Definition of the NDF for granular systems

Consider first a granular system of Np particles in the absence of a surrounding fluid phase. The dynamics of the nth particle can be described by the position of its center of mass X V 

Note that this system has a total of Ad = Ap(3 + 3 + M) degrees of freedom. 

For each set of initial conditions, Eqs. (4.1)-(4.3) can be solved to And X , U , and The initial conditions are randomly selected from known distribution functions, and we can assume that there is an infinite number of possible combinations. Each combination is called a realization of the granular flow, and the set of all possible realizations forms an ensemble. Note that, because the particles have finite size, they cannot be located at the same point thus X 4 X for n 4 m. Also, the collision operator will generate chaotic trajectories and thus the particle positions will become uncorrelated after a relatively small number of collisions. In contrast, for particles suspended in a fluid the collisions are suppressed and correlations can be long-lived and of long range. We will make these concepts more precise when we introduce fluid-particle systems later. While the exact nature of the particle correlations is not a factor in the definition of the multi-particle joint PDF introduced below, it is important to keep in mind that they will have 

The probability statement P[AnBnC] is interpreted as the probability that events A, B, and C occur together. The probability is computed as the fraction of all realizations in the ensemble for which the statement Ar BC C is true. 

The probability on the right-hand side is again defined with respect to the ensemble of all realizations of the granular flow. The dynamical behavior of the NDF will be determined by the right-hand sides of Eqs. (4.1)-(4.3), and we will return to this aspect later. First, however, we will look at methods to estimate the NDF that are based on a single realization of the granular flow. 

---