Passive objects in a chiral active bath: from micro to macro

We present a detailed derivation of the Langevin dynamics obeyed by a massive rigid body immersed in a chiral active bath. We show how the antisymmetric nature of the noise leads to an unusual relationship between the Langevin equation describing stochastic trajectories and the Fokker-Planck equation governing the time-evolution of the probability density. The chirality of the bath endows the object dynamics with odd diffusivity, odd mobility, and rotational ratchet effects that depend on the object symmetries. For rotationally-symmetric objects, we show that a hidden time-reversal symmetry leads to separate effective equilibrium descriptions for the translational and rotational degrees of freedom. Finally, starting from the bath dynamics, we construct a multipole expansion to quadrupolar order that allows predicting the far-field current and density modulation induced by the object on the bath.