Hydrodynamic Fluctuations and Damped Oscillating Friction Lead to Active Motion

The fractional Langevin equation is used to analyze the erratic motion of a particle in a complex environment. Hydrodynamic backflows and restoring harmonic forces can work concurrently on the probe particle in addition to the constraints imposed by the complex environment, whose characteristics are represented in the friction force the particle encounters. The friction force, which is used for the modelling, corresponds to a damped oscillating function and it is expressed by a one-parameter Mittag-Leffler function. Analytical solutions for the relaxation functions are extracted, and through them observables like the position autocorrelation function, velocity autocorrelation function and mean square displacement are explicitly given. The competition between hydrodynamic fluctuations and frictional forces establishes a memory whose time scale is larger than the diffusional time scale, and during this time window the motion of the particle is superdiffusive.