Active Brownian Particles. From Individual to Collective Stochastic Dynamics

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.Comment: 161 pages, Review, Eur Phys J Special-Topics, accepte

Mathematics and biology: a Kantian view on the history of pattern formation theory

Driesch’s statement, made around 1900, that the physics and chemistry of his day were unable to explain self-regulation during embryogenesis was correct and could be extended until the year 1972. The emergence of theories of self-organisation required progress in several areas including chemistry, physics, computing and cybernetics. Two parallel lines of development can be distinguished which both culminated in the early 1970s. Firstly, physicochemical theories of self-organisation arose from theoretical (Lotka 1910–1920) and experimental work (Bray 1920; Belousov 1951) on chemical oscillations. However, this research area gained broader acceptance only after thermodynamics was extended to systems far from equilibrium (1922–1967) and the mechanism of the prime example for a chemical oscillator, the Belousov–Zhabotinski reaction, was deciphered in the early 1970s. Secondly, biological theories of self-organisation were rooted in the intellectual environment of artificial intelligence and cybernetics. Turing wrote his The chemical basis of morphogenesis (1952) after working on the construction of one of the first electronic computers. Likewise, Gierer and Meinhardt’s theory of local activation and lateral inhibition (1972) was influenced by ideas from cybernetics. The Gierer–Meinhardt theory provided an explanation for the first time of both spontaneous formation of spatial order and of self-regulation that proved to be extremely successful in elucidating a wide range of patterning processes. With the advent of developmental genetics in the 1980s, detailed molecular and functional data became available for complex developmental processes, allowing a new generation of data-driven theoretical approaches. Three examples of such approaches will be discussed. The successes and limitations of mathematical pattern formation theory throughout its history suggest a picture of the organism, which has structural similarity to views of the organic world held by the philosopher Immanuel Kant at the end of the eighteenth century