Signatures of irreversibility in microscopic models of flocking

Flocking in $d=2$ is a genuine nonequilibrium phenomenon for which irreversibility is an essential ingredient. We study a class of minimal flocking models whose only source of irreversibility is self-propulsion and use the entropy production rate (EPR) to quantify the departure from equilibrium across their phase diagrams. The EPR is maximal in the vicinity of the order-disorder transition, where reshuffling of the interaction network is fast. We show that signatures of irreversibility come in the form of asymmetries in the steady-state distribution of the flock's microstates. These asymmetries occur as consequences of the time-reversal symmetry breaking in the considered self-propelled systems, independently of the interaction details. In the case of metric pairwise forces, they reduce to local asymmetries in the distribution of pairs of particles. This study suggests a possible use of pair asymmetries both to quantify the departure from equilibrium and to learn relevant information about aligning interaction potentials from data.

Figure 1

[(a), (b)] Typical configurations for Model I and Model II at different $D$ values. [(c), (d)] Distribution of the number of interacting neighbors associated to the configurations above. Bimodality at moderate to small $D$ values indicates phase coexistence in Model I. For Model II the distribution has an almost invariant shape, hence the amount of irreversible reshuffling of the interaction network cannot be deduced from its static connectivity alone. Red dashed line: mean number of neighbors; green solid line: expected number of neighbors for a Poisson point pattern [$\pi \rho R^2+1$ in the metric case (c), 6 in (d)]. (e) Polar order parameter (OP) of Model I. The white line is the mean field transition point [40]. (f) EPR across the phase diagram of Model I. (g) EPR curves as a function of the rescaled distance from the mean field transition point, $D_{MF}=J\rho \pi R^2/2$. A possible explanation for the remaining density dependence is provided in the main text. Inset: Equivalence of formulas (7) and (8) is verified, guaranteeing stationarity of the observed processes. (h) EPR per particle (Model I). Color code is associated with the system size $N$. Strong finite-size effects are evident for small $N$ in the symmetry-broken phase (where coexistence is realized), but $\dot{\mathcal{S}}/N$ seems intensive as $N$ is increased. The dashed vertical line represents the spinodal (extracted from the number of neighbors' distribution for $N=8192$). (i) OP of Model II. (j) EPR across the phase diagram of Model II. The white line represents the alleged critical point. (k) Density dependence of the EPR curves (discussed in the main text). Inset: irreversibility is governed by a single control parameter, $v_0{\rho }^{1/2}$, at fixed $D$. (l) EPR per particle for Model II (different colors for different $N$ values). All curves perfectly collapse on a single master curve, which converges to the two equilibrium limits as predicted. Simulation parameters: (Model I) $R=1$, $J=1$, $v_0=0.5$; when not explicitly indicated $N=1024$, $\rho =1$. (Model II) $J=1$, $v_0=0.5$; when not explicitly indicated $N=2048$.

Figure 2

(a) The time-reversal (TR) operator acts on the system's variables by flipping velocities and keeping positions unchanged. Out of equilibrium, the two configurations cannot be equally probable. (b) Sample trajectories from a system of $N=1024$ particles ($D=3$, $J=1$, $v_0=0.5$, $\rho =1$, $\mathrm{\Phi }\simeq 0.21$) elucidating the dissipation mechanism in Model I. Black stretches indicate the portion of trajectory in which the two particles are at a distance smaller than the interaction radius; gray stretches those where particles are at a distance $r>R$ and do not interact. In (b.i) strongly antialigned particles start interacting with an initial phase difference of $0.95\pi$ (triangle, converging red arrows, corresponding to a positive EPR contribution ${\varepsilon }_{\mathrm{in}}\simeq 1.76$). Due to rotational diffusion, they later leave each other's neighborhood at a new relative phase and displacement angle (square, blue arrows), corresponding to ${\varepsilon }_{\mathrm{out}}\simeq -0.92$. In (b.ii) converging particles (cross) enter the interaction disk contributing to the EPR negatively (${\varepsilon }_{\mathrm{in}}=-0.34$). After the interaction, they diverge being more aligned than before (circle), so that the EPR contribution associated to this configuration surmounts the previous one: ${\varepsilon }_{\mathrm{out}}=0.38$. (c) Contour plot of Eq. (16).

Figure 3

First row: Reconstructed log distributions of pairs of particles at distance $R$ from numerical simulations of Model I ($N=1024$, $v_0=0.5$, $\rho =1$, $J=1$). Second row: Antisymmetric part of the log distributions: $-u_{-}(\alpha ,\phi )=u(\alpha +\pi ,\phi )-u(\alpha ,\phi )$. A positive correlation with $\varepsilon (\alpha ,\phi )$, shown in Fig. 2, is evident. Black lines represent the level curves of the fitted $\lambda \varepsilon (\alpha ,\phi )$ function, with $\lambda$ free parameter.

Figure 4

Scalar measure for the degree of asymmetry of the log distribution of particle pairs at distance $r$: $\parallel u_{-}{\parallel }_1\left(r\right)=\int d\alpha d\phi \left|u_{-}(\alpha ,\phi ;r)\right|$. Curves peak at $r=R$ (dashed line) for all the considered $D$ values. Lowering the rotational diffusion coefficient, the decay to zero at large $r$ gets slower because of the presence of heterogeneous structures affecting the pair distribution. The trend of these curves is qualitatively reminiscent of $-{\partial }_{r}n\left(r\right)=\delta (r-R)$. All data are collected form simulations of systems of $N=2048$ particles.

Figure 5

(a) Subsample of trajectories of diffusing particles in different regimes. All the trajectories span the same time interval. In the symmetry broken phase the trajectories are shown in the reference frame of the center of mass; mutual diffusion occurs mainly in the transverse direction with respect to collective motion (black arrow). At $D=2$, where reshuffling is mostly efficient, the flock is disordered but the motion of the particles is still persistent. Persistence is reduced as the rotational diffusion coefficient $D$ is increased. In all cases, $N=1024$, ${\rho }_0=1$. (b) Autocorrelation function of the adjacency matrix of the flock, $C_{\text{net}}\left(t\right)$, defined as in Eq. (A14). The color map refers to $D$ values. The maximum point of the EPR, $D=2$, is marked in red and corresponds to the curve with the fastest decay. Dashed lines are the fitted curves from Eq. (A15). [(c),(d)] Parametric plot of fitted parameters, $a$ and $c$, vs EPR. The figure shows a positive correlation both for the effective diffusion coefficient $c$ and exponent $a$. Close to the transition point (marked by the red dot) reshuffling is the most efficient, and the system is in the farthest condition from an equilibrium one.

Figure 6

(a) Time series of the average EPR of the flock (gray) and of the contribution of a sample single particle, properly rescaled (red), computed as in Eq. (A19) (top) or as in Eq. (A20) (bottom). The two time series have comparable fluctuations, especially when the stochastic heat is considered (top). This fact shows that all the particles contribute in the same way to the global EPR. (b) Probability density of the single-particle stochastic heat [left, from Eq. (A19)] or irreversible work of fictitious reshuffling forces [right, from Eq. (A20)]. The histogram is unimodal, even when particles are organized into polar clusters, in clear contrast with the MIPS phenomenology [59].