Velocity and Speed Correlations in Hamiltonian Flocks

We study a 2D Hamiltonian fluid made of particles carrying spins coupled to their velocities. At low temperatures and intermediate densities, this conservative system exhibits phase coexistence between a collectively moving droplet and a still gas. The particle displacements within the droplet have remarkably similar correlations to those of birds flocks. The center of mass behaves as an effective self-propelled particle, driven by the droplet’s total magnetization. The conservation of a generalized angular momentum leads to rigid rotations, opposite to the fluctuations of the magnetization orientation that, however small, are responsible for the shape and scaling of the correlations.

Figure 1

Hamiltonian flocks. (a) Sketch of the phase diagram in the ($\varphi$, $T$) plane, for small $K$ and $N$. The red and blue arrows indicate the densities studied here, $\varphi =0.55$ and $\varphi =0.14$. (b),(c) Snapshots of the displacements and their fluctuations within a flock ($N=512$, $K=0.03$, $T={10}^{-2}$). (d),(e) Spatial correlations $C_d$ ($C_{\mathrm{sp}}$) of the displacement (the speed) fluctuations, against distance normalized by the typical droplet size, for three system sizes ($T={10}^{-2}$). The black lines are our theoretical predictions.

Figure 2

Dynamics and structure of the homogeneous phases across the Curie line and hexatic transition. (a) Mean magnetization modulus and (b) hexatic order parameter versus $T$. The arrows indicate the $T$ values at which the MSD curves in (c)–(f) are recorded. (c) MSD and (d) MSAD of the relative displacements versus time, in log-log scale. (e) MSD of the center of mass (the dashed black line indicates $v_G=K$), and (f) MSAD of the magnetization vector versus time, in log-log scale. (g),(h) Trajectories of the center of mass, with time going from light to dark, for two temperatures corresponding to the (g) mauve and (h) light gray curves in (c)–(f). $N=128$, $K=0.23$, $\varphi =0.55$ [red arrow in Fig. 1].

Figure 3

AVOUP dynamics. $(a){\tau }_m$ (green) and $D_r$ (orange) versus $T$ in lin-log scale. $(b)D_G$ against $T$ (black), together with the ABP (red) and AVOUP (blue) estimations. Inset: trajectories, with time going from light to dark, obtained by integrating the AVOUP equations of motion over $t=10000$, with an integration step $\delta t=1$, and fixed values $Km=0.1$, $D_r=1$. ${\tau }_m=5000$ (gray), and ${\tau }_m=500$ (mauve).

Figure 4

Spontaneous rigid rotations. Relative (a) MSD and (b) MSAD for $N=128$, $K=0.03$, $\varphi \approx 0.14$ [blue arrow in Fig. 1] and $T$ increasing from black to red. The coexistence line is crossed between the green and teal lines. (c) Typical ${\omega }_m={\dot{\theta }}_m$ (orange) and $N^{-1}\sum {\mathit{r}}_i^{\star }\times {\mathit{v}}_i^{\star }$ (green), time-averaged over ${\tau }_w={10}^2$, versus time ($T\approx {10}^{-2}$). (d) Relative displacements color coded according to their orientation and rescaled by their mean ($T\approx {10}^{-2}$, $\tau \approx {10}^4$), together with a sketch of the spontaneous rotation dynamics ($\mathit{m}$ in orange and ${\mathit{v}}_G$ in green). Upper inset: the rotation of $\mathit{m}$ from the gray to the black arrow, following the orange curved arrow.