Critical behaviour of a class of Ornstein-Zernike systems on a lattice

Summary The critical behaviour of a class of Ornstein-Zernike systems,i.e. systems in which the direct correlation function has the range of the potential, is studied for a lattice gas with nearest-neighbour interaction. The generic critical behaviour is equal to that of the mean spherical model. The Percus-Yevick equation falls in a special class due to the vanishing of a suitable expansion coefficient at the critical point. This special class has classical (mean field) critical exponents but a non-classical and nonuniversal scaling function.

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Riassunto Si presenta lo studio del comportamento critico di una classe di sistemi Ornstein-Zernike (sistemi in cui la funzione di correlazione diretta ha il raggio d'azione del potenziale) nel caso di gas reticolare con interazione a primi vicini. L'analisi mostra che, in generale, il comportamento critico presentato da questa classe risulta uguale a quello del modello sferico. L'equazione di Percus-Yevick cade in una speciale sottoclasse a causa dell'annullarsi al punto critico di un particolare coefficiente di sviluppo. Questa sottoclasse speciale ha esponenti critici classici (campo medio), ma funzioni di scala non classiche e non universali.

     

Medium-range fluctuations accompanying the metal-nonmetal transition in expanded fluid hg

We have carried out small angle x-ray scattering experiments of expanded fluid Hg in the metal-nonmetal (M-NM) transition region around 9.0 g cm(-3). Increase of small angle scattering intensity following the Ornstein-Zernike equation is clearly observed in the M-NM transition region as well as near the liquid-vapor critical point at 5.8 g cm(-3). The short-range correlation length, R, becomes twice as large in the M-NM transition region as in the critical region. The enhancement of R in expanded fluid Hg suggests a new type of fluctuations reflecting a first-order M-NM transition.     

A method of solution of the ornstein-zernike equation in one and three dimensions

An elegant analytic method is presented to derive a modified form of the Ornstein-Zernike equation which not only solves the statistical mechanical problem for hard rods and spheres but can be used to find solutions for arbitrary non-vanishing direct correlation functions.     

Mean spherical approximation for a model liquid metal potential

The solution of the Ornstein-Zernike equation for a direct correlation function c(x) with damped oscillations and a hard core condition imposed upon the total correlation function h(x) has been proposed by Cummings as a means of treating a simple model potential for liquid metals in the mean spherical approximation [1]. Here some numerical results are given for this model and their significance is discussed. The solution of the Ornstein-Zernike equation is also extended; the hard core condition is generalized to a soft core condition, and Yukawa terms are added to the oscillatory c(x). Ways in which these extensions can be incorporated into more accurate liquid metal models, as well as into more accurate approximations for these models, are discussed. Finally, it is shown that our solution of the Ornstein-Zernike equation, after a change in the core condition, yields the structure of a spin glass model considered by Høye and Stell in the MSA-like approximation they propose [22].     

Properties of hard disc fluids by Baxter’s method

The radial distribution function and the equation of state for hard disc fluids have been calculated at various densities by solving Ornstein-Zernike equation using Baxter’s method.     

Inequalities among static and dynamic critical-point exponents

The mode-mode coupling theoretical expression for the decay rate of order-parameter fluctuations in simple fluids near the critical point is evaluated with the aid of the recently extended Ornstein-Zernike theory. This gives that ψ?ν for the critical-point exponent ψ which characterize the behavior of the thermal conductivity. Also, the results for sound-wave absorption and dispersion near the gas-liquid critical point need not depend on a single reduced frequency variable since scaling may be violated.     

Phase diagrams of Ising fluids with Yukawa-Lennard-Jones interactions from an integral equation approach

An integral equation approach is developed to investigate phase coexistence properties of Ising spin fluids with Yukawa ferromagnetic and Lennard-Jones nonmagnetic interactions in the presence of an external field. The calculations are carried out on the basis of the Duh and Henderson closure with a specific Duh-like partitioning of the total potential. The coupled set of the Ornstein-Zernike equation, the closure relation and the external field constraint are solved using an efficient numerical algorithm. The phase diagrams are evaluated in a wide range of varying the external field and the ratio of strengths of Yukawa to Lennard-Jones interactions. Different types of the phase diagram topology as well as various external field dependencies of critical temperatures and densities are identified. The complexity with respect to simple Lennard-Jones fluids is explained by coupling between spatial and spin degrees of freedom in the system. A comparison of the obtained theoretical results with simulation data is made and a good agreement is observed.     

A hard sphere fluid model for superionic conductors

We develop a theory for the mobile constituent of a superionic conductor using the Ornstein-Zernike integral equation for the pair correlation function of an inhomogeneous fluid. We solve this equation in the Percus-Yevick approximation using a simple decoupling procedure and hard core potentials. Comparison is made with molecular dynamics calculations on α-AgI.     

Explicit partial static structure factors of neutral Yukawa fluids

An explicit expression for the partial static structure factors for neutral Yukawa liquids is presented, starting from Blum and Ginoza's MSA solution of the Ornstein-Zernike equation for the factorizable Yukawa closure.     

The compressibility theorem for quantum simple fluids at equilibrium

An extension of the compressibility theorem for quantum simple fluids within the pathintegral approach is presented. First, it is demonstrated that in the absence of quantum exchange, the isothermal compressibility can be formulated in an exact manner with the use of the pair radial correlation function of the path-integral centroids corresponding to the particles of the fluid. This adds up to the two known formulations based on the pair correlations between true quantum particles, namely the instantaneous and the pair linear response correlations. To complement this extension, an exact Ornstein-Zernike equation for pair centroid correlations is derived, which permits accurate estimates for the isothermal compressibility to be obtained. Several fluids are studied, new numerical results for the latter quantity are reported to support the theoretical points, and some difficulties present in this sort of calculation are discussed. The systems studied are the following: the quantum hard sphere fluid with and without attractive Yukawa interaction, liquid helium-4 and liquid para-hydrogen. Finally, the possibilities of extending the theorem to deal with quantum exchange are considered, and it is shown that the extension and its computational Ornstein-Zernike scheme also hold for a Bose fluid.     

Statistical mechanics of linear molecules I: Percus-Yevick and convolution-hypernetted-chain approximations

It is shown how the Percus-Yevick and convolution-hypernetted-chain approximations for dense fluids of linear molecules can be brought into practically applicable forms by using cartesian tensor formalism and pertubation theory. The possibility of expanding the intermolecular potential and the correlation functions is discussed in detail. Also, exact reductions for the Ornstein-Zernike relation in real space and for the Percus-Yevick equation are given. Some other approaches to the statistical mechanics of linear molecules are critically discussed. It is shown in particular, that the mean spherical model is an inconsistent type of perturbation theory. The necessary cartesian tensor formulae are given in appendices as well as formally exact expressions for the second virial coefficient and for the Fourier-transformed Ornstein-Zernike relation.     

Fluids of pseudo-hard bodies II. Reference models for water,methanol, and ammonia

Pseudo-hard body fluids resulting from extended primitive models of water, methanol, and ammonia have been investigated both by computer simulations and theory for a number of geometrical parameters. It is shown that none of the existing equations of state and integral equations for the site-site correlation functions is able to describe the properties of the pseudohard body fluids reasonably accurately. For the equation of state an accurate semi-empirical method is proposed and for the site-site correlation functions the reference average Mayer function perturbation theory has been found to perform at least qualitatively correctly, which is not the case with Ornstein-Zernike equation based theories.     

Susceptibility and magnetisation of a random ising model

The susceptibility of a bond disordered Ising model is calculated by configurationally averaging an Ornstein-Zernike type of equation for the two spin correlation function. The equation for the correlation function is derived using a diagrammatic method due to Englert. The averaging is performed using bond CPA. The magnetisation is also calculated by averaging in a similar manner a linearised molecular field equation. Part of the work was done, while one of the authors (DK) was visiting International Centre for Theoretical Physics, Trieste, Italy.     

Mixtures of polar and nonpolar molecules

The site-site Ornstein-Zernike (SSOZ) equation with mean spherical approximation closure is solved analytically for a mixture of hard dumbbells and polar hard dumbbells. The solution reduces to that of the pure polar hard dumbbell fluid at the polar species density rather than the total density. The thermodynamic properties of the mixture are obtained using the zero-pole approximation (ZPA) to the free energy. The mixture is shown to separate into two mixed phases, one rich in the nonpolar species and the other rich in the polar species. This phase separation terminates in an upper critical solution temperature. The excess thermodynamic functions are presented and the mixture exhibits both positive and negative values of the excess volume. The negative values of the excess volume occur in mixtures rich in the polar component.     

Direct and indirect correlations in low density supercritical Lennard-Jones fluids

We have carried out a detailed comparison of the direct and indirect correlation functions obtained from canonical molecular dynamics (NVT-MD) simulation of supercritical Lennard-Jones fluids to the results obtained by solving the Ornstein-Zernike equation with the approximate Duh-Henderson (DH) closure. The variations of equilibrium correlations are studied as functions of density at two supercritical isotherms near and away from the critical point. The direct and indirect correlations predicted using the DH closure provides a very good agreement with our simulation results at low densities. However, a marked deviation is observed at higher densities. These results are correlated to the discrepancies between the density and temperature dependence of the underlying bridge function. The implication of these results on the calculation of chemical potential and the Krichevskii parameter is also presented.     

Investigation of multiple scattering contributions to the depolarization of light scattered by xenon and carbon dioxide in the critical region: II. Numerical calculations

With reference to the experimental data given in paper I of this series, we now present the results of numerical calculations on the depolarization factor in the critical region of fluids. Attenuation, Ornstein-Zernike effects and contributions from depolarized triple and polarized double scattering have been taken into account.     

Hard sphere correlation functions in the Percus-Yevick approximation

A reformulation of the Ornstein-Zernike equation permits rapid and simple numerical calculation of the total correlation functions for single and multi-component systems of hard spheres. Such correlation functions permit the application of sophisticated statistical mechanical perturbation theories to more complicated systems than has been possible heretofore.     

On the theory of multiple scattering II. Critical scattering

The systematic theory of multiple scattering which we gave in a previous paper is further elaborated for critical scattering. It is shown that in each order the multiple-scattering intensity near the critical point is in essence a contraction of consecutive uncorrelated single-scattering intensities. The anomaly of the critical depolarization factor is calculated and is found to be in quantitative agreement with recent experimental results. Double scattering corrections to the Ornstein-Zernike plot are discussed.     

Integral equation theory for fluids ordered by an external field: separable interactions

The structural and thermodynamical properties of classical fluids orientationally ordered by an external field are investigated by means of integral equation theories. A general theoretical framework for handling these theories is developed and detailed for the particular case of separable interactions between fluid particles. This approach is then illustrated for the case of two (off lattice) models: the ferromagnetic Heisenberg model and a simple liquid crystal model, for which the numerical solution of integral equations such as the Percus-Yevick, the hypernetted chain, and the reference hypernetted chain closure equations are presented and compared with Monte Carlo simulation results and the analytical solution of the mean spherical approximation. The zero-field case is also examined, and the spontaneous ordering is analyzed in detail, mainly in what concerns the appearance of infinite wavelength singularity in the Ornstein-Zernike equation and the relation with the one-body closure equations and the long range orientational ordering that occurs. In particular, it is shown that the Wertheim one-body closure equation appears as a sum rule compatible with the Ornstein-Zernike equation. The relation between the elastic constant and the long range tail of the pair correlation function is made explicit. In particular, the long range behavior of the various terms in the expansion of the pair correlation function is depicted. The numerical investigation of the two models shows that it is not possible to discriminate between the four integral equations, as to which one would be the most accurate in all cases. The general trends in the thermodynamical and structural properties seem to indicate that the Percus-Yevick approximation is generally better in the strong ordering case, whereas the reference hypernetted chain approximation might be better suited to the study of the isotropic phase and the low ordering regimes.