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, numerical simulations show the tracer to transition from an active dynamics, characterized by boundary accumulation and ratchet currents, to an effective equilibrium regime. To account for this analytically, we eliminate the bath degrees of freedom under the assumption of linear coupling to the tracer and show convergence, in the large density limit, to an equilibrium dynamics at the bath temperature. We then develop a perturbation theory to characterize the tracer's departure from equilibrium at large but finite bath densities, revealing an intermediate time-reversible yet 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 or a soft active solid, the cold tracer drives the entire bath out of equilibrium, leading to a long-ranged suppression of bath fluctuations.