  In striking contrast to equilibrium systems, inertia can profoundly alter the
structure of active systems. Here, we demonstrate that driven systems can
exhibit effective equilibrium-like states with increasing particle inertia,
despite rigorously violating the fluctuation-dissipation theorem. Increasing
inertia progressively eliminates motility-induced phase separation and restores
equilibrium crystallization for active Brownian spheres. This effect appears to
be general for a wide class of active systems, including those driven by
deterministic time-dependent external fields, whose nonequilibrium patterns
ultimately disappear with increasing inertia. The path to this effective
equilibrium limit can be complex, with finite inertia sometimes acting to
accentuate nonequilibrium transitions. The restoration of near equilibrium
statistics can be understood through the conversion of active momentum sources
to passive-like stresses. Unlike truly equilibrium systems, the effective
temperature is now density dependent, the only remnant of the nonequilibrium
dynamics. This density-dependent temperature can in principle introduce
departures from equilibrium expectations, particularly in response to strong
gradients. Our results provide additional insight into the effective
temperature ansatz while revealing a mechanism to tune nonequilibrium phase
transitions.
