The Fokker-Planck Approach to Complex Spatio-Temporal Disordered Systems

When the complete understanding of a complex system is not available, as, e.g., for systems considered in the real-world, we need a top-down approach to complexity. In this approach one may start with the desire to understand general multi-point statistics. Here such a general approach is presented and discussed based on examples from turbulence and sea waves. Our main idea is based on the cascade picture of turbulence, entangling fluctuations from large to small scales. Inspired by this cascade picture, we express the general multi-point statistics by the statistics of scale-dependent fluctuations of variables and relate it to a scale-dependent process, which finally is a stochastic cascade process. We show how to extract from empirical data a Fokker-Planck equation for this cascade process, which allows to generate surrogate data to forecast extreme events as well as to develop a non-equilibrium thermodynamics for the complex systems. For each cascade events an entropy production can be determined. These entropies fulfil accurately a rigorous law, namely the integral fluctuations theorem.