The Chemical Master Equation: From Reactions to Complex Networks

This project investigates the chemical master equation and its links to complex networks. The report is composed of two parts: an introduction, deriving the chemical master equation from some basic results of statistical mechanics and probability theory, and a second part, relating the formalism of master equations to growing network models and random walks on graphs. At the end of the first part, further analytical and numerical results about Markov processes are reported and discussed. 1 The Physics behind the Chemical Master Equation The mathematical modelling of chemically reacting gaseous systems, via the framework of Markovian stochastic processes, relies on some delicate hypotheses from statistical mechanics [3]. In this section, we review these basic results, with the aim of outlining a physically coherent approach to the mathematics of the chemical master equation for chemical kinetics. 1.1 Some Physical Premises Historically, the modelling of chemical reactions as stochastic processes was introduced in [2] and became increasingly popular in the 1950s and 1960s. However, it was only in the nineties, with the work of Gillespie [1], that a rigorous microphysical derivation of such approach was provided, in order to demonstrate its a priori modelling validity. Before that date, in fact, it was possible to perform such fidelity check only a posteriori, through comparisons with real or molecular dynamics experiments [2, 13].