Keeping speed and distance for aligned motion

The cohesive collective motion (flocking, swarming) of autonomous agents is ubiquitously observed and exploited in both natural and man-made settings, thus, minimal models for its description are essential. In a model with continuous space and time we find that if two particles arrive symmetrically in a plane at a large angle, then (i) radial repulsion and (ii) linear self-propelling toward a fixed preferred speed are sufficient for them to depart at a smaller angle. For this local gain of momentum explicit velocity alignment is not necessary, nor are adhesion or attraction, inelasticity or anisotropy of the particles, or nonlinear drag. With many particles obeying these microscopic rules of motion we find that their spatial confinement to a square with periodic boundaries (which is an indirect form of attraction) leads to stable macroscopic ordering. As a function of the strength of added noise we see—at finite system sizes—a critical slowing down close to the order-disorder boundary and a discontinuous transition. After varying the density of particles at constant system size and varying the size of the system with constant particle density we predict that in the infinite system size (or density) limit the hysteresis loop disappears and the transition becomes continuous. We note that animals, humans, drones, etc., tend to move asynchronously and are often more responsive to motion than positions. Thus, for them velocity-based continuous models can provide higher precision than coordinate-based models. An additional characteristic and realistic feature of the model is that convergence to the ordered state is fastest at a finite density, which is in contrast to models applying (discontinuous) explicit velocity alignments and discretized time. To summarize, we find that the investigated model can provide a minimal description of flocking.

Figure 1

Symmetric planar collision of two identical pointlike particles according to Eq. (2). (a) and (b) Illustration of the model. (c) and (d) Geometry of the collision and results. We integrate the equations of motion with the forward Euler method. Initial and final distances were $r=1000$. If the integration time step is reduced from $\Delta t={10}^{-3}$ to ${10}^{-4}$, the results change very little [see circles and crosses in panel (d)].

Figure 2

Transition from disorder to order in a rectangle with periodic boundaries and without noise ($\xi =0$). At $t=0$ the system is started from the disordered state. During the simulations particles self-propel to maintain their speed, and they repel each other [see Eq. (1) without noise]. (a) Convergence to the $E\approx 1$ ordered state is robust to changes in model parameters ($N,\tau ,v_0$) and integration parameters ($R,\Delta t$). (b) Initial state: Coordinates are random with distances above $L/\left(2\sqrt{N}\right)$. Directions are random, and speeds are $v_0$. (c) Final state: Coordinates and directions are ordered.

Figure 3

The length of simulation time $t$ needed by the system to reach the average efficiency $\langle E\rangle =1/2$ when running without noise ($\xi =0$) and started from the disordered state. The results show averages over 100 simulation runs. The density of particles $\rho =N/L^2$ is varied among data points. In both panels the black solid line was obtained with the reference parameter set shown in Fig. 2. The two panels show results for various (a) time constants of self-propelling $\tau$ and (b) preferred velocities of motion $v_0$. We conclude that the most rapid convergence to the ordered state occurs always at a finite density.

Figure 4

Transition from order [Fig. 2] to disorder [Fig. 2]. At $t=0$ the system is in the ordered state. The reference parameter set of Fig. 2 is modified only by varying the noise amplitude $\xi$. The midpoint integration method is applied by setting the length of the noise vector to $\xi \sqrt{\Delta t}$ and $\xi \sqrt{\Delta t/2}$ for the first and second steps of an update, respectively. Main panel: At low noise amplitudes the system remains in the ordered state, and at high noise amplitudes it leaves the ordered state. The inset: The average efficiency reaches 1.2 (from above) at a simulation time that diverges—as a function of the noise amplitude—at a finite $\xi ={\xi }_c$ value similar to a power law. The solid line is a fit to the $\xi =63\cdots 68$ data points with the parameters ${\xi }_c=53.9\pm 4.6$ (critical point) and $\gamma =-10.1\pm 4.1$ (exponent).

Figure 5

Searching for hysteresis. We use the reference parameter values [Fig. 2] or modify it by setting the noise amplitude. We find that at simulation time $t={10}^6$ systems started from the ordered state ($E\approx 1$) have a higher average efficiency than those started from the disordered state. Thus, the system shows hysteresis with this parameter set and at this simulation time. However, after longer simulations or with more particles the two $\xi =50$ curves (marked) may converge to the same final average $E$ value.

Figure 6

Efficiency ($E$ ordering) as a function of noise amplitudes ($\xi$). The reference parameter set of Fig. 2 is changed by varying $\xi$. Main panel: With longer simulation times lower noise amplitudes are sufficient for replacing the ordered state with the disordered state. A circle marks the parameter values used in the inset. The inset: At a finite simulation time and finite system size the bimodal distribution of the efficiencies of 1000 simulation runs shows the presence of hysteresis. The peak of the high efficiency (ordered) state and the peak of the low efficiency (disordered) state do not move. Only the few simulation runs (instances of the system) marked between the two peaks by a right-to-left arrow are transitioning from the peak of the ordered state to the peak of the disordered state.

Figure 7

Simulation time necessary for the $\mathrm{disorder}\to \mathrm{order}$ transition without noise (red squares, solid line) and the $\mathrm{order}\to \mathrm{disorder}$ transition with noise (green triangles, dashed line). The density of particles is constant. With growing system size noise can destroy the ordered state in constant time (triangles). On the other hand, in the absence of explicit noise ($\xi =0$) the time needed for order to emerge from disorder grows at least as fast as a power law (squares). We have tested that in the displayed system size range this speed of growth is below exponential.