There are important engineering and physical processes that appear random. A canonical example is the motion of Brownian particles, discussed in Chapter 11. In random or stochastic processes, in contrast to deterministic ones, there is not one single outcome in the time evolution of a system, even when initial conditions remain identical. Instead there may be different outcomes, each with a certain probability. In this chapter we present stochastic processes and derive a general framework for determining the probability of outcomes as a function of time.

We are starting the discussion with reacting systems away from the thermodynamic limit. Typically, reacting systems are modeled with ordinary differential equations that express the change of concentrations in time as a function of reaction rates. This continuous and deterministic modeling formalism is valid at the thermodynamic limit. Only when the number of molecules of reacting species is large enough can the concentration be considered a continuously changing variable. Importantly, the reaction events are considered to occur deterministically at the thermodynamic limit. This means that there is certainty about the number of reaction events per unit time and unit volume in the system, given the concentration of reactants.

On the other hand, if the numbers of reacting molecules are very small, for example in the order of O(10-22NA), then integer numbers of molecules must be modeled along with discrete changes upon reaction.