Trajectory Statistics Govern Mechanical Power Transfer in Active Baths

Abstract

The mechanical response of a moving probe in an active bath reflects both the dynamics of the bath and how the probe couples to it. For a probe moving at velocity $\mathbf V$ and interacting weakly with an ideal active bath, we show that the trajectory statistics of a free bath particle determine a density response from which the bath force on the probe follows. The probe-particle interaction determines how this response is weighted over wave vector $\mathbf q$, while the probe motion selects frequencies $ω_{\mathbf q}=\mathbf q\cdot \mathbf V$. This trajectory-response relation separates the bath dynamics from the wave-vector weighting, so neither the stationary state nor force correlations need to be recalculated for each probe interaction and velocity. Applying this relation to active-particle models, we prove that positive power transfer is excluded for active Ornstein--Uhlenbeck particles in all dimensions and for complete-reset run-and-tumble particles in $d\geq2$, while one-dimensional run-and-tumble particles and two-dimensional active Brownian particles possess modes that can transfer positive mechanical power. For a given probe, this response predicts drag reversal, spontaneous probe motion, and motion-induced density distortions, all in quantitative agreement with simulations. Varying the probe geometry changes how the same bath response is weighted, thereby selecting different moving states. Beyond the leading weak-probe, ideal-bath limit, higher orders in probe strength involve multi-interval trajectory statistics, while finite-density corrections involve interacting-particle dynamics. Our results connect the trajectory statistics of active bath particles to the mechanical response of a passive probe, providing a route to predict probe response from single-particle trajectory statistics measured in the absence of the probe.