Forces from coarse-graining nonequilibrium degrees of freedom: exact results

Abstract: We explore the relaxation dynamics of a tracer in a harmonic trap coupled to a non-equilibrium bath particle in stationary state, finding qualitative differences compared to the well known equilibrium case. These can be attributed to an additional position and time dependent force acting on the tracer, emerging when averaging the bath degree in the non-equilibrium stationary state conditioned to a certain tracer position. Specifically, we provide analytical results for an overdamped tracer coupled linearly to a bath particle in different nonequilibrium scenarios, namely, subjected to a different temperature than the tracer, or to active noise with Gaussian or non-Gaussian fluctuations. For the case of different temperatures, the conditioned tracer-bath force can be as large in magnitude as the force from the trapping potential. For an active bath particle with memory, even the bath noise takes a finite average under conditioning of the tracer. Further, if the noise of the bath particle is non-Gaussian, the relaxation function of the tracer can be non-monotonic as a function of time. We also compute the \emph{pinned relaxation}, proposing that measurement and comparison of conditioned and pinned relaxation allows determination of the non-equilibrium forces in experiments.