The Statistical Foundation of Population Balances

In view of the numerous issues involved for discussion, the organization of this chapter deserves some explanation. 

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. 

Second, Section 7.2 provides the derivation of equations in the master density for some of the particulate processes discussed in Chapter 3 and shows the combinatorial complexity of their solution. The objective of this section is to show how Monte Carlo simulations of a process eliminate the quantitatively less significant combinatorial elements of the solution by artificial realization. Thus, the discussion here is of largely conceptual value. 

Section 7.3 goes into the direct derivation of stochastic equations of population balance. These equations are also obtainable from averaging the master density equations of Section 7.2, but are best obtained by using the methodology of Section 7.3. Some applications of stochastic analysis are shown in this section, which are of focal interest to the subject of this chapter. 

Fourth, in Section 7.4 we examine the closure problem arising in aggregating systems as well as in random environments and some approximations that have been used to obtain closure. 

---

Chapter 7 is concerned with the statistical foundation of population balance models. The chapter deals with master density formulations leading to mean field equations for the average behavior of the system and fluctuations about average behavior. This represents the subject of stochastic population dynamics applicable to small systems the relevance of which to engineering is discussed. Departures of the mean field equations from population balance equations are demonstrated. The mean field equations so obtained suffer from lack of closure. Closure approximations are presented suggesting more complex mean field equations than population balance along with applications. 

---