A Cold Tracer in a Hot Bath: In and Out of Equilibrium*

We study the dynamics of a zero-temperature overdamped tracer in a bath of Brownian particles. As the bath density is increased, the passive tracer transitions from an effectively active dynamics, characterized by boundary accumulation and ratchet currents, to a bona-fide equilibrium regime. To account for this, we construct the generalized Langevin equation of the tracer under the assumption of linear coupling to the bath and show convergence to equilibrium in the large density limit. We then develop a perturbation theory to characterize the departure from equilibrium at large but finite bath densities, revealing an intermediate time-reversible but non-Boltzmann regime, followed by a fully irreversible one. Finally, we show that when the bath particles are connected as a lattice, mimicking a gel, the cold tracer drives the entire bath out of equilibrium, leading to a long-ranged suppression of bath fluctuations.