Random Changes in Particle State

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. 

Although it is possible to address this situation for the general particle state vector including internal and external coordinates, we shall take the route of establishing the results for the one-dimensional case and proceed to infer the generalization for the vectorial case without elaborate derivation. 

Consider again a population of particles distributed according to a scalar state variable x, which we shall take to vary over the entire real, line and let /i(x, t) be the number density. The scalar state x is presumed to vary in 

Consider any property of the population calculated by summing that associated with each particle in the population. Denote the property associated with a single particle of state x by (x). Then the property associated with the entire population is given by 

Suppose we are interested in how the property G t) is changing with respect to time. Then we may write 

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Broadly, Monte Carlo methods are based on artificial realization of the system behavior. Our primary concern will be the simulation of systems in which the particle state changes deterministically. Thus, the systems encountered in Section 2.10 with random changes in particle state will be excluded for the present from our discussion in favor of a sketchy treatment of this case at a later stage. 

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