Page 328 - Numerical Methods for Chemical Engineering

7 Probability theory and

stochastic simulation

We now consider probability theory, and its applications in stochastic simulation. First, we
define some basic probabilistic concepts, and demonstrate how they may be used to model
physical phenomena. Next, we derive some important probability distributions, in particular,
the Gaussian (normal) and Poisson distributions. Following this is a treatment of stochastic
calculus, with a particular focus upon Brownian dynamics. Monte Carlo methods are then
presented, with applications in statistical physics, integration, and global minimization
(simulated annealing). Finally, genetic optimization is discussed. This chapter serves as a
prelude to the discussion of statistics and parameter estimation, in which the Monte Carlo
method will prove highly useful in Bayesian analysis.

The theory of probability

A stochastic system is one whose behavior is not purely deterministic and predictable, but
rather has (assumed inherent) randomness. The theory of probability provides a mathemat-
ical framework for understanding and modeling such systems. Rather than provide abstract
definitions, we introduce the subject through an example: modeling the distribution of
polymer chain lengths in condensation polymerization.

Condensation polymers

Let us consider a reacting system comprising two chemical species. The first has a number α 1
of acid groups, e.g. −COOH, and the second has a number β 2 of base end groups, e.g. −OH
or −NH 2 . These acid and base end groups react with each other to form linkages, −CONH−
or −COO− and a condensate molecule (e.g. water) (Figure 7.1). An example is terephthalic
acid and ethylene glycol (Figure 7.2), which react to form poly(ethylene terephathalate),
PET, a common material found in soda bottles and clothing. As each molecule contains two
functional groups, the reaction produces linear polymer chains with no branching.
If one of the species contains more than two functional groups, the polymer chains are
not linear, but rather contain many branches, and at sufficiently high conversions (above
the gel point) a cross-linked network is formed (Figure 7.3). We wish to apply probability
theory to understand how the distribution of chain lengths depends upon the conversion of

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