How to build master equations for complex systems

Typical complex systems, e. g., complex chemical reactions, reaction-diffusion systems, and turbulent fluids are described on a macroscopic level, that is, neglecting fluctuations, with the help of deterministic equations for corresponding variables. In this article it is shown on a phenomenological level, that these systems can be described in terms of integer- or real-valued Markov processes as well, which are governed by master equations. The latter are constructed such that the macroscopic law and the fluctuations around it are reproduced correctly. Stochastic processes defined through master equations can easily be simulated. The efficiency, the stability and the parallelization of the algorithms for stochastic simulations are discussed for some examples. In the last part of the paper it is shown that the same phenomenological approach can be successfully applied to open quantum systems. The wave function is assumed to be a complex valued stochastic process in Hilbert space and the quantum master equation for the statistical operator is regarded as the equation of motion for the two-point correlation function.