High-resolution ensemble prediction of a polar low development

Noname manuscript No. (will be inserted by the editor) Spatiospectral concentration in the Cartesian plane

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The purpose of this paper is to review, clarify, and critically analyse modern mathematical cosmology. The emphasis is upon the mathemati-cal structures involved, rather than numerical computations. The opening section reviews and clarifies the Friedmann-Robertson-Walker models of General

First and foremost, I would like to thank my advisor, Prof. Jennifer E. Michaels. I am most appreciative of your guidance over the last four years, from the basics of elastic waves to both short- and long-term career advice. I have grown a tremendous amount, both profes-sionally and personally, and am deeply grateful for your mentorship, patience, and support. I would also like to thank my Ph.D. committee for their valuable insight and sugges-

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Abstract. 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 pow-ered 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 observ-ables 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 par-ticles. 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. 1

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