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Register a new userVelocity and Speed Correlations in Hamiltonian Flocks
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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 droplets 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.
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In their comment on our work (ArXiv:1912.07056v1), Cavagna textit{et al.} raise several interesting points on the phenomenology of flocks of birds, and conduct additional data analysis to back up their points. In particular, they question the existence of rigid body rotations in flocks of birds. In this reply, we first clarify the notions of rigid body rotations, and of rigidity itself. Then, we justify why we believe that it is legitimate to wonder about their importance when studying the spatial correlations between speeds in flocks of birds.
We study a model of flocking for a very large system (N=320,000) numerically. We find that in the long wavelength, long time limit, the fluctuations of the velocity and density fields are carried by propagating sound modes, whose dispersion and damping agree quantitatively with the predictions of our previous work using a continuum equation. We find that the sound velocity is anisotropic and characterized by its speed $c$ for propagation perpendicular to the mean velocity $<vec{v}>$, $<vec{v}>$ itself, and a third velocity $lambda <vec{v}>$, arising explicitly from the lack of Galilean invariance in flocks.
The effect of quenched (frozen) disorder on the collective motion of active particles is analyzed. We find that active polar systems are far more robust against quenched disorder than equilibrium ferromagnets. Long ranged order (a non-zero average velocity $langle{bf v}rangle$) persists in the presence of quenched disorder even in spatial dimensions $d=3$; in $d=2$, quasi-long-ranged order (i.e., spatial velocity correlations that decay as a power law with distance) occurs. In equilibrium systems, only quasi-long-ranged order in $d=3$ and short ranged order in $d=2$ are possible. Our theoretical predictions for two dimensions are borne out by simulations.
In a granular gas of rough particles the spin of a grain is correlated with its linear velocity. We develop an analytical theory to account for these correlations and compare its predictions to numerical simulations, using Direct Simulation Monte Carlo as well as Molecular Dynamics. The system is shown to relax from an arbitrary initial state to a quasi-stationary state, which is characterized by time-independent, finite correlations of spin and linear velocity. The latter are analysed systematically for a wide range of system parameters, including the coefficients of tangential and normal restitution as well as the moment of inertia of the particles. For most parameter values the axis of rotation and the direction of linear momentum are perpendicular like in a sliced tennis ball, while parallel orientation, like in a rifled bullet, occurs only for a small range of parameters. The limit of smooth spheres is singular: any arbitrarily small roughness unavoidably causes significant translation-rotation correlations, whereas for perfectly smooth spheres the rotational degrees of freedom are completely decoupled from the dynamic evolution of the gas.
Presenting simple coarse-grained models of isotropic solids and fluids in $d=1$, $2$ and $3$ dimensions we investigate the correlations of the instantaneous pressure and its ideal and excess contributions at either imposed pressure (NPT-ensemble, $lambda=0$) or volume (NVT-ensemble, $lambda=1$) and for more general values of the dimensionless parameter $lambda$ characterizing the constant-volume constraint.
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