  We study a model of Brownian particles which are pumped with energy by means
of a non-linear friction function, for which different types are discussed. A
suitable expression for a non-linear, velocity-dependent friction function is
derived by considering an internal energy depot of the Brownian particles. In
this case, the friction function describes the pumping of energy in the range
of small velocities, while in the range of large velocities the known limit of
dissipative friction is reached. In order to investigate the influence of
additional energy supply, we discuss the velocity distribution function for
different cases. Analytical solutions of the corresponding Fokker-Planck
equation in 2d are presented and compared with computer simulations. Different
to the case of passive Brownian motion, we find several new features of the
dynamics, such as the formation of limit cycles in the four-dimensional
phase-space, a large mean squared displacement which increases quadratically
with the energy supply, or non-equilibrium velocity distributions with
crater-like form. Further, we point to some generalizations and possible
applications of the model.
