An alternative construction of the Percus-Yevick equation based on the equilibrium BBGKY hierarchy

The existing derivations of the Percus-Yevick equation are not readily extendable into the nonequilibrium domain. In particular, the elegant Percus functional construction relies on a test particle theorem which lacks an exact nonequilibrium generalization. We propose here a new construction which utilizes some elementary ideas of functional expansions together with the equilibrium BBGKY hierarchy of equations. Also, we feel this new construction provides fresh insight into the physical basis of the equilibrium Percus-Yevick equation.This research was supported in part by a grant from the Faculty Research Award Program of the City University of New York.     

A multigroup approach to the non-linear extended boltzmann equation

Summary Following a procedure which is typical of linear (neutron) transport theory, a multigroup approach is proposed for the non-linear extended Boltzmann equation in the presence of removal, a background medium, an external source and an external force field. The relevant multigroup equations, corresponding to a discretization of the speed variable only, are derived and discussed, especially in connection with the so-called semi-discrete models recently introduced in kinetic theory.     

Space distribution and energy straggling of charged particles via Fokker-Planck equation

Summary The Fokker-Planck equation describing a beam of charged particles entering a homogeneous medium is solved here for a stationary case. Interactions are taken into account through Coulomb cross-section. Starting from the charged-particle distribution as a function of velocity and penetration depth, some important kinetic quantities are calculated, like mean velocity, range and the loss of energy per unit space. In such quantities the energy straggling is taken into account. This phenomenon is not considered in the continuous slowing-down approximation that is commonly used to obtain the range and the stopping power. Finally the well-known Bohr or Bethe formula is found as a first-order approximation of the Fokker-Planck equation. The authors of this paper have agreed to not receive the proofs for correction.     

Second-order Percus-Yevick theory for a confined hard-sphere fluid

A fluid of hard spheres confined between two hard walls and in equilibrium with a bulk hard-sphere fluid is studied using a second-order Percus-Yevick approximation. We refer to this approximation as second-order because the correlations that are calculated depend upon the position of two hard spheres in the confined fluid. However, because the correlation functions depend upon the positions of four particles (two hard spheres and two walls treated as giant hard spheres), this is the most demanding application of the second-order theory that has been attempted. When the two walls are far apart, this calculation reduces to our earlier second-order approximation calculations of the properties of hard spheres near a single hard wall. Our earlier calculations showed this approach to be accurate for the single-wall case. In this work we calculate the density profiles and the pressure of the hard-sphere fluid on the walls. We find, by comparison with grand canonical Monte Carlo results, that the second-order approximation is very accurate, even when the two walls have a small separation. We compare with a singlet approximation (in the sense that correlation functions that depend on the position of only one hard sphere are considered). The singlet approach is fairly satisfactory when the two walls are far apart but becomes unsatisfactory when the two walls have a small separation. We also examine a simple theory of the pressure of the confined hard spheres, based on the usual Percus-Yevick theory of hard-sphere mixtures. Given the simplicity of the latter approach the results of this simple (and explicit) theory are surprisingly good.     

Moments of the Percus-Yevick hard-sphere correlation function

A simple recursive relation is derived for the momentsM _n ,n=1, 2,..., of the Percus-Yevick correlation functionh(r) for identical hard spheres. TheM _n are rational functions of the volume fractionw occupied by the spheres; the first ten are given explicitly, and a single-term asymptotic form is obtained to suffice for the rest. Applications of theM _n(w) include testing different approximations forh by numerical integration ofh(r) r ⁿ . We compare exact moments with shell approximationsM _n [h ^s ] corresponding to integration fromr=0 tos+1 fors=3–8, and with hybrid approximationsM _n [h ^s +h ^a ] which supplement the shell approximations with integrals of an asymptotic tail froms+1 to . For a givens, the hybrid approximation is better forw increasing than the shell approximation, andM _n [h ³+h ^a ] is even better thanM _n [h ⁸]     

Global existence of solutions for a model Boltzmann equation

A model recently introduced by Ianiro and Lebowitz is shown to have a global solution for initial data having a finiteH-functional and belonging toL ¹ (L _x ). Methods previously introduced by Tartar to deal with discrete velocity models are used.     

Renormalized kinetic theory of nonequilibrium many-particle classical systems

The far-from-equilibrium statistical dynamics of classical particle systems is formulated in terms of self-consistently determined phase-space density response, fluctuation, and vertex functions. Collective and single-particle effects are treated on an equal footing. Two approximations are discussed, one of which reduces to the Vlasov equation direct interaction approximation of Orszag and Kraichnan when terms that are explicitly due to particles are removed.Work performed under the auspices of the U.S. Department of Energy.     

On the linearized relativistic Boltzmann equation. II. Existence of hydrodynamics

Solutions are analyzed of the linearized relativistic Boltzmann equation for initial data fromL ²(r, p) in long-time and/or small-mean-free-path limits. In both limits solutions of this equation converge to approximate ones constructed with solutions of the set of differential equations called the equations of relativistic hydrodynamics.     

Derivation of non-Markovian transport equations for trapped cold atoms in nonequilibrium thermal field theory

The non-Markovian transport equations for the systems of cold Bose atoms confined by a external potential both without and with a Bose-Einstein condensate are derived in the framework of nonequilibrium thermal field theory (Thermo Field Dynamics). Our key elements are an explicit particle representation and a self-consistent renormalization condition which are essential in thermal field theory. The non-Markovian transport equation for the non-condensed system, derived at the two-loop level, is reduced in the Markovian limit to the ordinary quantum Boltzmann equation derived in the other methods. For the condensed system, we derive a new transport equation with an additional collision term which becomes important in the Landau instability.     

From superoperator formalism to nonequilibrium Thermo Field Dynamics

Emphasizing that the specification of the representation space or the quasiparticle picture is essential in nonequilibrium quantum field system, we have constructed the unique unperturbed representation of the interaction picture in the superoperator formalism. To achieve it, we put the three basic requirements (the existence of the quasiparticle picture at each instant of time, the macroscopic causality and the relaxation to equilibrium). From the resultant representation follows the formulation of nonequilibrium Thermo Field Dynamics (TFD). The two parameters, the number distribution and excitation energy, characterizing the representation, are to be determined by the renormalization condition. While we point out that the diagonalization condition by Chu and Umezawa is inconsistent with the equilibrium theory, we propose a new renormalization condition as a generalization of the on-shell renormalization on the self-energy which derives the quantum transport equation and determines the renormalized excitation energy.     

Perturbative solution to order βɛ of the Percus-Yevick equation for triangular well potential forn=2

The radial distribution function for a fluid in which the molecules interactvia a triangular well potential is considered. Expanding the radial distribution function in pwoers of βɛ, where ɛ is the depth of the potential andβ=1/k _BT the first-order terms are calculated analytically using the Percus-Yevick theory in the Baxter’s formulation. The first-order terms in the direct correlation functionc(r) are also calculated. The first- and second-order terms in the free energy obtained from the energy equation of state are calculated and compared with other calculations. An erratum to this article is available at .     

A system of conservation equations arising in nonlinear dynamics of gas mixtures

Il Nuovo Cimento D - On the basis of appropriate hypotheses, concerning the collision frequencies, the scattering probability distributions and the initial conditions, a system ofN conservation...     

Electronic fluctuation,the nature of interactions and the structure of liquid metals

Summary Interactions controlling ionic motion and structure in liquid metals can be systematically developed beginning with the ultimate view of such systems as neutral assemblies of nuclei and electrons, and proceeding to standard reduced Hamiltonians reflecting assemblies ofions and electrons. The assumption of electronically rigid ion leads via response methods and pseudopotentials to statically screened ion-ion potentials and beyond. Fluctuations are introduced into this otherwise common viewpoint by relaxing the assumption of electronically rigid ion cores and also by treating electronic response beyond linear order. It is argued that both effects can lead to more attractive pair interactions and possibly to effects much larger than, for example, Friedel oscillations. These contributions are state (i.e. density) dependent and their presence might be expected on the basis of clustering behavior seen for some systems in theirvapour phases. This leads to two limiting viewpoints on the liquid-state structure of such systems, the first as an entirely monoatomic phase but with a pair interaction that is unusual, the second as a system supporting transient clusters whose presence reflects the complexities argued on the basis of a more extended treatment of the electron problem. Paper presented at the workshop ?Highlights on Simple Liquids?, held in Turin at ISI on 1– May, 1989.     

Recent developments in the theory of ionic melts

Summary A brief review is given of the present state of the art in the theory and numerical simulation of ionic melts, and in particular of molten alkali halides. Some recent developments concerning the theoretical evaluation of the static pair structure, the phase diagram and the microscopic dynamics in these melts are discussed. The importance of ionic polarizabilities is stressed. Some perspectives for future work are pointed out in the conclusion. Unité de Recherche Associée 1325 du CNRS.     

Mode-coupling approach to the dynamics of simple liquids

Summary A new derivation of the mode-coupling theory of liquid dynamics is presented which is applicable to both quantum and classical systems. A clear distinction between the basic underlying hyphotheses and the more technical approximations has been attempted. Asymptotic corrections to the dynamics of modes due to mode-coupling effects and selfconsistent mode-coupling theories emerge from the original idea as two markedly different branches of application. Paper presented at the workshop ?Highlights on Simple Liquids?, held in Turin at ISI on 1–3 May, 1989.     

A Weak Formulation of the Boltzmann Equation Based on the Fourier Transform

In this article we present an alternative formulation of the spatially homogeneous Boltzmann equation. Rewriting the weak form of the equation with shifted test functions and using Fourier techniques, it turns out that the transformed problem contains only a three-fold integral. Explicit formulas for the transformed collision kernel are presented in the case of VHS models for hard and soft potentials. For isotropic Maxwellian molecules, a classical result by Bobylev is recovered, too.     

The inverse conjecture for the revised Enskog equation

It is shown that the pair correlation function (which is by definition the high-density factor in the revised Enskog theory) is not always a well-defined functional of the local density. Moreover, for a finite system with periodic boundary conditions and in the space homogeneous case, this function, computed at the contact value, is bounded at the maximum allowed density (i.e., a densityn _max such that, in one dimension, 1/a–1/Ln _max<1/a; equality sign, which corresponds to the usual close-packing density for whichL/a is an integer, being included as a particular case). At least for the one-dimensional gas model this finite value is shown to approach infinity in the thermodynamic and in the hydrodynamic limits. A new form for the revised Enskog equation, which does not depend on the inverse conjecture, is finally given.     

Melting properties of group-V elements and InSb

Summary In this paper, we study the structural properties of liquid group-V elements (and a compound InSb) and we analyse the differences with their room temperature crystalline phases. Neutron diffraction experiments are performed using short-wavelength neutrons (0.7 ?). We show different melting behaviours of a light element (As) and a heavy element (Sb) under study. Whereas As keeps its coordination three upon melting, Sb is six-coordinated in the melt. Similarly, InSb has a coordination number increasing from 4 to about 6 upon melting. Large entropies of melting are related to important structural variations. Paper presented at the workshop ?Highlights on Simple Liquids?, held in Turin at ISI on 1–3 May, 1989.     

Coarsely grained stochastic Boltzmann equation and its moment equations

The stochastic Boltzmann equation is coarsely grained. The coarsely grained stochastic (CGS) Boltzmann equation has fluctuating terms in its collision term. On the basis of the CGS Boltzmann equation, reduced Grad’s 26 moment equations are derived. Coarsely grained moment equations obtained from the CGS Boltzmann equation show that fluctuating terms remain as nonvanishing terms owing to the nonlinearity in the collision term of the CGS Boltzmann equation. The Navier-Stokes-Fourier law obtained using the CGS Boltzmann equation indicates that the pressure deviator and heat flux include fluctuations of their one-order higher moments.