Transition to Collective Motion in Nonreciprocal Active Matter: Coarse Graining Agent-Based Models into Fluctuating Hydrodynamics

Two hallmarks of nonequilibrium systems, from active colloids to animal herds, are agent motility and nonreciprocal interactions. Their interplay creates feedback loops that lead to complex spatiotemporal dynamics crucial to understand and control the nonlinear response of active systems. Here, we introduce a minimal model that captures these two features at the microscopic scale while admitting an exact hydrodynamic theory valid also in the fully nonlinear regime. Using statistical mechanics techniques, we exactly coarse-grain our nonreciprocal microscopic model into a fluctuating hydrodynamics and use dynamical systems insights to analyze the resulting equations. In the absence of motility, we find two transitions to oscillatory phases occurring via distinct mechanisms: a Hopf bifurcation and a saddle node on invariant circle bifurcation. In the presence of motility, this rigorous approach, complemented by numerical simulations, allows us to quantitatively assess the hitherto neglected impact of interspecies nonreciprocity on a paradigmatic transition in active matter: the emergence of collective motion. When nonreciprocity is weak, we show that flocking is accelerated and bands tend to synchronize with a spatial overlap controlled by nonlinearities. When nonreciprocity is strong, flocking is superseded by a chase and rest dynamical phase, where each species alternates between a chasing state, when they propagate, and a resting state, when they stand still. Phenomenological models with linear nonreciprocal couplings fail to predict the chase and rest phase, which illustrates the usefulness of our exact coarse-graining procedure. Finally, we demonstrate how fluctuations in finite systems can be harnessed to characterize microscopic nonreciprocity from macroscopic time-correlation functions, even in phases where nonreciprocal interactions do not affect the thermodynamic steady state.