Showing papers on "Boltzmann constant published in 1975"

Semiclassical limit of quantum mechanical transition state theory for nonseparable systems

Abstract: The semiclassical limit of quantum mechanical transition state theory is derived by invoking the classical path approximation for the Boltzmann density operator and making use of the stationary phase approximation; separability of motion along a reaction coordinate is not assumed. The resulting expression for the rate constant bears an interesting similarity to that of conventional transition state theory, although all quantities in it refer to the full classical dynamics on the potential energy surface. In place of the vibrational frequencies of the ’’activated complex’’ which appear in the conventional theory, for example, the semiclassical expression contains characteristic frequencies related to the stability properties of a periodic classical trajectory. Conservation of total angular momentum is easily accounted for in a rigorous manner so that the semiclassical model can be applied to three−dimensional dynamical systems.

Exact solutions of the Boltzmann equation

Abstract: The exact solutions of the Boltzmann equation in explicit form are found. The treatment is confined to Maxwellian molecules with interaction potential U(r) = alpha/r/sup 4/. (JFP)

The Linearized Boltzmann Collision Operator for Cut-Off Potentials

Abstract: Boundedness and compactness of integral operators arising from the linearized Boltzmann collision operator are investigated for a wide class of angular and radial cut-off potentials.

Temperature relaxation in a binary gas. II. Time dependent solution

Abstract: The time dependent Boltzmann equations describing the temperature relaxation in a hard‐sphere gas are solved and the explicit time dependence of the distribution functions is determined. The time evolution of the perturbations of the distribution functions is studied in detail with a moment method. The range of validity of the earlier results obtained with a steady state assumption is determined. Provided that the intial temperature ratio is neither too large nor too small (102–10−2), it is found that the steady state assumption is valid only in the extreme disparate mass limit, i.e., m1/m2 of the order of 10−3–10−5. Qualitatively it appears that the ratio of the self‐relaxation times to the temperature equilibration time must be of the order of 10−3–10−4 or smaller for a steady state to occur. Since the temperature relaxation rate is slow for this range of mass and initial temperatures, there is an extremely small perturbation of the distribution functions from the Maxwellian form. Perturbations of the d...

Formal study of a chemical reaction by Grad expansion of the Boltzmann equation. I

Abstract: The Boltzmann equations for a formal bimolecular chemical reaction in homogeneous gas phase are transformed into an infinite system of quadratic differential equations, by expanding the distribution functions of the molecules into the Grad polynomials. The properties of these expanded Boltzmann equations reflect the macroscopic laws. In particular they enable the Onsager reciprocity relations to be derived from time-reversal invariance.

Complexity and Information Theory

Abstract: The concept of “information” appeared in Physics in connection with the concept of “entropy”. It was observed (Boltzmann, 1896) in the framework of statistical thermodynamics, that the entropy is proportional to the logarithm of the number of alternatives (or microscopic states) which are possible for a physical system knowing all the macroscopic information about it. The entropy gives, in other words, a measure of the total amount of missing information about the system.

The self-diffusion constant for large heavy particles

Abstract: We construct a closed-form expression for the self-diffusion constant, D, for a hard-sphere particle whose mass and radius are large compared to the corresponding bath-particle quantities. The expression yields the Stokes-Einstein law at high bath-particle densities and the Boltzmann form for low densities. In addition, the first density correction to D is obtained and the higher-order density corrections are shown to diverge. The second density correction diverges as −log(k0R), where k0 is a cutoff wavevector and R is the radius of the particle.

Stimulated electron-phonon-photon interactions in nondegenerate semiconductors considering energy-dependent carrier relaxation time

Abstract: The energy−dependent relaxation time has been incorporated in the study of stimulated electron−phonon−photon interactions in nondegenerate semiconductors in the presence of the external dc electric field. The Boltzmann transfer equation has been used to obtain the gain constant. The results show that the percentage change in the gain constant is ∼15% (for ql<1, Eac=3 V/cm, Ω/ω=0.1, 〈vd〉=1.5 vs) and ∼30% (for ql≳1, Eac=30 V/cm, Ω/ω=2, 〈vd〉=2.0 vs) for moderate strengths of oscillating electromagnetic field (OEF). For weak OEF (Eac<ωvs/Ωμ for ql<1 and Eac

Implications of radiative equilibrium in neoclassical theory

Abstract: We discuss some implications of the description of spontaneous emission of radiation which has been offered by Jaynes and his collaborators in their "neoclassical" extension of semiclassical electrodynamics. In particular, we examine the thermal-equilibrium condition of radiation interacting with a tenuous gas of atoms. We argue that rate equations may be used to describe the interaction of such atoms with the chaotic thermal radiation field. For this situation the neoclassical spontaneous emission rate is incompatible with the well-secured laws of Boltzmann and Planck. Experimental evidence bearing on the accuracy of those laws as well as on the accepted level population dependence of the induced emission rate is reviewed.

Existence of weak shock wave solutions for higher−order theories of gas dynamics

Abstract: The existence of weak shock wave solutions for the higher order theories of gas dynamics generated by the Chapman−Enskog development of Boltzmann’s equation is proved.

Exchange symmetry and the quantum Boltzmann equations

Abstract: The quantum Boltzmann equations in the kinetic theory of a dilute gas are based on an assumed relation between the first and second order distribution functions. As usually formulated, this relation is inconsistent with exchange symmetry requirements. This paper discusses the changes which occur in the quantum Boltzmann equations and their collision integrals when the relation is modified to be consistent with the symmetry requirements for exchanges of whole molecules.

Mathematical aspects of kinetic model equations for binary gas mixtures

Abstract: A system of integrodifferential equations, which has a structure similar to the Boltzmann equations for a binary gas mixture and which qualitatively describes wave propagation, is investigated. The Oppenheim model is used and a linear initial−value problem is considered. The initial−value problem is shown to be well set mathematically with certain specifications on the initial distribution functions. Justification is made for the use of Fourier−Laplace transforms. A discussion is made of the dispersion relation and its analytic continuation. The roots σ (k) of the dispersion relation are shown to lie in three distinct regions of the σ plane: the hydrodynamic region, the semihydrodynamic region, and the rarefied region. It is established that the roots σ (k) are bounded by −1 + δ < Reσ ⩽ 0 under the assumption of plane−wave solutions which implies that the system is stable and that plane waves cease to exist if Reσ ⩽ −1 + δ.

Transport properties of disparate mass binary gases from a re-ordering of collision terms

Abstract: The present work extends an earlier study of consequences following from a re-ordering procedure applied to the collision integrals in the coupled Boltzmann equations for a binary gas. The re-ordering is that originally suggested by Grad for describing disparate-mass mixtures, and allows independent species temperatures and flow velocities. First order expressions for the heat flux and pressure tensor are obtained, and some evaluation procedures are discussed. It is shown that the approach leads to a physically unreasonable prediction for the coefficient of thermal conductivity, whereas the classical Chapman-Enskog prediction agrees with a simple mean-free path-estimate. This negative conclusion is shown to be unaffected by second-order corrections.

Note on Boltzmann-Langevin equation and hydrodynamic fluctuations

Abstract: A microscopic derivation presented of the generalization of the linearized Boltzmann equation with the Langevin fluctuation force, which has earlier been postulated by Bixon and Zwanzig and by Fox and Uhlenbeck in their kinematical discussions on the hydrodynamic fluctuations.

Derivation of equations of motion of a slightly rarefied gas around highly heated bodies from boltzmann's equation

Abstract: The motion of a slightly rarefied gas (K ≪ 1, where K is the Knudsen number) around highly heated bodies is examined. On the assumption that the characteristic macroscopic velocity of gas motion generated during contact with a highly heated body is on the order of or much greater than the velocity of the impinging stream, the corresponding hydrodynamic equations are derived from Boltzmann's equation by Hubert's method [1]. A qualitative study is made of the region of applicability of the equations obtained. A class of flows of a continuous medium in which the characteristic change in enthalpy is much larger than the characteristic kinetic energy was studied in [2]. The Navier-Stokes equations with boundary conditions of adhesion proved to be inadequate for a description of these flows since it was already necessary in the first basic approximation to take into account part of the Barnett terms and slippage. The authors of [2] suggest using simplified Barnett equations with the condition of creep, with the Barnett terms being on the same order as the inertial and Navier-Stokes terms. On the other hand, it is known that the Barnett equations are derived on the assumption that the additional terms are small in comparison with the Navier-Stokes and Eulerian terms. This makes it desirable to obtain equations describing this class of flows directly from Boltzmann's equation.

Temperature dependence of total free energy of activation for dislocation motion. I. Copper crystals after electron irradiation

Abstract: Thermodynamical analysis is made of plastic deformation to obtain the total reversible isothermal work (free energy) to overcome obstacles. Experiments were carried out to obtain the total free energy through this analysis, and revealed only two stage spectra in the obstacle strength, with transition region in between, after room temperature electron irradiation on copper. Corresponding spectra were also observed in the temperature dependence of the activation energy. A discussion is given in the light of spectrum theory of hardening. The total free energy is found to decrease with increasing electron dose, in a sharp contrast to the hardening after pile-neutron irradiation, which has been considered to be due to the Frank type dislocation loops. Activation entropies of several times of Boltzmann's constant at room temperature, tending to zero at 0 K, are obtained from the total free energies and compared with the analysis of Schoeck. Es wird eine thermodynamische Analyse der plastischen Deformation durchgefuhrt, um die gesamte reversible isotherme Arbeit (freie Energie) fur die Uberwindung von Hindernissen zu erhalten. Experimente wurden durchgefuhrt, um die gesamte freie Energie durch diese Analyse zu erhalten und ergaben an Kupfer nach Elektronenbestrahlung bei Zimmertemperatur nur Zwei-Stufen-Spektren der Hindernisstarke mit einem Ubergangsgebeit dazwischen. Entsprechende Spektren wurden auch in der Temperaturabhangigkeit der Aktivierungsenergie beobachtet und mit der Spektrentheorie der Aushartung diskutiert. Es wird gefunden, das die gesamte freie Energie mit zunehmender Elektronendosis abnimmt, im starken Gegensatz zur Aushartung nach Reaktor-Neutronenbestrahlung, was auf Franksche Versetzungsschleifen zuruckgefuhrt wird. Aktivierungsentropien in der Grose einiger Boltzmannkonstanten bei Zimmertemperatur, die bei etwa 0 K gegen null gehen, werden aus der gesamten freien Energie erhalten und mit der Analyse von Schoeck verglichen.

Derivation of the “fundamental” equation of radiation kinetics

Abstract: A large number of complex experimental cases are considered which in recent years have been studied with the aid of the admittedly simplified apparatus of the Holstein theory. The fundamental Holstein equation is derived, starting with a system of coupled Boltzmann equations for the photons and the density of excited states. The derivation process provides opportunities for obtaining some intermediate kinetic equations with broader ranges of applicability than the Holstein equation. Some remarks are made concerning the combined transport in space of photons and excited atoms.