Out-of-equilibrium response and fluctuation-dissipation violations across scales in flocking systems

Flocking systems are known to be strongly out of equilibrium. Energy input occurs at the individual level to ensure self-propulsion, and the individual motility, in turn, contributes to ordering, enhancing information propagation and strengthening collective motion. However, even beyond ordering, a crucial feature of natural aggregations is response. How, then, do off-equilibrium features affect the response of the system? In this work, we consider a minimal model of flocking and investigate response behavior under directional perturbations. We show that equilibrium dynamical fluctuation-dissipation relations between response and correlations are violated, both at the local and at the global levels. The amount of violation peaks at the ordering transition, exactly as for the entropy production rate. Entropy is always produced locally and connected to the local fluctuation-dissipation violation via Harada-Sasa relationships. However, cooperative mechanisms close to the transition spread off-equilibrium effects to the whole system, producing an out-of-equilibrium response on the global scale. Our findings elucidate the role of activity and interactions in the cost repartition of collective behavior and explain what is observed in experiments on natural living groups.