Pair correlation function in a fluid with density inhomogeneities: results of the Percus‒Yevick and hypernetted chain approximations for hard spheres near a hard wall

1986 ◽  
Vol 404 (1827) ◽  
pp. 323-337 ◽  
Keyword(s):  
Correlation Function ◽  
Pair Correlation Function ◽  
Hard Spheres ◽  
Pair Correlation ◽  
Hard Wall ◽  
Bulk Fluid ◽  
Pair Correlations ◽  
Hypernetted Chain ◽  
Density Inhomogeneities ◽  
Bulk Case

Solutions for the pair correlation function and density profile of a system of hard spheres near a hard wall are obtained by using the Percus‒Yevick and hypernetted chain approximations, generalized for inhomogeneous fluids. The Percus‒Yevick (PY) results are similar in accuracy to those obtained for the bulk fluid. The PY pair correlation function is generally too small near contact but quite good overall. The hypernetted chain (h. n. c.) results are difficult to obtain numerically and are poorer than in the bulk. Often the h. n. c. pair correlations are too small at contact, in contrast to the bulk case where they are too large, although there are configurations where the contact values of the pair correlation function are too large. Nearly always the error in the h. n. c. results is much worse than is the case for the bulk. Both approximations are qualitatively satisfactory in that they predict the correct asymmetries between the values of the pair correlation functions for pairs of hard spheres whose line of centres is parallel or normal to the surface of the wall.

Pair correlation function in a fluid with density inhomogeneities: parametrization for hard spheres near a hard wall

1985 ◽  
Vol 400 (1818) ◽  
pp. 163-174 ◽  
Keyword(s):  
Surface Tension ◽  
Correlation Function ◽  
Pair Correlation Function ◽  
Hard Spheres ◽  
Pair Correlation ◽  
Hard Wall ◽  
Sum Rule ◽  
Density Inhomogeneities ◽  
Semi Empirical ◽  
Good Agreement

The pair correlation function, g( z 1 , z 2 , r 12 ), of a system of hard spheres near a hard wall is approximated by a simple function that interpolates in a reasonable manner between the Percus shielding approximation for the case when the line of centres of the spheres is normal to the surface of the wall and the bulk pair correlation function, with a semi-empirical modification, for the case when the spheres are equidistant from the wall. This modification is chosen so that the surface tension, calculated from the pair correlation function, is given correctly. The resulting pair correlation function is in good agreement with computer simulations and satisfies a sum rule for the slope of the density profile fairly well.

On the pair correlation function for hard spheres at low density and temperature

1968 ◽  
Vol 46 (7) ◽  
pp. 879-888 ◽  
Author(s):  
M. S. Miller ◽  
J. D. Poll
Keyword(s):  
Correlation Function ◽  
Pair Correlation Function ◽  
Hard Spheres ◽  
Pair Correlation ◽  
Quantum Mechanical Calculation ◽  
Quantum Mechanical ◽  
Low Density ◽  
Free Particles ◽  
Low Density Limit ◽  
Hard Sphere Diameters

A quantum-mechanical calculation of the pair correlation function for hard spheres in the low-density limit has been made. This calculation is, therefore, valid at low temperatures, where quantum-mechanical diffraction and symmetry effects are important. Results are given for various temperatures and hard-sphere diameters. The pair correlation function is presented in the form g = gB + gS, where gB is the correlation function for Boltzmann particles and gS describes the symmetry effects. It is found that gS(R) for any value of the separation R is always smaller than the corresponding value for free particles.

Monte Carlo Simulation of the Interfaces Liquid-Crystal and Liquid-Rigid Wall

1978 ◽  
Vol 33 (12) ◽  
pp. 1557-1561 ◽  
Author(s):  
B. Borštnik ◽  
A. Ažman
Keyword(s):  
Monte Carlo ◽  
Correlation Function ◽  
Liquid Crystal ◽  
Pair Correlation Function ◽  
Hard Spheres ◽  
Pair Correlation ◽  
Rigid Wall ◽  
Bulk Liquid ◽  
Lennard Jones ◽  
The Monte Carlo Method

Abstract The structure of liquids at liquid-crystal and liquid-rigid wall interfaces was studied by the Monte Carlo method on systems consisting of either 128 Lennard-Jones atom s or 128 hard spheres. The resulting density profile can serve as a reference for the approximative methods based on the BGYB hierarchy of integral equations. The pair correlation function close to the rigid wall is found to deviate appreciably from the bulk liquid pair correlation function. The maxima and minima of g(r) are more pronounced in the first two layers of atom s close to the rigid wall.

Sum rules for the pair-correlation functions of inhomogeneous fluids: results for the hard-sphere–hard-wall system

1987 ◽  
Vol 410 (1839) ◽  
pp. 409-420 ◽  
Keyword(s):  
Sum Rules ◽  
Hard Spheres ◽  
Pair Correlation ◽  
Distribution Functions ◽  
Hard Wall ◽  
Chain Equation ◽  
Hypernetted Chain ◽  
Chain Approximation ◽  
Derivatives Of

Starting from well-known relations for the derivatives of the radial distribution functions of a mixture of fluids, and allowing the diameter of one particle to become exceedingly large, three sum rules for a fluid with density inhomogeneities are obtained. None of these sum rules are new. However, the relation between the Lovett–Mou–Buff–Wertheim and the Born–Green hierarchy of equations seems not well known. The accuracy of a recent parametrization of the pair correlation of hard spheres near a hard wall and of the solutions of the Percus–Yevick and hypernetted-chain equation for this same function are examined by determination of how well these functions satisfy these sum rules and the accuracy of their surface tension, calculated from the sum rule of Triezenberg and Zwanzig. Generally speaking, the Percus–Yevick theory gives the best results and the hypernetted-chain approximation gives the worst results with the parametrization being intermediate.

Pair correlations of the leVeque sequence on the polydisc

2008 ◽  
Vol 145 (1) ◽  
pp. 197-203
Author(s):  
R. NAIR
Keyword(s):  
Correlation Function ◽  
Pair Correlation Function ◽  
Resonance Condition ◽  
Pair Correlation ◽  
Natural Numbers ◽  
Pair Correlations ◽  
The Real ◽  
Image Position

AbstractWe consider a system of “forms” defined for ẕ = (zij) on a subset of $\Bbb C^d$ by where d = d1 + ⋅ ⋅ ⋅ + dl and for each pair of integers (i,j) with 1 ≤ i ≤ l, 1 ≤ j ≤ di we denote by $(v_{ij}(k))_{k=1}^{\infty}$ a strictly increasing sequence of natural numbers. Let ${\Bbb C}_1$ = {z ∈ ${\Bbb C}$ : |z| < 1} and let ${\underline X} \ = \ \times _{i=1}^l \times _{j=1}^{d_i}X_{ij}$ where for each pair (i, j) we have Xij = ${\Bbb C}\backslash {\Bbb C}_1$. We study the distribution of the sequence on the l-polydisc $({\Bbb C}_1)^l$ defined by the coordinatewise polar fractional parts of the sequence Xk(ẕ) = (L1(ẕ)(k),. . ., Ll(ẕ)(k)) for typical ẕ in ${\underline X}$ More precisely for arcs I1, . . ., I2l in $\Bbb T$, let B = I1 × ⋅ ⋅ ⋅ × I2l be a box in $\Bbb T^{2l}$ and for each N ≥ 1 define a pair correlation function by and a discrepancy by ΔN = $\sup_{B \subset \Bbb T^{2l}}${VN(B) − N(N−1)leb(B)}, where the supremum is over all boxes in $\Bbb T^{2l}$. We show, subject to a non-resonance condition on $(v_{ij}(k))_{k=1}^{\infty}$, that given ε > 0 we have ΔN = o(N$(log N)^{l + {1\over 2}}$(log log N)1+ε) for almost every $\underline x(\underline z)\in \Bbb T^{2l}$. Similar results on extremal discrepancy are also proved. Our results complement those of I. Berkes, W. Philipp, M. Pollicott, Z. Rudnick, P. Sarnak, R Tichy and the author in the real setting.

Computer simulations of polyatomic molecules - I. Monte Carlo studies of hard diatomics

1976 ◽  
Vol 348 (1655) ◽  
pp. 485-510 ◽  
Keyword(s):  
Monte Carlo ◽  
Correlation Function ◽  
Pair Correlation Function ◽  
Hard Spheres ◽  
Pair Correlation ◽  
Harmonic Series ◽  
The Monte Carlo Method ◽  
Monte Carlo Studies ◽  
High Degree ◽  
Orientational Structure

The Monte Carlo method has been used to study a model system of 256 hard diatomic molecules, each consisting of two fused hard spheres of diameter σ with centres separated by reduced distance L = L/σ of 0.2, 0.4 and 0.6, at densities typical of the liquid state. The orientational structure of dense, hard diatomic fluids has been studied by calculating up to sixteen terms in the expansion of the total pair correlation function, g ( r 12 , ω 1 , ω 2 ), in spherical harmonics. The coefficients g u'm ( r 12) the series have been calculated as ensemble averages in the simulation. At short distances, the system exhibits a high degree of angular correlation, which increases with increasing density and elongation; however, this correlation is relatively short ranged at all densities and elongations, and in no case is there significant angular structure at distances greater than twice the major diameter of the molecule. In the nearest neighbour shell there is a strong preference for 'T-shaped’ pair orientations. At low elongations and densities the spherical harmonic coefficients are in close agreement with those predicted both by the ‘blip function’ theory and the solution of the Percus-Yevick equation for hard diatomics. The harmonic series for the total pair correlation function, is rapidly convergent at distances greater than L + σ , but slowly convergent at smaller distances. The results are suitable for use as a non-spherical reference system for perturbation calculations.

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