Entropy production rate in thermodynamically consistent flocks

We study the entropy production rate (EPR) of aligning self-propelled particles which undergo a flocking transition towards a polarized collective motion. In our thermodynamically consistent lattice model, individual self-propulsion is the exclusive source of irreversibility. We derive the fluctuating hydrodynamics for large system sizes using a controlled coarse-graining: our procedure entails an exact correspondence between the EPR evaluated at the hydrodynamic and particle-based levels. We reveal that EPR is maximal when the system adopts a homogeneous configuration, either apolar or polar, and reduced in the non-homogeneous state where a polar band travels in a apolar background {due to strong spatial EPR modulations.} By analyzing the {latter} we also show that asymmetric energetic exchanges occur at the trailing and leading edges, which we map into a thermodynamic cycle in density-polarization space. Finally, we demonstrate that the regime of weak self-propulsion features a singular scaling of EPR, and a non-analyticity of the travelling band profiles.