Brownian Particles far from Equilibrium

TL;DR

This paper models energy-pumped Brownian particles with non-linear friction, revealing new dynamic behaviors like limit cycles and crater-like velocity distributions, through analytical solutions and simulations.

Contribution

It introduces a velocity-dependent friction function based on an internal energy depot, extending traditional Brownian motion models to include energy pumping effects.

Findings

Formation of limit cycles in phase space

Quadratic increase in mean squared displacement with energy supply

Non-equilibrium velocity distributions with crater-like shape

Abstract

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.