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In this section we describe Monte Carlo simulations of the pair distribution function of dense discrete random media with multiple sizes of particles. Two Monte Carlo methods, namely, the Metropolis technique and the sequential addition of particles, are commonly used. Particle positions are generated randomly in a cubic box. The number of particles used is between 400 and 4000. In the Metropolis Monte Carlo technique the particles are next shuffled to create different realizations. On the other hand, the sequential addition method adds particles serially and randomly into the box. Thus, in the sequential addition method the particles are not shuffled. In positioning the particles, we assume that there are no interparticle forces except that interpenetration is not allowed. Many realizations are used. We have used up to 3000 realizations. The pair distribution functions are calculated by using the definition of joint probability density functions and by counting the occurrence of pair separation of particles as a function of separation distance. The counting is averaged over these realizations. The edge effects of the cubic box are taken into account by adopting the periodic boundary conditions. The Monte Carlo results for the pair distribution function are illustrated and compared with the results of the Percus-Yevick approximation for the single-size case and for the case of multiple sizes. They are found to be in good agreement. The Monte Carlo simulations show that the Percus-Yevick approximations of single size and multiple sizes can be used for macroscopic objects such as particles in geophysical terrain and composite materials. The computer experiments are performed on a system with N number of particles. These N spheres with different sizes are placed inside a cubic cell of side length fJ.. Let N i be the number of particles of radius ai. Then number

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density ni and fractional volume Ii for particles of radius ai are given by Ni ni = 3 (8.3.1)

(8.3.2) (8.3.3)

.

(8.3.4)

where I is the total fractional volume occupied by the particles. We note that the pair distribution function gij (1') approaches unity as l' becomes large, meaning that the particle positions are independent if they are far apart. For a fractional volume of less than 40%, the pair distribution functions are practically equal to unity for pair separations larger than five diameters. Thus taking N between 400 and 4000 is generally sufficient to simulate the pair distribution function. In the simulations, periodic boundary conditions are employed. The N particles of interest are placed inside the central cell, and this cental cell is taken to be surrounded by the periodic images of itself. Each image cell contains N particles with exactly the same geometric arrangement as that in the central primary cell.

We use the technique of Metropolis et al. [1953] for both the 3-D case and 2-D case. The 2-D case is described in 9, Section 2. For the 3-D case, we consider a cubic box containing N particles. Initially, the N spheres are placed randomly inside the primary cell with no overlap. To generate new realizations, the particles are shuffled as follows. In each cycle every particle is subject to be randomly displaced once. The acceptance of its new position is according to whether it overlaps another sphere or not. The displacement is random and is not governed by any interparticle force except that it cannot penetrate other particles. We should note that the displacement of particles introduced in the simulation is for the purpose of creating new configurations and realizations. It does not mean that the particles move physically. The steps are as follows.

Step 1: Set an initial configuration of the system. The initial configuration may be produced by assigning the particle coordinates within the cell. All the coordinates of spheres lie in the range [0, ]. The initial configuration

may be obtained by placing all the particles in a periodic spatial function in the box. Step 2: Change the system configuration by random displacement of particles. The particles are displaced sequentially in the simulation. The displacement of a particle to a new trial position is determined according to the following prescription:

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