Nucleation-induced transition to collective motion in active systems

While the existence of polar ordered states in active systems is well established, the dynamics of the self-assembly processes are still elusive. We study a lattice gas model of self-propelled elongated particles interacting through excluded volume and alignment interactions, which shows a phase transition from an isotropic to a polar ordered state. By analyzing the ordering process we find that the transition is driven by the formation of a critical nucleation cluster and a subsequent coarsening process. Moreover, the time to establish a polar ordered state shows a power-law divergence.

Figure 1

Illustration of the lattice gas model. Filaments of length $L$ move on a hexagonal lattice with constant speed performing a persistent random walk.

Figure 2

(a) The boundary for our lattice gas model is a reflecting hexagon with a side length $L_{\mathrm{box}}=1000$. The circle indicates the ROI with a radius of $L_{\mathrm{box}}/10$. Starting from a disordered state, first small clusters emerge (b) which subsequently evolve into bands (c) (see also the videos in the Supplemental Material [44]). The directions of cluster movement are indicated by the arrows. (d) The polarization ${\mathcal{P}}_{\mathrm{ROI}}$ (orange, light gray) is calculated over all particles in the ROI, while the polarization ${\mathcal{P}}_{\mathrm{SIM}}$ (blue, dark gray) is determined over all particles in the entire simulation box. The bold and dashed arrows correspond to (b) and (c), respectively. (e) Orientational correlation function $C(\tau ;t)$ as a function of time $\tau$ for a series of initial times $t$ as indicated in the graph. Simulation results refer to the parameters $\alpha =5$, $\epsilon =0.5$, and $\rho =0.6$. All simulations were typically run for $\sim$$2\times {10}^4$ time steps.

Figure 3

(a) Time traces of $\Gamma$ for $\rho =0.6$, and a set of values: $\epsilon =0,0.5,1$ with $\alpha =5$, and $\epsilon =1$ with $\alpha =1$, from bottom to top. Close to the phase boundary from the isotropic (ISO) to the polar ordered state (PO) [$\epsilon =1$; (red)], a lag phase exists, where the systems wait for the nucleation of a cluster that triggers the emergence of polar order; $T_{\mathrm{nuc}}$ and $T_{\mathrm{stat}}$ are indicated by vertical dotted or dashed lines. Inset: For $\alpha =1$, no order develops, irrespective of the value of $\epsilon$ ($\epsilon =0,5,100$ from bottom to top). (b) Free volume gain $\Phi =\rho -\Gamma$ (triangles) for $\alpha =5$ and $\epsilon =0.5$ exhibits a “jump” at ${\rho }_c\approx 0.4$ (dashed line) and then is significantly above the reference curve $\rho -{\Gamma }_{\mathrm{hom}}\left(\rho \right)$ (black $+$'s are simulation results). $T_{\mathrm{stat}}$ diverges for densities slightly above ${\rho }_c$ (squares). (c) Phase diagram as a function of $\rho$ and $\epsilon$ for $\alpha =5$ (solid) and $\alpha =10$ (dashed).

Figure 4

(a) Time traces of $m_{\max }$ (red solid), $\langle m\rangle$ (green dashed), and $\delta m$ (blue dotted) of the $P\left(m\right)$ for $\rho =0.6$. (b) Cluster mass distribution $P\left(m\right)$ for $\rho =0.6$ during the lag phase and in the stationary regime of the polar ordered state sampled over more than ${10}^4$ realizations. For reference, power-law distributions $P\left(m\right)\propto m^{-\delta }$ with $\delta =4,5$ are indicated by dashed lines. (c) Mean values of the largest cluster, $\langle M_{\max }\rangle$ (red squares) and $\langle M_{\mathrm{nuc}}\rangle$ (blue dots), serve as lower and upper bounds for critical nucleation cluster mass. In the transition window (shaded gray area) from the ISO to the PO phase, the fraction of realizations $\mathcal{S}$ which develop polar order (solid line) increases from 0 to 1. Parameters for all graphs are $\alpha =5$ and $\epsilon =0.5$.