An active fractional Ornstein-Uhlenbeck particle: diffusion and dissipation

In this work, we consider a particle subjected to active fluctuations due to the presence of an active bath. The active fluctuation driving the particle is modeled as an Ornstein-Uhlenbeck process with memory, due to the recent experimental result in that a bacterial bath under a chemotactic mechanism exhibits a large distribution of persistence time, which may result in long-range persistence of a probe particle that cannot be captured by the conventional Ornstein-Uhlenbeck model for active noise. As a paradigmatic model, we consider introducing a fractional Ornstein-Uhlenbeck process with a power-law kernel, which includes a fractional order variable 0 < alpha <= 1 , to model an active noise acting on a particle. We present the relevant properties of the active noise and its implications for the diffusion of free and harmonically confined single particles. This is also supported by the derived active Smoluchowski description. More importantly, we find that the effect of alpha can either reduce or increase the effective temperature at long time, depending on the competition between the harmonic timescale and persistence time. Lastly, the effects of presented active noise on the spectral density of the energy dissipation rate and the dissipation energy from both the thermal and active bath are explored.