HOW TO BUILD MASTER-EQUATIONS FOR COMPLEX-SYSTEMS

Typical complex systems, e. g., complex chemical reactions, reaction-d iffusion systems, and turbulent fluids are described on a macroscopic level, that is, neglecting fluctuations, with the help of deterministi c equations for corresponding variables. In this article it is shown o n a phenomenological level, that these systems can be described in ter ms of integer- or real-valued Markov processes as well, which are gove rned by master equations. The latter are constructed such that the mac roscopic law and the fluctuations around it are reproduced correctly. Stochastic processes defined through master equations can easily be si mulated. 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 phenomenologi cal 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 oper ator is regarded as the equation of motion for the two-point correlati on function.