Using a powerful combination of projection-operator method and path-space response theory, we derive the fluctuation dynamics of a slow inertial probe coupled to a steady nonequilibrium medium under the assumption of time-scale separation. The nonequilibrium is realized by external nongradient driving on the medium particles or by their (athermal) active self-propulsion. The resulting friction on the probe is an explicit time correlation for medium observables and is decomposed into two terms: one entropic, proportional to the noise variance as in the Einstein relation for equilibrium media, and a frenetic term that can take both signs. As an illustration, we give the exact expressions for the linear friction coefficient and noise amplitude of a probe in a rotating run-and-tumble medium. We find a transition to absolute negative probe friction as the nonequilibrium medium exhibits sufficient and persistent rotational current. There, the run-away of the probe to high speeds realizes a nonequilibrium-induced acceleration. Simulations show that its speed finally saturates, yielding a symmetric stationary probe-momentum distribution with two peaks.

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• Fluctuation-dissipation theorem
• Fluctuations & noise
• Nonequilibrium statistical mechanics
• Coarse graining
• Fokker–Planck equation