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Single-particle and two-particle probability density functions are obtained from N-particle probability density function by integration over the remaining variables
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In terms of the Dirac delta function, the single-particle or number density function n(1)(r) of the medium is defined as
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and the integration of n(1)(r) over the whole medium gives the total number of particles:
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The average number density, (n(1) (r)), is equal to
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n(l) (r)p(rl' r2, .. , rN)
N L p(r)
= Np(r) (4.5.18)
where p(r) is the single-particle probability density function of (4.5.14). If the medium is statistically homogeneous, the distributions of particles are equally probable everywhere within the volume V. Hence the single particle
probability density function is p(r) = 1/V. The average number density, no, will be denoted by N no= V A two-particle number density function n(2) (r, r ' ) can be defined as
n(2)(r, r') =
L I:>5(r - ri)8(r' - rj)
This function is zero unless there are two particles simultaneous at two different positions ri and rj. The integration of n(2)(r, r ' ) over the variable r' will give
n(2) (r,r')dr' = (N -l)n(1)(r)
where n(l)(r) is given by (4.5.16). It is noted that n(2)(r, r ' ) i= n(1) (r)n(l) (r' ), because the occupation of position r ' can be strongly influenced by that of r when particles are densely packed. The average of two-particle number density function (n(2) (r, r ' )) is equal to
(n(2)(r, r'))
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II dr
I:: I::p(r, r') = N(N j=l j#l
l)p(r, r ' )
where p(r, r' ) is the two-particle probability density function of (4.5.15). In terms of the two-particle probability density function, the pair distribution function g(r, r ' ) is defined by
(n(2)(r, r ' )) = (n(l)(r))(n(l)(r'))g(r, r')
Thus the pair distribution function g(r, r ' ) is proportional to the two-particle probability density function p(r, r ' ). For a homogeneous random medium, (n(1)(r)) = (n(1)(r' )) = no, then
(n(2) (r, r')) = n6g(r, r')
If Ir-r/l ---t 00, then, p(r, r ' ) ---t p(r)p(r' ), and it is expected that g(r, r ' ) ---t 1. The definitions and formulae for the number densities and pair distribution function can be generalized to the cases of mixture or multispecies of particles and adhesive particles. In condensed matter physics and molecular liquid physics, the pair distribution functions of atom or molecule positions
6 Gaussian Rough Surface and Spectral Density
have been obtained from theoretical models, computer simulations, and experimental measurements [Ziman, 1979]. The applications of these methods to derive the pair distribution functions for discrete random medium study will be described in Volume II. For the geophysical remote sensing application, snow sections prepared stereologically have been analyzed to determine a family of pair distribution functions that can be used to calculate the radar backscatter from snowcover [Zurk et al. 1997].
Gaussian Rough Surface and Spectral Density
Since the characterizations of the terrain surface of interest are frequently very difficult to obtain from field measurements, various geoscience remote sensing applications require the use of random rough surface models. In a random surface model, the elevation of surface, with respect to some mean surface, is assumed to be a stochastic process. To characterize a random process of surface displacement, it generally requires a multivariate probability density function of surface heights. For naturally occurring surfaces, it is reasonable to assume a Gaussian (or normal) height distribution, and to be stationary, meaning that its statistical properties are invariant under the translation of spatial coordinates. Although realistic rough surface profiles are not necessarily Gaussian, the use of Gaussian statistics greatly reduces the complexities associated with such random processes. A complete description of the Gaussian random process is given by its mean and covariance function alone.