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We review the concept of nonlinear Brownian motion, originally introduced by Klimontovich, and consider several applications to real systems, including e.g. atoms, molecules or ions laser cooling fields, charged grains in plasmas and interdisciplinary problems. In particular, we also discuss recent developments in the field of active Brownian particles. After summarizing the basic properties of active Brownian particle models, solutions of the corresponding Fokker-Planck equation are analyzed for free motions as well as for motions in confining fields. Furthermore, we study the distributions for finite systems of self-confined particles, interacting via Morse and Coulomb potentials. Finally, applications to clusters of atoms subject to laser cooling as well as to clusters of charged grains in dusty plasmas are discussed.

Klimontovich's contributions to the kinetic theory of nonlinear Brownian motion and new developments