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We investigate the steady-state entropy production rate (EPR) in theHydrodynamic Vicsek Model (HVM) and Diffusive Flocking Model (DFM). Both modelsdisplay a transition from an isotropic gas to a polar liquid (flocking) phase,in addition to traveling polar clusters and microphase-separation in themiscibility gap. The phase diagram of the DFM, which may be considered anextension of the HVM, contains additional structure at low densities where wefind a novel crystal phase in which a stationary hexagonal lattice ofhigh-density ridges surround low density valleys. From an assessment of thescaling of the EPR at low noise, we uncover that the dynamics in this limit maybe organised into three main classes based on the dominant contribution. Trulynonequilibrium dynamics is characterised by a divergent EPR in this limit, andsustains global time-reversal symmetry (TRS) violating currents at zero noise.On the other hand, marginally nonequilibrium and effectively equilibriumdynamics have a finite EPR in this limit, and TRS is broken only at the levelof fluctuations. For the latter of these two cases, detailed balance isrestored in the small noise limit and we recover effective Boltzmann statisticsto lowest nontrivial order. We further demonstrate that the scaling of the EPRmay change depending on the dynamical variables that are tracked when it iscomputed, and the protocol chosen for time-reversal. Results acquired fromnumerical simulations of the dynamics confirm both the asymptotic scalingrelations we derive and our quantitative predictions.

Time-reversal symmetry violations and entropy production in field theories of polar active matter