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

## TL;DR

This paper introduces the mean drift concept to adapt mean field theory for analyzing middle-sized population processes with bounded agents, extending traditional models to more realistic scenarios.

## Contribution

It develops a systematic method to derive mean drift and constructs new differential equations for analyzing finite population processes.

## Key findings

- Mean drift provides a tailored approximation for bounded population sizes
- New differential equations improve analysis accuracy for middle-sized systems
- Propagation of chaos underpins the derivation of mean drift

## Abstract

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 is derived through a systematic application of 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.