If the master equation possesses nonlinear transition rates, it may be impossible to solve it analytically. The system size expansion utilises the ansatz that the variance of the steady-state probability distribution of constituent numbers in a population scales like the system size. This ansatz is used to expand the master equation in terms of a small parameter given by the inverse system size.
Specifically, let us write the , the copy number of component , as a sum of its "deterministic" value (a scaled-up concentration) and a random variable , scaled by :
The probability distribution of  can then be rewritten in the vector of random variables :
Consider how to write reaction rates  and the step operator  in terms of this new random variable. Taylor expansion of the transition rates gives:
The step operator has the effect  and hence :
We are now in a position to recast the master equation.
This rather frightening expression makes a bit more sense when we gather terms in different powers of . First, terms of order  give
These terms cancel, due to the macroscopic reaction equation
The terms of order  are more interesting:
which can be written as
where
and
The time evolution of  is then governed by the linear Fokker–Planck equation with coefficient matrices  and  (in the large- limit, terms of  may be neglected, termed the linear noise approximation). With knowledge of the reaction rates  and stoichiometry , the moments of  can then be calculated.
The approximation implies that fluctuations around the mean are Gaussian distributed. Non-Gaussian features of the distributions can be computed by taking into account higher order terms in the expansion.