We investigate the motion of Brownian particles which have the ability to take up energy from the environment, to store it in an internal depot, and to convert internal energy into kinetic energy. The resulting Langevin equation includes an additional acceleration term. The motion of the Brownian particles in a parabolic potential is discussed for two different cases: (i) continuous take-up of energy and (ii) take-up of energy at localized sources. If the take-up of energy is above a critical value, we found a limit-cycle motion of the particles, which, in case (ii), can be interrupted by stochastic influences. Including reflecting obstacles, we found for the deterministic case a chaotic motion of the particle. [S0031-9007(98)06328-5] PACS numbers: 05.40. + j, 05.45. + b, 05.60. + w, 87.10. + eActive motion is based on energy consumption. For biological systems, an external supply of energy is crucial, e.g., to maintain metabolism and to perform movement [1]. For a spatially inhomogeneous supply of energy, the organism needs to store energy internally, in order to overcome periods of starvation, e.g., during the search for new sources. But even provided the homogeneous supply of energy, the organism needs to convert the energy taken up from the environment into kinetic energy. Dependent on the level of biological organization, the take-up, storage, and conversion of energy is a rather complex process.In the following, we consider the motion of microscopic biological objects, such as cells or bacteria, which can be sufficiently described by a Langevin dynamics. Stochastic differential equations have long been used to describe the motion of organisms [2,3]. In order to derive a simplified model of active biological motion, we study Brownian particles with an internal energy depot. The motion of simple Brownian particles in a space-dependent potential, U͑r͒ can be described by the Langevin equation:where g 0 is the friction coefficient of the particle at position r, moving with velocity y. F ͑t͒ is a stochastic force with strength S and a d-correlated time dependenceRecently, Brownian motion models attracted much attention for describing nonequilibrium transport on the microscale [4]. In addition to the dynamics described above, the Brownian particles discussed here are active particles [5] to the effect that they have the ability to take up energy from the environment and to store it in an internal depot, which is considered a new element of the model. Further, the particles are able to convert internal energy into kinetic energy. Considering also internal dissipation, the resulting balance equation for the internal energy de-pot, e, of an active particle is given by d dt e͑t͒ q͑r͒ 2 c e͑t͒ 2 d͑y͒ e͑t͒ .( 3) q͑r͒ is the space-dependent take-up of energy and c describes the internal dissipation assumed to be proportional to the depot energy. d͑y͒ is the rate of conversion of internal into kinetic energy which should be a function of the actual velocity of the particle. A simple ansatz for d͑y͒ reads: d͑y͒ d 2 y 2 ; ...
