Lattice Model to Derive the Fluctuating Hydrodynamics of Active Particles with Inertia

We derive the hydrodynamic equations with fluctuating currents for the density, momentum, and energy fields for an active system in the dilute limit. In our model, nonoverdamped self-propelled particles (such as grains or birds) move on a lattice, interacting by means of aligning dissipative forces and excluded volume repulsion. Our macroscopic equations, in a specific case, reproduce a transition line from a disordered phase to a swarming phase and a linear dispersion law accounting for underdamped wave propagation. Numerical simulations up to a packing fraction $\sim 10\%$ are in fair agreement with the theory, including the macroscopic noise amplitudes. At a higher packing fraction, a dense-diluted coexistence emerges. We underline the analogies with the granular kinetic theories, elucidating the relation between the active swarming phase and granular shear instability.

Figure 1

Sketch of APs in a lattice. Particles $A–D$ show hopping (black arrows) to neighbor sites according to the directions of their active velocity (blue arrows), including periodic conditions ($A$) and excluded volume ($C$,$D$). Next neighbors interact ($E$,$F$), with a force which aligns velocities (from blue to red arrows). Self-propulsion acts in the direction of the velocity and brings the speed toward a fixed value ($G$, from blue to green arrow). The velocity of a particle can be modified also by external noise ($H$, from blue to green).

Figure 2

Typical configurations in the three main stable states observed in the simulations. Left: A uniform disordered state. Middle: A uniform swarming state. Right: A nonuniform clustered state with a converging radial velocity field.

Figure 3

Swarming phase parameter $r$ (which goes from 0 in the disordered phase to 1 in the swarming phase; see the legend on the right) as a function of relative noise amplitude $\mathrm{\Gamma }$ and rescaled dissipation rate $\varphi \gamma$ at three different average densities $\varphi$. The solid lines indicate the theoretical transition, Eq. (14).

Figure 4

Amplitude of fluctuations of the hopping current (symbols) and its theoretical prediction Eqs. (16) and (17) (solid line), as a function of $\varphi$ for various sizes $L$. Here $\varphi \gamma =0.25$, $\mathrm{\Gamma }=4$, and the system is prepared at temperature $T_0=1$.