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

We investigate the steady-state entropy production rate (EPR) in the hydrodynamic Vicsek model
(HVM) and diffusive flocking model (DFM). Both models display a transition from an isotropic
gas to a polar liquid (flocking) phase, in addition to travelling polar clusters and
microphase-separation in the miscibility gap. The phase diagram of the DFM, which may be
considered an extension of the HVM, contains additional structure at low densities where we find
a novel crystal phase in which a stationary hexagonal lattice of high-density ridges surround low
density valleys. From an assessment of the scaling of the EPR at low noise, we uncover that the
dynamics in this limit may be organised into three main classes based on the dominant
contribution. Truly nonequilibrium dynamics is characterised by a divergent EPR in this limit, and
sustains global time-reversal symmetry (TRS) violating currents at zero noise. On the other hand,
marginally nonequilibrium and effectively equilibrium dynamics have a finite EPR in this limit,
and TRS is broken only at the level of fluctuations. For the latter of these two cases, detailed
balance is restored in the small noise limit and we recover effective Boltzmann statistics to lowest
nontrivial order.We further demonstrate that the scaling of the EPR may change depending on the
dynamical variables that are tracked when it is computed, and the protocol chosen for
time-reversal. Results acquired from numerical simulations of the dynamics confirm both the
asymptotic scaling relations we derive and our quantitative predictions.