We review theoretical models of individual motility as well as collective dynamics and pattern formation of active particles. We focus on simple models of active dynamics with a particular emphasis on nonlinear and stochastic dynamics of such self-propelled entities in the framework of statistical mechanics. Examples of such active units in complex physico-chemical and biological systems are chemically powered nano-rods, localized patterns in reaction-diffusion system, motile cells or macroscopic animals. Based on the description of individual motion of point-like active particles by stochastic differential equations, we discuss different velocity-dependent friction functions, the impact of various types of fluctuations and calculate characteristic observables such as stationary velocity distributions or diffusion coefficients. Finally, we consider not only the free and confined individual active dynamics but also different types of interaction between active particles. The resulting collective dynamical behavior of large assemblies and aggregates of active units is discussed and an overview over some recent results on spatiotemporal pattern formation in such systems is given. © 2012 EDP Sciences and Springer.
Romanczuk, P., Bär, M., Ebeling, W., Lindner, B., & Schimansky-Geier, L. (2012, March). Active Brownian particles: From individual to collective stochastic dynamics: From individual to collective stochastic dynamics. European Physical Journal: Special Topics. https://doi.org/10.1140/epjst/e2012-01529-y