Master density function

First, the master density function is introduced in Section 7.1. The scalar particle state is discussed in detail in Section 7.1.1 with directions for generalization to the vector case in Section 7.1.2. Coupling with the continuous phase variables is ignored in the foregoing sections, but the necessary modifications for accommodating the environmental effect on the particles are discussed in Section 7.1.3. Thus, from Section 7.1, the basic implements of the stochastic theory of populations along with their probabilistic interpretations become available. These implements are the master density, moment densities that are called product densities, and the resulting mathematical machinery for the calculation of fluctuations. 

Since the master density function is insensitive to the permutation of the particle state arguments, the integrals in the inner sum are all the same so that... 

Notice in particular the first term on the right-hand side where a Dirac delta function appears because of integration with respect to z. The second term on the right-hand side has been identified as such based on the assumption that z z since Zj z while integrating the previous equation. We now make use of the symmetry properties of the master density function. In both terms on the right-hand side of the preceding equation, the summands within the inner sums are independent of the index of summation so that we may write... 

We now investigate the product densities arising out of this master density function. 

A similar derivation is possible for the master density function of an aggregation process but is left to the reader. Instead, we will consider the derivation of equations for an aggregation process in Section 7.3 directly using product densities. 

We now define a more general master density function than that defined in Section 7.1.1. Again, we confine the treatment to the scalar particle state, denoted x, since the extension to the vector case, as seen in Section 7.1.2, is straightforward. 

The integral U((7m) represents the area under the engineering stress-strain master curve up to the maximum stress (7. The static stress is a function of the energy density function U([Pg.18]

One of the most important and elusive quantities in molecular djmamics calculations of reacting system is the intermolecular energy transfer probability density function P(E ,E) which is used in master equation calculations of reaction rate coefficients [1]. 

The master density should be concerned not only with the population of particles and their states, but also with the state of the environment. Thus, we define the function 2 0 such that... 

The unfilled parentheses are either for the master density as in (7.3.12) or for the product density functions as in the other equations. The vector case is now fully identified. In Section 7.4.2, we discuss an application of the foregoing development. 

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