Active Ornstein-Uhlenbeck model for self-propelled particles with inertia

Self-propelled particles, which convert energy into mechanical motion, exhibit inertia if they have a macroscopic size or move inside a gaseous medium, in contrast to micron-sized overdamped particles immersed in a viscous fluid. Here we study an extension of the active Ornstein-Uhlenbeck model, in which self-propulsion is described by colored noise, to access these inertial effects. We summarize and discuss analytical solutions of the particle's mean-squared displacement and velocity autocorrelation function for several settings ranging from a free particle to various external influences, like a linear or harmonic potential and coupling to another particle via a harmonic spring. Taking into account the particular role of the initial particle velocity in a nonstationary setup, we observe all dynamical exponents between zero and four. After the typical inertial time, determined by the particle's mass, the results inherently revert to the behavior of an overdamped particle with the exception of the harmonically confined systems, in which the overall displacement is enhanced by inertia. We further consider an underdamped model for an active particle with a time-dependent mass, which critically affects the displacement in the intermediate time-regime. Most strikingly, for a sufficiently large rate of mass accumulation, the particle's motion is completely governed by inertial effects as it remains superdiffusive for all times.

Automated summary

This study investigates the inertial effects of self-propelled particles by extending the active Ornstein-Uhlenbeck model. Analytical solutions of the particle's mean-squared displacement and velocity autocorrelation function are summarized and discussed for different scenarios. The role of initial particle velocity in a nonstationary setup is taken into account, and the observed dynamical exponents range from zero to four. In most cases, the behavior reverts to that of an overdamped particle after a certain inertial time, except for harmonically confined systems where inertia enhances overall displacement. Additionally, an underdamped model with time-dependent mass is considered, showing significant influence on particle displacement in the intermediate time-regime. For a sufficiently large mass accumulation rate, the particle's motion is completely governed by inertial effects and remains superdiffusive for all times.