Transitions in a self-propelled-particles model with coupling of accelerations

We consider a three-dimensional, generalized version of the original self-propelled-particles (SPP) model for collective motion. By extending the factors influencing the ordering, we investigate the case when the movement of the SPPs depends on both the velocity and the acceleration of the neighboring particles, instead of being determined solely by the former one. By changing the value of a weight parameter $s$ determining the relative influence of the velocity and the acceleration terms, the system undergoes a kinetic phase transition as a function of a behavioral pattern. Below a critical value of $s$ the system exhibits disordered motion, while above it the dynamics resembles that of the SPP model. We show that the critical value of the strategy variable could correspond to an evolutionary optimum in the sense that the information exchange between the units of the system is maximal in this point.

Figure 1

(Color online) Average velocity $⟨\phi ⟩$ as a function of the strategy variable $s$. The data points were obtained by averaging the results of 12 simulations, using the parameters $L=100$, $\rho =0.16$, and $\eta =1∕9$, with a relaxation time of $T_{\mathit{relax}}=10000$ time steps and averaging over an additional $T_{\mathit{avr}}=10000$ time steps.

Figure 2

(Color online) Probability density function (PDF) of the order parameter $\phi$. Parameters as in the previous plot. The one-humped distributions in a range of $s$ values close to the critical one $(s_c\simeq 0.59)$ suggest a second-order phase transition.

Figure 3

(Color online) The Binder cumulant $G$ as a function of the strategy variable $s$. The dotted lines at $2∕3$ and $4∕9$ indicate the theoretical value in the case of the ordered and totally disordered states, respectively. Parameters as in the previous plots. The absence of a sharp minimum indicates a second-order phase transition.

Figure 4

(Color online) Average velocity values as a function of the distance from the critical point. The fitted line, with $s_c=0.590\pm 0.002$, had a slope of $\beta =0.35\pm 0.05$. Parameters as in the previous plots.

Figure 5

(Color online) Border surface between the ordered and disordered phases in the $s$, $\rho$, $\eta$ parameter space. The ordered $(⟨\phi ⟩⩾0.01)$ and disordered $(⟨\phi ⟩<0.01)$ states are indicated by heavy and light dots, respectively. Squares show the critical noise values ${\eta }_c(s,\rho )$ obtained from the plot of $\log ⟨\phi ⟩$ as a function of $\log [({\eta }_c-\eta )∕{\eta }_c]$ for each $s$ and $\rho$ pair. $L=100$.

Figure 6

(Color online) The curvature of trajectories, $\lambda$, as a function of the strategy variable $s$. The dotted line at $s_c=0.590$ indicates the position of the critical point for ordering. Parameters as in the previous plots.

Figure 7

Shown are the positions and trajectories of particles for different strategy values. These images were obtained by using perspective projection of the three-dimensional data onto two dimensions. This is the view we have if we look at the transparent simulational box with its closest edge being vertical and centered at the observational field. Subfigures show the typical behavior (a), (d) below, (b), (e) near, and (c), (f) above the critical point, respectively. (a)–(c) Positional data of the particles shown as birds. (d)–(f) Each curve shows the trajectory of a particle over 60 time steps after reaching steady state $(t=20000)$. Different shades of gray indicate the time past, with darker tones denoting more recent positions. (a), (d) Below the critical point, at $s=0.5$, the particles move in circles. (b), (e) Near the critical point, at $s=0.53$, the particles move sinuously. (c), (f) Above the critical point, at $s=0.9$, the particles move in the same direction. $L=20$, $\eta =1∕6$, other parameters as in the previous plots.

Figure 8

(Color online) Information exchange between particles $\left(\psi \right)$ as a function of the strategy variable $s$. The curve for $\mu$ was very similar (not shown). Parameters as in Fig. 1.

Figure 9

(Color online) The order parameter ($⟨\phi ⟩$, × marks), the Binder cumulant ($G$, triangles), the information exchange between particles ($\psi$, circles), and the average curvature of trajectories ($\lambda$, squares) as a function of the strategy variable $s$ in case of the AC-SPP model with SVA. Compared to the OVA in the case of the SVA, the critical strategy value is at lower $s$, but the behaviors of the $⟨\phi ⟩$, $G$, and $\lambda$ functions are very similar. The information exchange between particles has its maximum value near the critical point, but it is much more sensible for $s$. Parameters as in Fig. 1.