Pair-correlation function

The relativistic pair correlation function g is defined in complete analogy to its nonrelativistic counterpart, 

The series representation (B.68) of gx ° is not very suitable for actual applications of the RWDA. A simple and reasonably accurate fit to gx [92] is based on its form for z = 0, 

The Fourier transform of g(k) gives the pair correlation function  

Equation (S.40) indicates the physical significance of represents the correlation length of the concentration fluctuation. 

Experimentally, the correlation length can be determined from the behaviour of g(A) in the small k region  

The definition of this apparent correlation length can be extended to dilute regimes, in which case app gives g/V3 (see eqn (2.72)). The behaviour of app is entirely different in dilute solutions and concentrated solutions. In dilute solutions, lapp dincreases with the molecular weight and the excluded volume, while in concentrated solutions app — I is independent of the molecular weight and decreases as a function of concentration and excluded volume (see eqn (5.36)). The reason can be easily understood from Fig. 5.1 once polymers overlap each other, the excluded volume interaction tends to make the concentration homogeneous. 

The simple theory given above is valid only at rather high concentration or at small excluded volume, i.e., near the 0 condition. At both these limits there are additional difficulties. At very high concentration the precise form of the potential matters. Also, near the 0 conditions the precise details of the interaction, in the sense of a cluster expansion passed to the two-body term, can also matter. In both of these limits it is possible to make an appropriate improvement, and the results have been found to be in good agreement with experiments.  

In most applications, it has been found more useful to employ the pair correlation function, defined below, rather than the pair distribution function itself. 

The symbol n may be read as and, i.e., the combination in (2.37) means that the first and the second events occur.  

Two events are called independent whenever the probabihty of their intersection is equal to the product of the probabilities of the two events. In general, the two separate events given in (2.37) are not independent the occurrence of one of them may influence the likelihood, or the probability, of occurrence of the other. For instance, if the separation R = I R — R I between the two elements is very small (compared to the molecular diameter of the particles), then fulfilling one event strongly reduces the chances of the second. 

We now invoke the following physically plausible contention. In a fluid, if the separation R between two elements is very large, then the two 

For any finite distance R, the factorization of (X, X ) into a product may not be valid. We now introduce the pair correlation function, which measures the extent of deviation from (2.39), and is defined by 

For any finite distance R, factoring of ) into a product may not be valid. 

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In general, it is diflfieult to quantify stnietural properties of disordered matter via experimental probes as with x-ray or neutron seattering. Sueh probes measure statistieally averaged properties like the pair-correlation function, also ealled the radial distribution function. The pair-eorrelation fiinetion measures the average distribution of atoms from a partieular site. 

Unlike the solid state, the liquid state cannot be characterized by a static description. In a liquid, bonds break and refomi continuously as a fiinction of time. The quantum states in the liquid are similar to those in amorphous solids in the sense that the system is also disordered. The liquid state can be quantified only by considering some ensemble averaging and using statistical measures. For example, consider an elemental liquid. Just as for amorphous solids, one can ask what is the distribution of atoms at a given distance from a reference atom on average, i.e. the radial distribution function or the pair correlation function can also be defined for a liquid. In scattering experiments on liquids, a structure factor is measured. The radial distribution fiinction, g r), is related to the stnicture factor, S q), by... 

It follows that the exact expression for the pair correlation function is... 

Hemmer P C 1964 On van der Waals theory of vapor-liquid equilibrium IV. The pair correlation function and equation of state for long-range forces J. Math. Phys. 5 75... 

Figure A2.4.11. Water pair correlation functions near the Pt(lOO) surface. In each panel, the frill curve is for water molecules in the first layer, and the broken curve is for water molecules in the second layer. From [30].

Hwang L-P and Freed J H 1975 Dynamic effects of pair correlation functions on spin relaxation by translational diffusion in liquids J. Chem. Rhys. 63 4017-25... 

Figure 2 Pair correlation functions of 0-0 and O-H at ( puted with the parameters of the SPC water model.

Figure 12 Pair correlation functions of Cl—H and N—H along the reaction path.

It is clear that Eq. (85) is numerically reliable provided is sufficiently small. However, a detailed investigation in Ref. 69 reveals that can be as large as some ten percent of the diameter of a fluid molecule. Likewise, rj should not be smaller than, say, the distance at which the radial pair correlation function has its first minimum (corresponding to the nearest-neighbor shell). Under these conditions, and if combined with a neighbor list technique, savings in computer time of up to 40% over conventional implementations are measured for the first (canonical) step of the algorithm detailed in Sec. IIIB. These are achieved because, for pairwise interactions, only 1+ 2 contributions need to be computed here before i is moved U and F2), and only contributions need to be evaluated after i is displaced... 

Although stratification, according to the plot in Fig. 10, occurs continuously as increases, it is accompanied by a curious structural reorganization in transverse directions (i.e., parallel to the planar substrate). A suitable measure of transverse structure is the pair correlation function defined in Eq. (62). However, for simplicity we are concerned only with the in-plane pair correlation function defined as [see Eq. (62)]... 

Fig. 11(a) displays plots of the in-plane pair correlation function for s = 2. and 3.0 well outside the regime where K exhibits its first maximum (see Fig. 12). The plots indicate that the transverse structures of one- and two-layer fluids (see Fig. 10) are essentially identical and typical of dense Lennard-Jones fluids. However, the transverse structure of a two-layer fluid is significantly affected as the peak of K is approached, as can be seen in Fig. 11(b) where g (zi,pi2) is plotted for s = 2.55 and 2.75, which points...

FIG. 11 The in-plane pair correlation function g (zi,Pi2) as a function of inter-... 

B. Gotzelmann, S. Dietrich. Density profiles and pair correlation functions of hard spheres in narrow slits. Phys Rev E 55 2993-3005, 1997. 

A comparison of the predictions for the cavity pair correlation function of some first-principles theories with computer simulation results [72] at a fairly high density has been presented by Stell (see Fig. 2 in Ref. 69). The... 

The pair correlation functions can be expressed directly in terms of the computed coefficients from Eq. (61) in particular, the number-number pair distribution function gN ir) and the number-number structure factor SNN k). Thus,... 

Finally, we relate the gradient of the local density ViPq(fi) to the pair correlation functions. For this purpose we take the gradient of the expansion (5)... 

In any relation given above, the knowledge of the total or direct pair correlation functions yields an equation for the density profile. The domain of integration in Eqs. (14)-(16) must include all the points where pQ,(r) 0. In the case of a completely impermeable surface, pQ,(r) = 0 inside the wall... 

A set of equations (15)-(17) represents the background of the so-called second-order or pair theory. If these equations are supplemented by an approximate relation between direct and pair correlation functions the problem becomes complete. Its numerical solution provides not only the density profile but also the pair correlation functions for a nonuniform fluid [55-58]. In the majority of previous studies of inhomogeneous simple fluids, the inhomogeneous Percus-Yevick approximation (PY2) has been used. It reads... 

Far from the surface, the theory reduces to the PY theory for the bulk pair correlation functions. As we have noted above, the PY theory for bulk pair correlation functions does not provide an adequate description of the thermodynamic properties of the bulk fluid. To eliminate this deficiency, a more sophisticated approximation, e.g., the SSEMSA, should be used. 

However, before proceeding with the description of simulation data, we would like to comment the theoretical background. Similarly to the previous example, in order to obtain the pair correlation function of matrix spheres we solve the common Ornstein-Zernike equation complemented by the PY closure. Next, we would like to consider the adsorption of a hard sphere fluid in a microporous environment provided by a disordered matrix of permeable species. The fluid to be adsorbed is considered at density pj = pj-Of. The equilibrium between an adsorbed fluid and its bulk counterpart (i.e., in the absence of the matrix) occurs at constant chemical potential. However, in the theoretical procedure we need to choose the value for the fluid density first, and calculate the chemical potential afterwards. The ROZ equations, (22) and (23), are applied to decribe the fluid-matrix and fluid-fluid correlations. These correlations are considered by using the PY closure, such that the ROZ equations take the Madden-Glandt form as in the previous example. The structural properties in terms of the pair correlation functions (the fluid-matrix function is of special interest for models with permeabihty) cannot represent the only issue to investigate. Moreover, to perform comparisons of the structure under different conditions we need to calculate the adsorption isotherms pf jSpf). The chemical potential of a... 

First we are looking for the adsorption of a fluid consisting of particles of species m, in a slit-like pore of width H. The pore walls are chosen normal to the z axis and the pore is centered at z = 0. Adsorption of the fluid m, i.e., the matrix, occurs at equihbrium with its bulk counterpart at the chemical potential The matrix fluid is then characterized by the density profile, p (z) and by the inhomogeneous pair correlation function A (l,2). The structure of that fluid is considered... 

The simulations are repeated several times, starting from different matrix configurations. We have found that about 10 rephcas of the matrix usually assure good statistics for the determination of the local fluid density. However, the evaluation of the nonuniform pair distribution functions requires much longer runs at least 100 matrix replicas are needed to calculate the pair correlation functions for particles parallel to the pore walls. However, even as many as 500 replicas do not ensure the convergence of the simulation results for perpendicular configurations. 

The correlation functions of the partly quenched system satisfy a set of replica Ornstein-Zernike equations (21)-(23). Each of them is a 2 x 2 matrix equation for the model in question. As in previous studies of ionic systems (see, e.g.. Refs. 69, 70), we denote the long-range terms of the pair correlation functions in ROZ equations by qij. Here we apply a linearized theory and assume that the long-range terms of the direct correlation functions are equal to the Coulomb potentials which are given by Eqs. (53)-(55). This assumption represents the mean spherical approximation for the model in question. Most importantly, (r) = 0 as mentioned before, the particles from different replicas do not interact. However, q]f r) 7 0 these functions describe screening effects of the ion-ion interactions between ions from different replicas mediated by the presence of charged obstacles, i.e., via the matrix. The functions q j (r) need to be obtained to apply them for proper renormalization of the ROZ equations for systems made of nonpoint ions. 

A. M. Walsh, R. D. Coalson. Lattice field theory for spherical macroions in solution Calculation of equilibrium pair correlation function. J Chem Phys 700 1559-1566, 1994. 

Figure. 3 (a) Partial pair correlation function.s gij(B.) in liquid K-Sb alloys, (b) Total, partial, and local electronic densities of states in liquid Ko.soSbo.so- Cf. text. 

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