The Fokker–Planck Approach to Complex Spatiotemporal 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 desire to understand general multipoint 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 multipoint 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 the generation of surrogate data to forecast extreme events as well as to develop a nonequilibrium thermodynamics for the complex systems. For each cascade event, an entropy production can be determined. These entropies accurately fulfill a rigorous law, namely the integral fluctuations theorem.