  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 intertial 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.

Soft Condensed Matter;Statistical Mechanics