Systematic derivation of a macroscopic equation and the Fokker-Planck equation by scaling expansion method in stochastic processes

The coarse-graining method for a gain-loss type master equation which describes time evolution of a system following a Markov process, are discussed by introducing a scaling concept. The master equation expressed in new variables scaled by two characteristic sizes of a system, is expanded in Ω-1 and Ωsup(-1/2). Thus, not only a macroscopic equation of motion and the Fokker-Planck equation, but also a continuity equation of hydrodynamics, are automatically obtained through the Kramers-Moyal expansion