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The statement of the mean field approximation theorem in the mean field theory of Markov processes particularly targets the behaviour of population processes with an unbounded number of agents. However, in most real-world engineering applications one faces the problem of analysing middle-sized systems in which the number of agents is bounded. In this paper we build on previous work in this area and introduce the mean drift. We present the concept of population processes and the conditions under which the approximation theorems apply, and then show how the mean drift can be linked to observations which follow from the propagation of chaos. We then use the mean drift to construct a new set of ordinary differential equations which address the analysis of population processes with an arbitrary size.

The Mean Drift: Tailoring the Mean Field Theory of Markov Processes for Real-World Applications