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Two hallmarks of non-equilibrium systems, from active colloids to animalherds, are agents motility and nonreciprocal interactions. Their interplaycreates feedback loops leading to complex spatiotemporal dynamics crucial tounderstand and control the nonlinear response of active systems. Here, weintroduce a minimal model that captures these two features at the microscopicscale, while admitting an exact hydrodynamic theory valid also in thefully-nonlinear regime. Our goal is to account for the fact that animal herdsand colloidal swarms are rarely in the thermodynamic limit where particlenumber fluctuations can be completely ignored. Using statistical mechanicstechniques we exactly coarse-grain a nonreciprocal microscopic model into afluctuating hydrodynamics and use dynamical systems insights to analyze theresulting equations. In the absence of motility, we find two transitions tooscillatory phases occurring via distinct mechanisms: a Hopf bifurcation and aSaddle-Node on Invariant Circle (SNIC) bifurcation. In the presence ofmotility, this rigorous approach, complemented by numerical simulations, allowsus to quantitatively assess the hitherto neglected impact of inter-speciesnonreciprocity on a paradigmatic transition in active matter: the emergence ofcollective motion. When nonreciprocity is weak, we show that flocking isaccelerated and bands tend to synchronize with a spatial overlap controlled bynonlinearities. When nonreciprocity is strong, flocking is superseded by aChase & Rest dynamical phase where each species alternates between a chasingstate, when they propagate, and a resting state, when they stand still.Finally, we demonstrate how fluctuations in finite systems can be harnessed tocharacterize microscopic non-reciprocity from macroscopic time-correlationfunctions, even in phases where nonreciprocal interactions do not affect thethermodynamic steady-state.

The transition to collective motion in nonreciprocal active matter: coarse graining agent-based models into fluctuating hydrodynamics