Pattern Formation in Self-Propelled Particles with Density-Dependent Motility

We study the behavior of interacting self-propelled particles, whose self-propulsion speed decreases with their local density. By combining direct simulations of the microscopic model with an analysis of the hydrodynamic equations obtained by explicitly coarse graining the model, we show that interactions lead generically to the formation of a host of patterns, including moving clumps, active lanes, and asters. This general mechanism could explain many of the patterns seen in recent experiments and simulations.

Figure 1

(A) Phase diagram in the $(ϵ,\lambda )$ plane, for $N=3000$, $L=10$, $\gamma =0.16$, $v_0=2$, and $v_1=0.1$. Blue filled circles on the phase boundary correspond to peaks in the variance of the particle density, while green squares separate states with zero and nonzero mean orientation. Phases are labeled as per discussion in the text. Horizontal and vertical red lines indicate linear instabilities toward clustering and ordering, respectively. (B) Snapshots of the stripy (b), aster (e), moving clumps [d(i)], and lane [d(ii)] patterns. The crosses in A correspond to the snapshots in B. Particles are color coded by direction, with blue (darker gray) horizontal and red (lighter gray) vertical.

Figure 2

(a) Phase boundary for the flying $XY$ model when $\lambda =0$, showing the critical value of $ϵ$ as a function of $\gamma$. Blue (darker gray) points for $v=2.0$, red (lighter gray) for $v=0.5$. Inset: data for $v=2.0$ for smaller values of $\gamma$. (b) Phase boundary for $ϵ=5$, $\gamma =0.16$. In all cases $L=10$ and $N=1000$.

Figure 3

Patterns found for $\lambda \ne 0$ in the microscopic simulations (left column) and in the numerical solution of the hydrodynamic equations (right). Tables show dimensionless parameter values: $\tilde{\lambda }=\lambda {\rho }_0$, ${\tilde{v}}_1=v_1/v_0$, $\tilde{\gamma }=\gamma {\rho }_0/ϵ$, $\tilde{D}=Dϵ/v_0^2$. Arrows show the $\mathbf{W}$ field, colors the density [red (gray): high; blue (dark gray): low; yellow (light gray): intermediate]. In the right column, only a fraction of the simulated system is shown for clarity.