Pair correlation function normalized

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... 

In sections 3.5 and 4.2 we inferred from the normalization condition on the pair correlation function that the pair correlation function must have different behaviors in an open (O) and in a closed (C) system. In this appendix, we further elaborate on this aspect of the pair correlation function. 

For most practical purposes, when we are interested in the behavior of the pair correlation function itself, the difference between (G.3) and (G.6) is negligible for macroscopic system, where N-s 1023. However, this small difference becomes important when we integrate over macroscopic volumes. This is clear from the following two exact normalization conditions (see section 3.5)... 

However, it is possible that the two conditions may be equivalent.) For the sake of clarity of discussions in the following sections we assume this stronger condition i.e., existence of the convergent series for F p, and the cell-pair correlation function which can be obtained from F p - Note, however, that this assumption can always be replaced by a (possibly) weaker assumption [for example, the condition (b) for /4 p ] in the ensuing discussion. Similar Taylor expansion for the normalization constant C in (4) can be carried out however, we shall not discuss it here explicitly. Only the final result of the expansion will be presented at the end of the derivation. 

Fig. 15. Anisotropic oxygen-oxygen pair correlation functions gooip, for the adsorbate molecules (left), the molecules in the second layer (middle), and the molecules in the bulk-like center of the water lamina (right) between Pt(lOO) surfaces, p is the transversal and z the normal part of the interatomic distance.

The pair distribution function clearly has dimensions (density), and it is normal to introduce the pair correlation function g(Aj, A2) defined by... 

The autocorrelation function is sometimes referred to simply as the correlation function. Among those working in crystal structure analysis, the autocorrelation function is known as the Patterson function. Many of the distribution functions obtained from scattering intensity data are in the nature of the correlation function, with possible differences in the normalization constant or a constant term. Functions in this vein include the pair correlation function or the radial distribution function (and its uniaxial variant cylindrical distribution function), discussed in Chapter 4. 

In this equation the first term describes the so-called direct correlations between the molecules 1 and 2 while the second term describes the correlations via a third particle. The direct correlation fimction C 2(o i, W2) generally describes short-range intermolecular correlations and decays rapidly with increasing intermolecular separation while the full pair correlation function 32( 1,012) normally has a long-range tail. Note that from a mathematical point of view Ci(w) and C2(wi,w2) are functions of w and, simultaneously, can be considered as functionals of the density p(w). For example, the functional derivative 5Ci oj )/5p(uj2) = C2(wi, W2)- Thus it is more precise to use the forms Ci(w, [p(w)]) and C2(wi, W2, [p(wi)], [ ( 2)]), which will be used below. 

In Fig. 2.26 we show the pair correlation function g(R) for the square-well potential (Fig. 2.10b) and for the primitive model (Fig. 2.10d). These two should represent the normal liquid and the water-like liquid. Note that in the case of the normal liquid (Fig. 2.26a) the height of the first peak of (R) increases as we increase the density of the liquid. We have seen this behavior in Sec. 1.4, Fig. 1.28. 

In these relatively simple solvable models we also introduce several concepts which normally appear in the context of the theory of liquids, such as the analogue of the solvation process, the pair correlation function and potential of average force, triplet... 

The site-site intramolecular and intermolecular pair correlation functions play the same role in ISM fluids as the radial distribution function does in atomic fluids. The site-site intramolecular distribution function, coay(r), is the probability that two sites a and y are a distance r apart (normalized so that f dreoay(T) = 1). The site-site intermolecular pair distribution function, gay(r) is defined via... 

Fig. 24.2. Single-molecule recording of T4 lysozyme conformational motions and enzymatic reaction turnovers of hydrolysis of an E. coli B cell wall in real time, (a) This panel shows a pair of trajectories from a fluorescence donor tetramethyl-rhodamine blue) and acceptor Texas Red (red) pair in a single-T4 lysozyme in the presence of E. coli cells of 2.5mg/mL at pH 7.2 buffer. Anticorrelated fluctuation features are evident. (b) The correlation functions (C (t)) of donor ( A/a (0) Aid (f)), blue), acceptor ((A/a (0) A/a (t)), red), and donor-acceptor cross-correlation function ((A/d (0) A/d (t)), black), deduced from the single-molecule trajectories in (a). They are fitted with the same decay rate constant of 180 40s. A long decay component of 10 2s is also evident in each autocorrelation function. The first data point (not shown) of each correlation function contains the contribution from the measurement noise and fluctuations faster than the time resolution. The correlation functions are normalized, and the (A/a (0) A/a (t)) is presented with a shift on the y axis to enhance the view, (c) A pair of fluorescence trajectories from a donor (blue) and acceptor (red) pair in a T4 lysozyme protein without substrates present. The acceptor was photo-bleached at about 8.5 s. (d) The correlation functions (C(t)) of donor ((A/d (0) A/d (t)), blue), acceptor ((A/a (0) A/a (t)), red) derived from the trajectories in (c). The autocorrelation function only shows a spike at t = 0 and drops to zero at t > 0, which indicates that only uncorrelated measurement noise and fluctuation faster than the time resolution recorded (Adapted with permission from [12]. Copyright 2003 American Chemical Society)...

Another function often denoted g r) is called the pair distribution function. It is normalized so that, as r -r oo, g(r) 1 and has the property that for r shorter than the distance of closest approach of pairs of atoms g r) becomes zero. It is closely related to the pair density function, p(r) = pog(r). Clearly, p(r) oscillates about, and then asymptotes to, the average number density of the material, po at high-r and becomes zero as r 0. The relationship between these correlation functions is given by ... 

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