David J. Vezzetti

ON FLUIDS OF PARTICLES WITH SHORT RANGE REPULSION AND WEAK LONG RANGE ATTRACTIVE INTERACTION.

We have obtained an appreciably simplified derivation of the expansion in inverse range of the thermodynamic functions of a system of particles with short‐range repulsion and weak long‐range attractive interaction. We assume all properties of the system with only the short‐range repulsive interaction to be known and consider this system as a reference system for the purpose of a perturbation expansion. We derive corrections to the van der Waals equation and to the Maxwell construction. The terms in our expansion are integrals over products of the modified Ursell functions of the reference system only and factors which are the transformed long‐range potential. A simple rule is given to determine the order, in the inverse range of the attractive potential, of each term in the expansion. The simplification of the derivation is achieved by avoiding the expansion and subsequent resummation of any functions pertaining to the reference system.

On the Ising Model with Long‐Range Interaction

We have obtained an expansion of the free energy per spin of an Ising model with long‐range interaction in the absence of an external field, for temperatures above the Curie temperature of the Weiss‐Bragg‐Williams approximation (BWCP). We use as expansion parameter the reciprocal (γ) of an effective number of neighbors. Terms through order γ2 are obtained by extracting factors from a representation of the partition function as an average over random fields. For terms of higher order, we give a diagrammatic series in which all terms through order γn are contained in the diagrams with not more than 2(n − 1) bonds. The terms of order γ3 are given explicitly. For temperatures below the BWCP we have calculated terms through order γ. Since after a few finite terms the coefficients in this γ expansion become infinite at the BWCP, we exhibit a modification of the random field representation which avoids this difficulty. We have compared our results with those of previous authors wherever available—that is, throug...

Refractive Index, Attenuation, Dielectric Constant, and Permeability for Waves in a Polarizable Medium

A new method is presented for calculating the refractive index, attenuation, dielectric constant, and permeability for electromagnetic waves in a medium of polarizable particles. It is similar to the method of Yvon and Kirkwood for finding the static dielectric constant. The main merit of the method is that it avoids the statistical hypotheses used in such calculations by Lorentz, Reiche, Hoek, Rosenfeld, and other authors. In addition, it permits the calculations to be continued to any degree of accuracy. We first use the method to obtain the dispersion equation as a power series in the molecular polarizability. The nth term in this series involves the distribution function of n + 1 particles. The terms of first and second degree are written out explicitly in terms of the two‐ and three‐particle distribution functions. When terms of second and higher degree are omitted and the result specialized to particles with a scalar electric polarizability and zero magnetic polarizability, the dispersion equation a...

Role of Droplets and Bubbles in the “Condensation” of the One‐Dimensional Gas of Kac, Uhlenbeck, and Hemmer

The functional integral representing the grand canonical partition function of the Kac, Uhlenbeck, and Hemmer model is approximated, in a systematic way, to give a much simpler integral representing one‐dimensional Brownian motion with absorption. It is shown that the paths which make the dominant contributions to this integral are those which correspond to alternating regions of high and low density in the fluid. An interpretation of these paths as representing configurations of droplets and bubbles is given. The pressure of the system is obtained by summing over the droplet‐bubble configurations, and it is shown that there is no phase transition for any finite value of the range of the attractive potential. A number of properties of the droplets are determined. The relation of this work to inverse range expansions and other recent works is discussed.

ON THE ISING MODEL WITH LONG RANGE INTERACTION II: CRITICAL REGION ANALYSIS.

The high‐temperature expansion for the free energy in powers of γ (the reciprocal of the range of interaction) developed in a previous paper is studied in the critical region. In terms of the graphology introduced in the previous paper, it is proved that the ring diagrams give the dominant contribution in the critical region, if and only if the integral R(νγ)=(12π)D ∫ 2π… ∫ 0g(ω)dDω1−νγg(ω) diverges no worse than logarithmically at [the Curie‐Weiss (CW) point] νCW = J/kTCW = [γg(0)]−1, where D denotes dimensionality and γg(ω) the Fourier transform of the interaction potential. The results are in agreement with various model results, and it is conjectured that the above condition is also a necessary and sufficient condition for the existence of a phase transition.

A new derivation of some fluctuation theorems in statistical mechanics

A simple derivation is given of some of the fluctuation theorems of statistical mechanics which relate integrals of molecular distribution functions to thermodynamic properties. The derivation employs the generating function for the probability Pω(n) that a domain ω contains n particles. Various forms of the generating function are derived, and each leads to a different form of the fluctuation theorems.

Fluids of Particles with Short‐Range Repulsion and Weak Long‐Range Attractive Interaction. II. The Two‐Particle Distribution Function

In Paper I, we presented an expansion of the pressure and density in grand canonical form and corrections to the Maxwell rule for a system of particles with short‐range repulsion and weak long‐range attraction. These expansions can be ordered in powers of γ, the inverse range of the attractive potential. It was assumed that the thermodynamic functions and the molecular distribution functions of the reference system, i.e., the system with only the repulsive interaction, are given. In the present paper we have calculated the γ expansion of the pair distribution function, under the same assumption. The result is obtained by functional differentiation of the series for the pressure and presented in the form of a series of diagrams. The dominant order in γ of each diagram is the same as the order of that diagram in the series for the pressure, from which it is derived.

Approach of the statistical theory of light scattering to the phenomenological theory

A fully statistical treatment of the spectrum of light scattered by a simple fluid is given. The results are shown to be in close accord with the phenomenological theory of the same process.

Comments on renormalization group series

In previous articles by Nauenberg and by Niemeijer and Ruijgrok, the renormalization group approach has been used to obtain series expansions for the free energies of certain one- dimensional spin systems. We investigate here the question of whether this method provides a fundamentally new way of approximating the largest eigenvalue of a transfer matrix or kernel. For the models mentioned above, this is not the case. The partial sums of the series are essentially equivalent to the free energies of a sequence of finite models of increasing size.

Depolarized light scattering from simple fluids

It is shown that even in single scattering approximation, the light scattered by a simple fluid exhibits a depolarized component which arises from including additional terms in the current density. A new way of measuring the shear viscosity of a fluid is proposed.