Van Kampen's system-size expansion

The van Kampen's expansion is a systematic expansion of the master equation in powers of the system size (volume, number of particles). The leading order terms give deterministic equations which govern the evolution of infinite systems where stochastic fluctuations can be neglected. Second-order terms give rise to a Fokker-Planck equation which describes fluctuations for large but finite systems.

Though it may seem to be yet another text-book method which never gives any useful results in practice, I must say that I have had a pleasure of using the van Kampen's expansion recently and, surprisingly, it has worked very well even for a problem of oscillations in an idealised gene-expression model.

In my talk I will first explain the method on a very simple example which can also be solved exactly without the van Kampen, and then I will show how well it works in more complicated situations.