are studied. The time-correlation function is shown to obey a kinetic equation identical to that for the singlet density. It is also shown that the description of density fluctuations based on the Langevin equation is equivalent to the simplest decoupling of the rigorous hierarchy of equations for correlation functions.

Wave propagation in a binary gas mixture as described by kinetic models of the Boltzmann equations

Abstract: The initial‐value problem defined by the linear one‐dimensional Oppenheim model, which qualitatively describes wave propagation in a binary gas mixture and which is representative of other kinetic models, is examined. An asymptotic study of the dispersion relation and its analytic continuation is carried out to determine the structure of the solution.

A Boltzmann-type equation in the kinetic theory of vehicular traffic

Abstract: Abstract In the theory of vehicular traffic flow on a highway the traffic interaction process is often considered as a collision similar to the particles' interaction in the kinetic theory of gases. This concept leads to a Boltzmann-type nonlinear intergro-differential equation which governs the traffic density function. The purpose of this paper is to present a constructive method for the determintation of a solution for this type of equation under certain boundary and initial conditions. Our method is by successive approximation which yields existence of both global and local solutions of the problem.