Average values of pairwise quantities

A pairwise quantity is a function that is expressible as a sum of terms, each of which depends on the configuration of a pair of particles, namely 

All the steps taken in (3.4) are quite simple. It is instructive, however, to go through them carefully since this is a standard procedure in the theory of classical fluids. In the first step on the rhs of (3.4), we have merely interchanged the signs of summation and integration. The second step is important. We exploit the fact that all particles are equivalent thus each term in the sum has the same numerical value, independent of the indices /, j. Hence, we replace the sum over N(N — 1) terms by N(N — 1) times one integral. In the latter, we have chosen the (arbitrary) indices 1 and 2, 

A simpler version of (3.5) occurs when the particles are spherical so that each configuration X consists of only the locational vector R. This is the most frequent case in the theory of simple fluids. The corresponding expression for the average in this case is 

A common case that often occurs is when the function /(R, R ) depends only on the separation between the two points R = R — R. In addition, for homogeneous and isotropic fluids, (R, R ) also depends only on R, This permits the transformation of (3.6) into a one-dimensional integral, which, in fact, is the most useful form of the result (3.6). To do this, we first transform to relative coordinates 

Since the integrand in (3.8) depends on the scalar R, we can integrate over all the orientations to get the final form 

Clearly, due to the equivalence of the particles, we could have chosen any other two indices. The third and fourth steps make use of the definition of the pair distribution function defined in section 5.2.2. 

The derivations carried out in this chapter apply to systems of simple spherical particles. We shall also point out the appropriate generalizations for non-spherical particles that do not possess internal rotations. For particles with internal rotations, one needs to take the appropriate average over all conformations. An example of such an average is discussed in chapter 7. 

There are some steps common to most of the procedures leading to the relations between thermodynamic quantities and the pair distribution function. Therefore, in the next section we derive a general theorem connecting averages of pairwise quantities and the pair distribution function. 

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The use of these formulas involves subdividing the visible spectrum into L (at least 16) intervals. The subscript n labels these wavelength intervals. / characterizes the intensity of illumination in interval n, while Rn is the average reflection factor for this interval. The quantities xn, yn, and zn are found in tables (contained in standards) for typical illumination conditions and for an observer with normal vision. If two colored samples are of the same color, their X, Y, and Z values are equal pairwise. If two objects have the same color under a given illumination, this... 

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