Hamiltonian flocks: Time-Reversal Symmetry and its consequences

Abstract

The fluctuation-dissipation theorem is a hallmark of equilibrium system that stem from their time-reversal symmetry. In many non-equilibrium systems, in particular active ones, extensions and explicit violations of this theorem are used to assess their ''distance'' to equilibrium. In Hamiltonian flocks, conservative yet non-Galilean models of polar liquids, previous work reported collective motion without the activity that usually underlies it. In this paper, we show that this model obeys a generalized time-reversal symmetry that yields a fluctuation-dissipation theorem that mixes position and polarity degrees of freedom. Due to the oddness of spin under time reversal, the system also obeys Onsager-Casimir reciprocity rather than standard Onsager relations. The coupling also induces rich spin orientation dynamics, including a non-trivial diffusion constant at long times. Finally, we show that considering the naïve time-reversal operation rather than the generalized one that leaves the system invariant leads to a spurious entropy production rate, that could be wrongly interpreted as a distance to equilibrium. Our findings suggest looking for possible extensions of time-reversal symmetry in active-looking systems, which may lead to yet unknown generalizations of the fluctuation-dissipation theorem.

Figures (8)

• Figure 1: Single-particle trajectories. Example trajectories for single particles following the dynamics defined by Eqs. \ref{['eq:Langevin_p']} and \ref{['eq:Langevin_omega']} for $\gamma_t = \gamma_r = \gamma$. In the top row, $v_0 = 0$ with (a) $K = 0$ (Brownian particle) and (b) $\sqrt{\beta }K = 5$. In the bottom row, $\bm{v}_0 = v_0 \hat{\bm{e}}_x$ with $\sqrt{\beta}v_0 = 1$, and (c) $\sqrt{\beta}K = 0.5$ or (d) $\sqrt{\beta}K = 10$. Insets of the bottom row show the same trajectories in the frame moving at $\bm{v}_0$. Throughout panels, a gradient of colors represents time, flowing from black ($t=0$) to pink ($\gamma t= 50$), and a few snapshots of the spin are shown as green arrows.
• Figure 2: Overdamped trajectories. Example trajectories for single particles following the overdamped dynamics defined by Eqs. \ref{['eq:Langevin_r_overdamped']} and \ref{['eq:Langevin_theta_overdamped']} for $\gamma_t = \gamma_r = \gamma$. In the top row, $v_0 = 0$ with (a) $K = 0$ (Brownian particle) and (b) $\sqrt{\beta }K = 5$. In the bottom row, $\bm{v}_0 = v_0 \hat{\bm{e}}_x$ with $\sqrt{\beta}v_0 = 1$, and (c) $\sqrt{\beta}K = 0.5$ or (d) $\sqrt{\beta}K = 10$. Insets of the bottom row show the same trajectories in the frame moving at $\bm{v}_0$. Throughout panels, a gradient of colors represents time, flowing from black ($t=0$) to pink ($\gamma t= 50$), and a few snapshots of the spin are shown as green arrows.
• Figure 3: Undamped and noiseless trajectories. Conservative single-particle trajectories obtained by integrating Eqs. \ref{['eq:undamped_p']} and \ref{['eq:undamped_omega']}. (a) Galilean particle ($K = 0$) with finite $p$ and $\omega$. (b) Circular trajectory for $K > 0$, $p = 0$, $\omega >0$. (c) Stable-oscillation regime of the spin for $K > 0$, $p>0$, and $\omega < \omega_c$. (d) Winding regime of the spin for $K > 0$, $p>0$, and $\omega > \omega_c$. Throughout the figure, time flows from dark purple to light pink, and green arrows indicate a few points of measurement of the spin.
• Figure 4: Einstein-Smoluchowski-Sutherland Relations. We numerically integrate Eqs. \ref{['eq:Langevin_p']}, \ref{['eq:Langevin_omega']}, \ref{['eq:Langevin_r']}, \ref{['eq:Langevin_theta']} and compute the mean-square displacements (\ref{['eq:def-Delta2']}) across $N$ trajectories then compute averages over noise and equilibrium initial conditions. (a) MSD in the frame co-moving at $\bm{v}_0$, $\Delta r^2_{\text{fluct}}$, against time, both dimensionless. Colored symbols present data across values of $v_0$ (symbols) and $K$ (colors). All curves fall very close to a Brownian prediction (black line), Eq. (\ref{['eq:Delta2r-Brownian']}), with a ballistic regime at short times and a diffusive one at long times (gray dashed guide line). (b) MSAD against dimensionless time for $v_0 = 0$ and across $K$ values (colors). Symbols indicate numerical data, and solid lines theoretical predictions from Eq. \ref{['eq:msad_v0=0']}. We here set $\gamma_r = \gamma_t = \gamma$ and average over $N=10^6$ trajectories.
• Figure 5: Oscillatory behavior. MSAD against dimensionless time for $v_0 = 0$, analogous to Fig. \ref{['fig:ESSR']}(b), but for $\sqrt{\beta}K = 20$ (red) and $\sqrt{\beta}K = 100$ (dark blue). Open symbols indicate numerical measurements from integrating the full Langevin dynamics and averaging over $N = 10^6$ trajectories, solid lines indicate the analytical prediction from Eq. \ref{['eq:msad_v0=0']}.