Boltzmann and hydrodynamic description for self-propelled particles

We study analytically the emergence of spontaneous collective motion within large bidimensional groups of self-propelled particles with noisy local interactions, a schematic model for assemblies of biological organisms. As a central result, we derive from the individual dynamics the hydrodynamic equations for the density and velocity fields, thus giving a microscopic foundation to the phenomenological equations used in previous approaches. A homogeneous spontaneous motion emerges below a transition line in the noise-density plane. Yet, this state is shown to be unstable against spatial perturbations, suggesting that more complicated structures should eventually appear.

Figure 1

Schematic view of the dynamics of the model: (a) self-diffusion events, (b) binary collisions with alignment interactions—see text for notations.

Figure 2

(Color online) Numerical simulations with multiple-particle interactions, from Ref. 10 [dashed line in (a), (b), and (d)] and with binary collisions (full line). (a) Average order parameter; (b) Binder cumulant; (c) distribution of the order parameter for binary collisions; (d) density profile along the coordinate $x_{\perp }$.

Figure 3

Phase diagram of the model in the plane $(\rho ∕\lambda ,\sigma )$. A transition line (full line, ${\sigma }_0=\sigma$; dashed line, ${\sigma }_0=1$) indicates the linear instability threshold of the state $w=\mid \mathbf{w}\mid =0$.