When does an active bath behave as an equilibrium one?

Active baths are characterized by a non-Gaussian velocity distribution and a quadratic dependence with active velocity \(v_0\) of the kinetic temperature and diffusion coefficient. While these results hold in over-damped active systems, inertial effects lead to normal velocity distributions, with kinetic temperature and diffusion coefficient increasing as \(\sim v_0^\alpha\) with \(1