Equilibrium Pair Research Articles

Spin-lattice relaxation and molecular reorientations near Tc with application to NH4Cl

Reorientations of the ammonium tetrahedra around C2 axes in the neighborhood of the structural phase transition in a NH4Cl crystal are described by a master equation. Within each group of two sterically different tetrahedral configurations characterized by a pseudospin variable σ=± 1 distinction is made of a=12 sterically equivalent positions. Cooperative dynamical behavior is introduced by extending Glauber's theory of the dynamics of an Ising system to the tetrahedral motion in molecular crystals. The (approximate) solution of the master equation yields a critical slowing down behavior for the pseudospin correlation function. This explains quadrupole spin-lattice relaxation T1Q experiments. In addition the decay time τ for the probability P(σa;t) of finding a tetrahedron in a given position σ, α∈ 12 is found to depend on the equilibrium pair correlation function g=<σiσi+1>. By relating this quantity to the thermal expansion, the proton spin-lattice relaxation time T1P can be expressed in terms of the anomalous thermal expansion. The theory explains the anomalous behaviour found experimentally for T1P around the phase transition at low pressure and high pressure.

A Dynamic Model of Firm Entry: Comment

Extending Quandt and Howrey's analysis of firm entry, Myers and Weintraub [1] proved local stability of the equilibrium pair of number of firms and labour hire. This note is intended to derive conditions for global stability of the equilibrium in Myers-Weintraub model. We adopt Myers-Weintraub's notation for which the reader is referred to [1]. Labour per unit of capital (or per firm) and number of firms producing a homogeneous good change according to

Electron Correlations at Metallic Densities. V

In this paper we present a modification of an earlier theory of Singwi et al. of electron correlations at metallic densities. The modification consists in allowing for the change of the pair correlation function in an external weak field via the density derivative of the equilibrium pair correlation function. This results in a new expression for the local-field correction. The present theory has the merit of satisfying almost exactly the compressibility sum rule and of giving a satisfactory pair correlation function. Results of self-consistent numerical calculations for the static pair correlation function, correlation energy, compressibility, and plasmon dispersion relation for the electron liquid in the metallic-density range are presented. For those interested in the application of the results of the present paper, numerical values of the local-field correction as a function of wave number have been tabulated in the density range ${r}_{s}=1\ensuremath{-}6$.

Numerical Study of Nonequilibrium Diagram Theory

In this paper we numerically analyze the first few diagrams in a Boltzmann-like collision operator that occurs in Severne's exact kinetic equation for the singlet distribution function. A similar analysis was used by Allen and Cole in deriving their singlet and doublet kinetic equations. Our analysis shows that the diagrams neglected by Allen and Cole in their kinetic equations are not negligible and these should be incorporated into dense fluid theories. The Allen–Cole kinetic transport coefficients and equilibrium pair correlation function are presented and calculated for dense argon. These results are not promising.

Lattice dynamics and transport equations for quantum crystals including singular interactions

The technique of non-equilibrium Green's functions is used to describe equilibrium properties and to derive transport equations for quantum crystals. The resulting equations are of the same structure as in anharmonic perturbation theory involving, however, an effective pair potential. This is formed by means of the equilibrium pair correlation function and is also defined for singular interactions.

Theory of Phenomenological Coefficients in Solid-State Diffusion. III. Limiting Law Conductance for Pure Ionic Crystals

The cycle diagrams of the previously derived cluster expansions are studied in detail for pure ionic crystals containing Schottky defects. The chain diagram approximation for the equilibrium pair correlation function is employed throughout. Some diagrams are then zero and the remainder yield, in the continuum limit, the Onsager limiting law for the relaxation effect for the conductance at zero frequency. The coefficient of the correction of lowest order in κa to this limiting law which arises when the cycle diagrams are evaluated allowing for the discreteness of the lattice is determined for the NaCl structure. The predictions of this correction are compared with exact numerical evaluations of the cycle diagrams. The corrections from the time-dependent correlation terms are well approximated by the analytic result and are very close to those predicted by the Pitts ion-size correction for electrolyte solutions. However, the remaining corrections from the time-independent correlation terms are but poorly represented by the analytic result and tend to cancel the other corrections. It is concluded that the major approximation in the use of the Onsager limiting law is the use of the chain diagram approximation for the pair correlation function (equivalent to the Debye–Hückel linearized result in the continuum limit) and this is probably relatively more important than effects arising from the discreteness of the lattice or corrections arising from the dynamics of vacancy pairs, triplets, etc. Further numerical study of this point is required. The generalization of the results to Frenkel defects is considered briefly but without detailed results.

Inverse Problem in Classical Statistical Mechanics

Examples of inverse solutions for the effective additive pair potential from guessed equilibrium pair distribution are given for the Percus-Yevick, hypernetted-chain, and Stillinger-Buff (with an inexact mean-density closure relation) equations. The found potentials depend on density and temperature and the potential for a given guessed distribution predicted by the Percus-Yevick theory is always smaller than that from the hypernetted chain equation. The exact inverse problem of determining the potential from the nonuniform pair distribution is formulated via the Stillinger-Buff equation and the difficulties in solving this problem are indicated. In a related inverse problem we characterize the possible equations of state consistent with the assumption that the potential of mean force of the fluid is directly proportional to the, possibly density dependent, pair potential.

Statistical-Mechanical Derivation of the Partial Molecular Stress Tensors in Isothermal Multicomponent Systems

Commencing with the general equations of molecular dynamics as used by Bearman and Kirkwood, the hydrodynamic equations of motion are derived for each component in an isothermal multicomponent system consisting of sensibly spherical nonreacting molecules. It was the primary purpose of the investigation of which this is a report to attempt to obtain a microscopic statistical basis for the macroscopic phenomenologic development of Snell and Spangler, especially in regard to their new formulation for the partial molecular stress tensors. An expansion of the number densities in pair space of perturbed nonequilibrium states alternate to that used by Bearman and Kirkwood is employed. Our expansion esplicitly recognizes the position vector arguments of both the number densities in singlet space and the local mean velocities of the individual components, as well as for the equilibrium pair correlation function. The quantities are then expanded about the central position vector in a Taylor's series. The results are not only corroborative of the previous phenomenologic development when the flows of the individual components are solenoidal and attention is confined to systems in which the kinetic contribution to the stress tensor is negligible in comparison to the contribution arising from intermolecular interaction, but they also provide a new microscopic theory of hydrodynamical flow in multicomponent systems.

Onsager Relations and the Spectrum of Critical Opalescence

The necessity of extending the hydrodynamic equations near the critical point is investigated on the basis of Onsager's assumption concerning the regression of fluctuations. It is shown that it follows from the Onsager relations that in general the new equations are nonlocal. The most important coefficient corresponding to the inverse compressibility in the original equations is obtained from the Onsager relations and expressed in terms of the equilibrium pair correlation function. Fixman's extension of hydrodynamics follows when the pair correlation is given by the Ornstein—Zernike formula. The consequences for the spectrum of light scattering near the critical point are studied and the width and relative intensity of the central scattering peak are expressed in terms of the equilibrium pair correlation function.

A perturbation approach to Bose-condensed pairs

AbstractA perturbation method is developed, and is used to obtain approximate expressions for the expectation values of one-particle and two particle operators in the quasi-chemical equilibrium (pair correlation) approximation to statistical mechanics, for the case of non-extreme Bose-Einstein condensation of the correlated pairs. To lowest order, the approximate results reproduce the results obtained previously for the case of extreme Bose-Einstein condensation.

Expectation values of operators in the quasi-chemical equilibrium theory part II

AbstractExpectation values of one-particle and two-particle operators are evaluated in the quasi-chemical equilibrium (pair correlation) approximation to statistical mechanics. Earlier work was restricted to the case of extreme Bose-Einstein condensation of the correlated pairs; the new formulas are not so restricted, but are correspondingly more complicated to evaluate practically. However, a simple result can be obtained for the expectation value of the number of particles.

On the generalization of the boltzmann equation to general order in the density

An expansion of the pair distribution function as a functional of the first distribution function in powers of the density is derived for a system not in equilibrium. This is achieved, using cluster expansions that are analogous to the Ursell-Uhlenbeck-Kahn- and the Husimi-expansions known from equilibrium statistical mechanics. On the basis of this procedure it is shown, that the density expansions of the non-equilibrium- and the equilibrium pair distribution function can be characterized by the same diagrams, where in the non-equilibrium case the first distribution function takes the role of the density in the equilibrium case. The general term in a systematic generalization of the Boltzmann equation to higher densities follows immediately from the expansion.

On the Kinetic Theory of Simple Dense Fluids. XI. Experimental and Theoretical Studies of Positive Ion Mobility in Liquid Ar, Kr, and Xe

We report in this paper measurements of the mobility of positive ions in liquid Ar, Kr, and Xe at various temperatures and pressures, and an extension of the theory of Rice and Allnatt to the description of ionic motion. It is found that (a) The magnitude, the pressure dependence, and the temperature dependence of the positive ion mobility in liquid Ar, Kr, and Xe can be quantitatively accounted for by an extension of the theory of Rice and Allnatt [S. A. Rice and A. R. Allnatt, J. Chem. Phys. 34, 2144 (1961)]. As in our previous studies of self-diffusion in these same liquids [J. Naghizadeh and S. A. Rice, J. Chem. Phys. 36, 2709 (1962)], it is found that negative contributions to the momentum autocorrelation function are of dominant importance. (b) Modification of the simple Nernst—Einstein relationship between the charge mobility and the self-diffusion constant of the parent atom by inclusion of the effects of polarization leads to quantitative agreement between theory and experiment. (c) The effect of the long-range Coulomb potential is to increase the local density about the ion and thereby to increase the frictional forces arising from the short range interaction potential. The rate of dissipation of energy directly by the long-range Coulomb potential is small compared to the rate of energy dissipation by the short range potential. (d) The agreement between experiment and theory is very satisfactory if the positive ion is Ar2+, Kr2+, or Xe2+ and it is suggested that this is the ionic species present. Agreement is much poorer if a different ionic species (say Ar+) is postulated. (e) If the ions are regarded as probes to alter and study the liquid structure it can be deduced that the current theory of liquids seriously miscalculates the change in the equilibrium pair correlation as a function of temperature and pressure. We believe the error to arise from either a premature truncation of a slowly convergent series used in the solution to an integral equation or, at worst, that the assumed series expansion is invalid [T. L. Hill, Statistical Mechanics (McGraw-Hill Book Company, Inc., New York, 1956)].

Expectation values of operators in the quasi chemical equilibrium theory

AbstractExpectation values of one-particle and two-particle operators are evaluated in the quasi-chemical equilibrium (pair correlation) approximation to statistical mechanics. Certain reductions, corresponding to the “quenching” of interactions by the Pauli exclusion principle, are carried out quite generally. More specific reductions, which lead to immediately useful expressions, are possible on the assumption of extreme Bose-Einstein condensation of the correlated pairs.

Transport coefficients of dense fluids of molecules interacting according to a square well potential

A simplified kinetic theory is described for spherical molecules interacting according to a square well potential. The statistical premises of this theory correspond closely to those of Longuet-Higgins' and Pople's theory of dense fluids of rigid spheres. The thermal conductivity λ, shear viscosity η and bulk viscosity κ are predicted to be in the ratio where k is Boltzmann's constant and m the mass of a molecule. To calculate any of these coefficients absolutely requires independent information about the equilibrium pair distribution. This may be obtained from the self-diffusion constant D, whose value is also given by the theory. For argon the measured value of D leads to a theoretical value for η which is in good agreement with experiment; for this liquid however the observed ratio κ:η is very much smaller than predicted.