by Werner Ebeling, Humboldt University, Berlin

Theory of Brownian motion revisited - from Einstein-Smoluchowski to swarm dynamics

We begin with a survey of the 100 years of development after the early work of Einstein, Smoluchowski, Langevin, Fokker and Planck on the theory of Brownian motion. Next we consider several approaches to the statistical foundation of the stochastic equations. Further we give – based on a recent book with Sokolov - an overview on several extensions as random walk approaches, and in particular to driven Brownian motion. In the last part we explore the collective motions of interacting driven particles forming swarms, study the modes and the attractors of motion. Several applications of the dynamics of swarms in physics and ecology are given.