Potential Of Average Force Research Articles

Expansion Formulae of the Free Energy and Distribution Functions in Powers of the One Particle Distribution Function

A new method is proposed which enables one to convert the formulas for a uniform multi-component system to those for a non-uniform one. As an application of the method, the expansion formulas of the free energy, potentials of average force and distribution functions for a non-uniform multi-component system are derived in a simple and systematic way. The formulas are obtained both for a continuous and for a lattice system. The formula obtained for the free energy for a continuous system coincides with that given by Arinstein, and the first few terms of the formulas for the free energy and the pair distribution function for a lattice system are identical to those given by Yvon. A procedure to convert the formulas for a continuous system to those for a lattice system and vice versa is given.

On the Kirkwood Superposition Approximation

The Kirkwood superpositIOn approximation is investigated by.making use of the expan­ sion theorem for the potential of average force and its correction term is found in a form of the expansion in powers of particle number density p. The lowest order term of this correction is calculated for a special configuration of three particles in a fluid consisting of hard spheres and the Kirkwood approximation is shown to overestimate the distribution function of triplets in this case. The Kirkwood integral equation with correction is solved by expanding the radial distribution function in powers of p. The solution thus obtained is shown to be exact up to the order of p2, while the Kirkwood approximation itself does not yield the correct term of the order of p2. It is shown that the distribution function of triplets can, in principle, be expanded in terms of radial distribtution functions and a few terms of this expansion are calculated explicitly, of which the first term just corresponds to the Kirkwood superposition approximation.

On Mayer's ionic solution theory

This theory, which is based upon McMillan and Mayer's cluster expansion of the grand partition function, is extended to include all of the components of the potentials of average force between sets of solute molecules or ions in the solvent at infinite dilution. The resulting cluster integral sum, which is a sum of integrals over an infinite number of infinite series of Mayer prototype graphs, is transformed into the sum of a single infinite series of integrals on graphs involving new types of bonds. These bonds are defined in a way first suggested by Meeron. There results a compact expression for the cluster integral sum.

Expression in Terms of Molecular Distribution Functions for the Entropy Density in an Infinite System

An expression for the entropy in a finite subvolume of an infinite fluid, originally proposed by Strato-novich on the basis of information theory, is rewritten in terms of functions related to the potentials of average force. The result exhibits a correction to the corresponding expression obtained by H. S. Green for a finite system, in the form of a series in powers of the concentration, of which the first terms have been investigated by Richardson. By neglect of correlations of three or more molecules and powers of the density higher than the third, the result is specialized to the case of a gas of low density, for which the configurational entropy per molecule is calculated with use of the radial distribution obtained by Kirkwood for hard-sphere and hard-core Lennard-Jones potentials.

Errata: Theory of Potentials of Average Force and Radial Distribution Functions in Ionic Solutions

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Nodal Expansions. II. The General Expansion of nth Order

The first-order nodal expansion of potentials of average force and distribution functions, obtained in a previous publication, is summed over again and converted into a second-order expansion. The method is generalized, and recursion formulas are given for nth order expansions of potentials of average force in terms of ``sheaf potentials.'' These latter are seen to conform exactly to the Kirkwood superposition principle. The feasibility of actual calculations is briefly discussed. Possible applicability to quantum-mechanical problems (Feynman diagrams) is noted.

Theory of Potentials of Average Force and Radial Distribution Functions in Ionic Solutions

The procedure employed by Mayer for the calculation of osmotic pressure and activity coefficients of ions in solution from the Fuchs expansion of the grand potential in multicomponent systems is applied to the recently obtained expansion of the potential of average force in these systems. The resulting multiply infinite series are summed to give a series analogous to the original expansion, involving the Debye-Hückel potential of average force in a remarkably simple way. A similar expansion for the radial distribution function is presented. Explicit expressions for both quantities, exact through terms of first order in the ionic concentration, are given. An expansion for the osmotic pressure, analogous to the virial expansion, is presented. The presence of the Debye-Hückel potential of average force in all these expansions is discussed, and the possibility of obtaining the potential of average force exact through terms of second order in the ionic charging parameters is indicated. Possible applications to ionic conductivity, diffusion, and viscosity are discussed. Additional applications to the theory of defects in ionic crystals, impurity conduction in semiconductors, etc., are suggested. The significance of the individual terms in the new expansions is discussed. Proof of an n-fold convolution theorem for Fourier transforms is presented.

Nodal Expansions. Distribution Functions, Potentials of Average Force, and the Kirkwood Superposition Approximation

The customary density power series expansions of potentials of average force and distribution functions are converted into a new class of expansions, defined in terms of topological nodes. The physical meaning of the terms in the new expansions is discussed, and arguments are presented to show that the new expansions can be expected to converge considerably more rapidly than the customary density expansions. The probable occurrence of phase transitions is discussed. Possible further developments are suggested. The Kirkwood superposition approximation is shown to be valid for a large number of terms in the expansions of potentials of average force and distribution functions. The customary comparison of the exact value of the third virial coefficient with that obtained via the Kirkwood superposition approximation is shown to provide no proof of the validity or invalidity of the latter.

Erratum : Series Expansion of Distribution Functions in Multicomponent Fluid Systems

The equations of Kirkwood and Salzburg for distribution functions are generalized to multicomponent systems. The generalized equations serve to derive a recurrence relation between the coefficients of powers of particle number densities in a Maclaurin expansion of the distribution functions. This recurrence relation is then used to derive the general term in the expansion of the distribution functions in terms of modified irreducible integrals in multicomponent systems, which includes the original one‐component expansion of Mayer and Montroll as a special case.The corresponding expansion of the potentials of average force is derived. The use of the new expansion is illustrated by a relatively simple derivation of the Fuchs expansion of the grand potential in multicomponent systems.Possible applications to ionic solutions, impurities in solids and x‐ray diffraction in solutions of several solutes are briefly discussed.Some new formulas in the theory of cumulants (Thiele semi‐invariants) are presented.

Series Expansion of Distribution Functions in Multicomponent Fluid Systems

The equations of Kirkwood and Salzburg for distribution functions are generalized to multicomponent systems. The generalized equations serve to derive a recurrence relation between the coefficients of powers of particle number densities in a Maclaurin expansion of the distribution functions. This recurrence relation is then used to derive the general term in the expansion of the distribution functions in terms of modified irreducible integrals in multicomponent systems, which includes the original one-component expansion of Mayer and Montroll as a special case. The corresponding expansion of the potentials of average force is derived. The use of the new expansion is illustrated by a relatively simple derivation of the Fuchs expansion of the grand potential in multicomponent systems. Possible applications to ionic solutions, impurities in solids and x-ray diffraction in solutions of several solutes are briefly discussed. Some new formulas in the theory of cumulants (Thiele semi-invariants) are presented.

Mayer's Treatment of Ionic Solutions

Mayer's expressions for the ionic activity coefficient and osmotic pressure of an electrolyte solution are shown to involve known and accurately tabulated functions. Mayer's infinite series of integrals is summed to give a single integral involving the Debye-Huckel potential of average force. With the help of the new form, the use of the single cut-off parameter for binary electrolytes is discussed from a quantitative standpoint.

Theory of Protein Solutions. I

A summary of a few results of the McMillan-Mayer solution theory is given in Sec. II, in order to have available necessary equations in a form required in later sections. It is pointed out in Sec. III that binding equilibria in a solution can in principle be discussed equally rigorously either implicitly or explicitly. The definition of ``binding'' in an explicit formulation is, strictly speaking, somewhat arbitrary but the arbitrariness disappears for practical purposes if the binding forces are very strong. However, the purely thermodynamic results are rigorously independent of the exact definition used. With the aid of Secs. II and III, it is possible to formulate in Sec. IV a general theory of protein solutions, including the effect of binding of ions or molecules on protein molecules. The discussion here is restricted to a single protein species and a single type of molecule capable of being bound, but the generalization to any number of protein or bound species is easy and will be published later. The topics discussed are osmotic pressure virial expansion, number of bound molecules per protein molecule expanded in powers of the protein concentration, distribution functions for sets of protein molecules (including the radial distribution function), potentials of average force on sets of protein molecules, superposition approximation in relation to the osmotic pressure and the Born-Green-Yvon integrodifferential equation for the distribution functions, and, finally, the relation to the recent Kirkwood—Shumaker theory. The theory applies equally well to polyelectrolyte, colloidal, and other solutions, but the ``protein'' language is used throughout for definiteness.

Statistical Thermodynamics of Solutions of Electrolytes and Non-Electrolytes

In this paper the generalized Born-Green integral equations for solutions of electrolytes and non-electrolytes are derived according to the method previously shown. The assumption of super­ position for the potentials of average force of solute molecules or ions is employed. throughout the present paper. The integral equations may be used, if the potentials of average force at infinite dilution are known, to compute the deviation from perfect solution behavior.

The Theory of Ionic Solutions

The virial development for the osmotic pressure of a solution may be used, if the potentials of average force of the solute molecules at infinite dilution are known, to compute the deviations from perfect solution behavior. The expression for the logarithm of the activity coefficient can thus be obtained as a sum of coefficients multiplied by powers of the concentration. For an ionic solution, with 1/R2 forces, the series is only conditionally convergent. By summing certain additive terms occurring in the coefficients over all powers of concentration convergence can be attained. The integrations necessary to obtain terms correct up to and including c32 are performed. The results are given in terms of certain functions which can readily be computed.

Integral Equations between Distribution Functions of Molecules

We may define distribution functions for given number, n, of molecules as being proportional to the probability density of finding n molecules in a certain configurational position, averaged over all positions of the other molecules of the system. These functions are the exponent of minus the potential of average force between molecules.. devided by kT. Mayerll derived a set of integralequations which relate variations in the potentials of average force at two different fugacities. On the other hand, Kirkwood!!} evolved an integral equation for the radial distribution function, and Born and Green3) obtained a similar equation. 'But the relations between these integral equations and Mayer's general ,variational method are apparently not known. The purpose of this paper is to attempt to clarify these relations, at the same time, to derive new integral equation which is a generalization of the Born-Green equation. We will deal with a multicomponent system, because the final equations are in no way more complicated if a system of many components is considered than if a single component system is handled.

Integral Equations between Distribution Functions of Molecules

Integral equations are derived that relate variations in the potentials of average force between molecules of a system at two different densities or activities. These permit the calculation of the change in thermodynamic properties, or of the change in the distribution of molecules in space, in a liquid or crystalline phase, if either the temperature is varied, or if the the mutual potentials between the molecules is assumed to change. The equations are in a somewhat complex, but still distinguishable, matrix form. A matrix operates on the variations in potential occurring at one activity and transforms them into those occurring at a second activity. The matrix elements are combinations of the distribution functions at the second activity, to which the transformation is made, multiplied by powers of the difference of the two activities. The matrix approaches the unit matrix in value as this activity difference approaches zero. The product of the two matrices, one which transforms from activity α to activity β, with that which transforms from activity β to α, is the unit matrix. This leads to an integral (matrix) equation between the distribution functions at any two activities. The calculation of one element of this matrix product leads to a cell type equation for the computation of the activity, or free energy, of a system in terms of the potential between a single molecule and the molecules that immediately surround it in the system. The transformation matrices have solutions, at certain values of the activity difference y, corresponding to variations in the potentials of average force, which are transformed into zero variations at the new activity. The activities at which such variations can occur are those of the phase transitions in the systems. The solutions at these special values of the activity are differences of combinations of the distribution functions in the two pure phases that can coexist at these activities.

The Statistical Thermodynamics of Multicomponent Systems

Distribution functions, Fn(z, {n}), for multicomponent systems are defined proportional to the probability density that n molecules in an infinite isothermal system of fugacity set z will occupy the configurational coordinates symbolized by {n}. All thermodynamic functions may be obtained as certain sums of integrals of these distribution functions. These sums are always convergent, but impractically slow in convergence for numerical use without further transformation. In particular, the grand-partition function, exp [VP(z)/kT], may be expanded in a power series in the fugacities z with coefficients given by integrals of the distribution functions Fn(0, {n}) at the fugacity set 0. As has been previously demonstrated for one component systems, this is shown to be a special case of a more general relation permitting the calculation of the distribution functions (and therefore the thermodynamic functions) for one fugacity set from those at another set. The function —kT ln Fn(z, {n}) is the potential of average force, Wn(z, {n}), of n molecules at the fugacity set z; at zero fugacities, the potential of average force is simply the potential energy. When only short range forces are present, by employing the procedure previously used for the imperfect gas, the integrals of the distribution functions may be simplified to integrals of lower order which become the coefficients in such expressions as the power-series expansion of the pressure increment in terms of the fugacity increments. These series have the advantage of a much higher rate of convergence than those involving integrals of the distribution functions, themselves; however, they do not converge when the initial and final fugacity sets bridge a phase transition. The general equations are applied to the imperfect multicomponent gas, to isotopic gas mixtures, and to condensed systems with the variables converted to the usual activities and activity coefficients. A change of independent variable with a corresponding change in coefficients then permits the evaluation of the pressure increment in terms of the solute concentrations. The equations for osmotic pressure are developed and found to be entirely analogous to those for the pressure of an imperfect gas. Finally, the osmotic pressure in a binary system and its behavior near phase transitions are discussed with emphasis on the critical mixing phenomenon, for which the existence of a region of anomalous first-order transition is shown to be probable.

Contribution to Statistical Mechanics

The method of the grand partition function may be used to calculate distribution functions Fn(z, {n}) proportional to the probability that n molecules in a system of fugacity z, and fixed temperature, occupy the position of their coordinates symbolized by {n}. The method makes use of the distribution functions Fn(0, {n}) at zero fugacity. The distribution functions may be written Fn(z, {n})=exp[−Wn(z, {n})/kT], in which Wn(z, {n}) is the potential of average force of n molecules at the fugacity z, which becomes equal to the ordinary potential energy at zero fugacity. The equations may be generalized to permit the calculation of the distribution functions at any fugacity assuming a knowledge of them at any other fugacity. Using methods previously employed for imperfect gases, the pressure, and also the density in molecules per unit volume, may be developed in a power series of difference of fugacity around any arbitrary fugacity. The coefficients of these developments are calculable at all fugacities by the same equations always employing the potentials of average force at the fugacity around which the development is made. The power series obtained represent functions which are regular on the real positive axis of fugacity except at the points characteristics of the phase transitions in the system.