Brownian particles far from equilibrium

We study a model of Brownian particles which are pumped with energy by mean s of a non-linear friction function, for which different types are discusse d. A suitable expression for a non-linear, velocity-dependent friction func tion is derived by considering an internal energy depot of the Brownian par ticles. In this case, the friction function describes the pumping of energy in the range of small velocities, while in the range of large velocities t he known limit of dissipative friction is reached. In order to investigate the influence of additional energy supply, we discuss the velocity distribu tion function for different cases. Analytical solutions of the correspondin g Fokker-Planck equation in 2d are presented and compared with computer sim ulations. 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 incr eases quadratically with the energy supply, or non-equilibrium velocity dis tributions with crater-like form. Further, we point to some generalizations and possible applications of the model.