System size expansion

The system size expansion, also known as van Kampen's expansion or the Ω-expansion, is a technique pioneered by Nico van Kampen used in the analysis of . Specifically, it allows one to find an approximation to the solution of a master equation with nonlinear transition rates. The leading order term of the expansion is given by the linear noise approximation, in which the master equation is approximated by a Fokker–Planck equation with linear coefficients determined by the transition rates and stoichiometry of the system.

Less formally, it is normally straightforward to write down a mathematical description of a system where processes happen randomly (for example, radioactive atoms randomly decay in a physical system, or genes that are expressed stochastically in a cell). However, these mathematical descriptions are often too difficult to solve for the study of the systems statistics (for example, the mean and variance of the number of atoms or proteins as a function of time). The system size expansion allows one to obtain an approximate statistical description that can be solved much more easily than the master equation.

Systems that admit a treatment with the system size expansion may be described by a probability distribution ${\displaystyle P(X,t)}$, giving the probability of observing the system in state ${\displaystyle X}$ at time ${\displaystyle t}$. ${\displaystyle X}$ may be, for example, a vector with elements corresponding to the number of molecules of different chemical species in a system. In a system of size ${\displaystyle \Omega }$ (intuitively interpreted as the volume), we will adopt the following nomenclature: ${\displaystyle \mathbf {X} }$ is a vector of macroscopic copy numbers, ${\displaystyle \mathbf {x} =\mathbf {X} /\Omega }$ is a vector of concentrations, and ${\displaystyle \mathbf {\phi } }$ is a vector of deterministic concentrations, as they would appear according to the rate equation in an infinite system. ${\displaystyle \mathbf {x} }$ and ${\displaystyle \mathbf {X} }$ are thus quantities subject to stochastic effects.

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