Stochastic Equations of Population Balance

Pr[Particle does not suffer breakage during interval (t — dt, t) but adds mass X(x ) 

Transposing the second term on the right to the left, dividing by dt, and letting dt tend to zero, we obtain 

Equation (7.3.1) yields the usual population balance equation 

The boundary condition at x = 0 is obtained by argument that the probability there is a particle (nucleus) between 0 and X(0) dt on the size coordinate during time t and t + dt can be obtained in two different ways. First, it is given by the left-hand side of the equation appearing below by definition of the first-order product density. Second, it is also given by the right-hand side by definition of the nucleation rate, which is the transition probability for the appearance of a nucleus in the time interval t, t + dt). Thus, 

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Third, 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. 

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. 

The population balance equations considered so far were for systems in which particles changed their states deterministically. Thus specification of the state of the particle and its environment was sufficient to determine the rate of change of state of that particle. Applications may, however, be encountered where the particle state may change randomly as determined, for example, by a set of stochastic differential equations. Since, however, the population balance equation is a deterministic equation, our desire is to seek the expected displacement of particles moving randomly in particle state space during an infinitesimal interval dt. 

In dealing with stochastic problems, it became clear from Section 7.3 that one is frequently faced with lack of closure, especially in situations where interaction occurs between particles or between particles and their environment. Such lack of closure arises because of the development of correlations between particle states promoted by preferential behavior between particle pairs of specific states or between the particle and its environment. The population balance equation, which generally comes about by making the crudest closure approximation, does not make accurate predictions in such cases of the average behavior of the system. The question naturally arises as to whether one can find other mean field descriptions by making more refined closure approximations on the unclosed product density equations. 

Because radical entry, exit, and termination are stochastic events, particles have a different number of radicals and the number of radicals in a given particle varies stochastically with time. Equation (14) gives ffie population balance of particles... 

The derivation of regular patterns of cellular behavior that can be assigned to certain defined subpopulations (or single cells) from measured data necessarily involves modeling approaches that allow assignment of mechanistic (e.g., metabohc) models to subpopulations with individual parameters or other variations. Further, such models must allow the tracking of individual cell s dynamic behavior even when they fluctuate between different populations. Two principal approaches can be distinguished systems of partial different equations (population balance systems, PBE) and stochastic cell ensemble models (CEMs)... 

The parameters appearing in the present stochastic model can be easily estimated from some correlations and determined by eonducting simple batch experiments. The transient behavior of molecules can be evaluated by analytically solving a system of the governing ordinary differential equations of the present stochastic model. The stochastic modeling, therefore, provides an easier approach than the deterministic population balance modeling in which a partial differential equation must be solved. These facts are very attractive from the practical point of view. 

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