Phase separation of self-propelled ballistic particles

Self-propelled particles phase-separate into coexisting dense and dilute regions above a critical density. The statistical nature of their stochastic motion lends itself to various theories that predict the onset of phase separation. However, these theories are ill-equipped to describe such behavior when noise becomes negligible. To overcome this limitation, we present a predictive model that relies on two density-dependent timescales: ${\tau }_F$, the mean time particles spend between collisions; and ${\tau }_C$, the mean lifetime of a collision. We show that only when ${\tau }_F<{\tau }_C$ do collisions last long enough to develop a growing cluster and initiate phase separation. Using both analytical calculations and active particle simulations, we measure these timescales and determine the critical density for phase separation in both two and three dimensions.

Figure 1

(a) A schematic representation of two key timescales involved in the triggering of phase separation for active self-propelled particles. For a single active particle (red), the free time ${\tau }_F$ considers the time a particle spends between collision events, while the collision time ${\tau }_C$ considers the lifetime of a collision. (b) A schematic representation of the incident angle of collision ${\theta }_0$, for Eq. (2).

Figure 2

The mean free time ${\tau }_F$ (blue triangles) and the mean collision time ${\tau }_C$ (red circles), as a function of area fraction $\varphi$, in both two (a) and three dimensions (b). Solid colored lines represent the predicted values calculated with Eqs. (1) and (3). The congestion density, being the minimum density required for phase separation, is the point where ${\tau }_C>{\tau }_F$ is ${\varphi }_{\text{con}}>0.25$ for two dimensions and ${\varphi }_{\text{con}}>0.18$ for three dimensions (marked with vertical solid lines). The critical density for phase separation found using simulations is ${\varphi }_{\text{crit}}$ (marked with the vertical dashed lines). Visualizations of the results at various densities are shown with particles colored by their cluster size. Local density histograms are also provided (using the same color scale), showing a transition from unimodal to bimodal distributions upon phase separation. For the 3D visualization, particles not belonging to a large cluster are reduced in diameter for easier viewing.

Figure 3

Timescales ${\tau }_C$ (solid red line), ${\tau }_F$ (solid blue line), and ${\tau }_C-{\tau }_F$ (dashed black line), as a function of time $\tau$ in two dimensions. The density is $\varphi =0.4$, well above the spinodal of phase separation at ${\varphi }_{\text{crit}}=0.28$. Insets show snapshots at two different times, with particles colored by their cluster size. Each data point is averaged over at least 10 000 measurements.

Figure 4

Average timescale measurements from simulations for clusters consisting of $n=3$ particles in both (a) two dimensions and (b) three dimensions. The average individual timescales for all particles from Fig. 2 are also shown for comparison. Larger clusters tend to have a crossover of ${\tau }_C^{\left(n\right)}={\tau }_F^{\left(n\right)}$ at a density higher than ${\tau }_C={\tau }_F$ (marked with vertical lines).