Brownian motion and beyond

Abstract

Roughly 190 years ago Robert Brown reported the rapid oscillatory motion of microscopic particles, the first systematic study of what we now call Brownian motion. At the beginning of the 20th century Albert Einstein, Marian Smoluchowski, and Pierre Langevin formulated the mathematical laws of diffusion. Jean Perrin's experiments 110 years ago then prompted a very active field of ever refined diffusion experiments. Despite the long-standing history of Brownian motion, I will report several new developments in the field of diffusion and stochastic processes. This new research has been fuelled mainly by novel insights into complex microscopic systems such as living biological cells, made possible by Nobel-Prize winning techniques in laser physics, superresolution microscopy, or through supercomputing studies. Topics covered include Brownian yet non-Gaussian diffusion in heterogeneous systems, the interplay of geometry- and reaction-control in molecular reaction kinetics, and anomalous diffusion with a power-law time dependence of the mean squared displacement. Such non-Brownian dynamics is non-universal, and questions of the precise underlying physical mechanisms become relevant. Both appropriate physical observables as well as effects of ergodicity and ageing will be discussed.