A Power Series Expansion of the Master Equation

In order to solve the master equation by a systematic approximation method, an expansion in powers of some parameter is needed. The appropriate parameter is the reciprocal size of the system, defined as the ratio of intensive and extensive variables. The lowest approximation yields the phenomenological law for the approach to equilibrium. The next approximation determines the mean square of the fluctuations about the phenomenological behavior. In equilibrium this approximation has the form of a linear Fokker–Planck equation. The higher approximations describe the effect of the non-linearity on the fluctuations, in particular on their spectral density. The method is applied to three examples: density fluctuations, Alkemade's diode, and Rayleigh's piston. The relation to the expansion recently given by Siegel is also discussed.