Collective motion of self-propelled particles interacting without cohesion

We present a comprehensive study of Vicsek-style self-propelled particle models in two and three space dimensions. The onset of collective motion in such stochastic models with only local alignment interactions is studied in detail and shown to be discontinuous (first-order-like). The properties of the ordered, collectively moving phase are investigated. In a large domain of parameter space including the transition region, well-defined high-density and high-order propagating solitary structures are shown to dominate the dynamics. Far enough from the transition region, on the other hand, these objects are not present. A statistically homogeneous ordered phase is then observed, which is characterized by anomalously strong density fluctuations, superdiffusion, and strong intermittency.

Figure 1

(Color online) Typical behavior across the onset of collective motion for moderate-sized models ($\rho =2$, $v_0=0.5$, $L=64$) with angular (black circles) and vectorial noise (red triangles). The reduced noise amplitude $\epsilon =1-\eta /{\eta }_t$ is shown on the abscissa (transition points estimated at ${\eta }_t=0.6144\left(2\right)$—vectorial noise—and ${\eta }_t=0.478\left(5\right)$—angular noise). (a) Time-averaged order parameter ${⟨\phi \left(t\right)⟩}_t$. (b) Binder cumulant $G$. (c) Variance $\sigma$ of $\phi$. (d) Order parameter distribution function $P$ at the transition point. Bimodal distribution for vectorial noise dynamics (red dashed line); unimodal shape for angular noise (black solid line). Time averages have been computed over $3\times {10}^5$ time steps.

Figure 2

(Color online) FSS analysis of angular noise dynamics ($\rho =2$, $v_0=0.5$, time averages computed over $2\times {10}^7$ time steps). Time-averaged order parameter (a) and Binder cumulant (b) as a function of noise for various system sizes $L$. (c) Piece of an order parameter time series close to the transition point ($L=256$, $\eta =0.476$).

Figure 3

(Color online) Transition to collective motion with short-range repulsive interactions. Left panels: angular noise. Right panels: vectorial noise. (a),(b) Order parameter vs noise amplitude at different system sizes. (c),(d) Binder cumulant $G$ as a function of noise amplitude. ($\rho =2$, $v_0=0.3$, time averages carried over ${10}^7$ time steps.)

Figure 4

(Color online) Transition to collective motion in three spatial dimensions. Left panels: angular noise. Right panels: vectorial noise. (a),(b) Time-averaged order parameter vs noise amplitude at different system sizes. (c),(d) Binder cumulant $G$ as a function of noise amplitude. ($\rho =0.5$, $v_0=0.5$, time averages carried over ${10}^5$ time steps.)

Figure 5

Correlation time $\tau$ of the order parameter near the transition point for vectorial noise dynamics with repulsion. System parameters are $\rho =2$, $v_0=0.5$, and $\eta \approx {\eta }_t$. (a) Time correlation function $C\left(t\right)$ at $L=128$. The lin-log inset shows the exponential decay. (b) Correlation time $\tau$ as a function of system size $L$. The dashed line marks linear growth with $L$. Correlation functions were computed on samples of $\approx {10}^6$ realizations for typically ${10}^3$ time steps.

Figure 6

(Color online) FSS analysis of vectorial dynamics with short-range repulsive force ($\rho =2$, $v_0=0.3$). Convergence of the finite-size transition points measured from different moments of the order parameter FSS to the asymptotic transition point ${\eta }_t$ (see Fig. 3). Upper panels: finite-size transition points estimated from (a) time average, (b) variance, and (c) Binder cumulant. The horizontal dashed line marks the estimated asymptotic threshold ${\eta }_{\text{t}}=0.5544\left(1\right)$. Lower panels: scaling of the finite-size reduced noise $\epsilon =1-\eta /{\eta }_t$ transition point. (d) Exponential convergence for the jump location in the time-averaged order parameter. (e) Power-law behavior of the variance peak position. (f) Power-law behavior of the Binder cumulant minimum. The dashed lines in (e) and (f) mark the estimated exponents ${\gamma }_{\sigma }={\gamma }_G=2$.

Figure 7

(Color online) Hysteresis in three spatial dimensions with vectorial noise. (a) Order parameter vs noise strength along the hysteresis loop observed with a ramp rate of $2\times {10}^{-6}$ per time step ($\rho =1/2$, $v_0=0.5$, $L=32$). Empty circles mark the path along the adiabatic increase of noise amplitude; full triangles for adiabatic decrease. (b) Nucleation times from the disordered phase to the ordered phase (${\tau }_{\uparrow }$, left curves) and vice versa (${\tau }_{\downarrow }$, right curves) for two system sizes [other parameters as in (a)]. Each point is averaged over 1000 realizations.

Figure 8

(Color online) Asymptotic phase diagrams for the transition to collective motion. (a) Two space dimensions: threshold amplitude ${\eta }_t$ for angular noise as a function of density $\rho$ at $v_0=0.5$. Inset: Log-log plot to compare the low-density behavior with the mean-field predicted behavior ${\eta }_{\text{t}}\sim \sqrt{\rho }$ (dashed red line). (b) As in (a), but with vectorial noise dynamics. (c) Noise-density phase diagram in three dimensions for vectorial noise dynamics at fixed velocity $v_0=0.5$. In the log-log inset the transition line can be compared with the predicted behavior ${\eta }_t\sim {\rho }^{1/3}$ (dashed red line). (d) Two space dimensions: threshold amplitude ${\eta }_t$ for angular noise as a function of particle velocity $v_0$ at fixed density $\rho =1/2$ (black circles) and 1/8 (red triangles). The horizontal dashed line marks the noise amplitude considered in Ref. [39] (see Sec. 3F).

Figure 9

(Color online) First-order transition for angular noise dynamics at high (left panels) and low (right panels) velocity $v_0$. Typical averaging time is $\approx {10}^6$ time steps. (a) Time-averaged order parameter and (c) Binder cumulant at large particle velocity for angular noise in two spatial dimensions at increasing velocities and $L\gtrsim L^{\ast }\left(\rho =2\right)$. (b) Time-averaged order parameter and (d) Binder cumulant for $v_0=0.05$ and two increasing system sizes $\left(\rho =1/2\right)$.

Figure 10

(Color online) Crossover system size $L^{\ast }$ above which the discontinuous character of the transition appears [as testified by the existence of a minimum in the $G\left(\eta \right)$ curve]. Black circles: angular noise. Red triangles: vectorial noise. (a) $L^{\ast }$ vs $\rho$ for $v_0=0.5$. Inset: the low-density behavior in log-log scales; the dashed line marks a power-law divergence proportional to $1/\sqrt{\rho }$. (b) $L^{\ast }$ vs $v_0$ at fixed density (open symbols, $\rho =1/2$; filled symbols, $\rho =2.0$).

Figure 11

(Color online) Typical snapshots in the ordered phase. Points represent the position of individual particles and the red arrow points along the global direction of motion. (a)–(c) Angular noise, $\rho =1/2$, $v_0=0.5$, $\eta =0.3$, and increasing system sizes, respectively, $L=64$, 256, and 1024. Sharp bands can be observed only if $L$ is larger than the typical bandwidth $w$. (d) Vectorial noise, $\rho =1/2$, $v_0=0.5$, $\eta =0.55$, and $L=64$: bands appear at relatively small system sizes for this type of noise. For clarity, only a representative sample of 10 000 particles is shown in (b) and (c). Boundary conditions are periodic.

Figure 12

(Color online) Same as Fig. 11 but in different geometries and boundary conditions or space dimensions. (a),(b) Vectorial noise ($\eta =0.325$, $\rho =1/8$, and $v_0=0.5$); boundary conditions are periodic along the $y$ (vertical) axis and reflecting in $x$. (a) A long single band travels along the periodic direction. (b) The domain size along the periodic direction is too small to accommodate bands, and a single band bouncing back and forth along the nonperiodic direction is observed. (c) Angular noise, repulsive force, and periodic boundary conditions ($\rho =2$, $\eta =0.23$, and $v_0=0.3$). (d) High-density sheet traveling in a three-dimensional box with periodic boundary conditions (angular noise with amplitude $\eta =0.355$, $\rho =1/2$, and $v_0=0.5$).

Figure 13

(Color online) (a) Typical density (black line) and order parameter (dashed red line) profiles for bands in two dimensions (vectorial noise, $\rho =2$, $\eta =0.6$, and $v_0=0.5$). (b) Tail of the density profile shown in (d) (black line) and its fit (blue dashed line) by the formula ${\rho }_{\perp }\left(x_{\parallel },t\right)\approx a_0+a_1\left(t\right)\text{exp}\left(-x_{\parallel }/w\right)$, with $w\approx 6.3$ (lin-log scale). (c),(d) Traveling sheet in three dimensions (angular noise, $\rho =1/2$, $\eta =0.355$, and $v_0=0.5$). (c) Projection of particle positions on a plane containing the global direction of motion (marked by red arrow). (d) Density (black line) and order parameter (dashed red line) profiles along the direction of motion $x_{\parallel }$.

Figure 14

(Color online) Emergence of high-density high-order traveling bands $\left(d=2\right)$ from a spatially homogeneous (uniformly distributed random positions) initial condition with all particle velocities oriented along the major axis of a $196\times 1960$ domain with periodic boundary conditions. Vectorial noise of amplitude $\eta =0.6$, density $\rho =2$, and $v_0=0.5$. (a) Space-time plot of the density profile. Time is running from left to right from $t=0$ to 12 000, while the longitudinal direction is represented on the ordinates. Color scale from blue (low values) to red (high values). (b) Same as (a) but at later times (from $t=148000$ to 160 000). (c) Spatial Fourier power spectrum $S$ of an early density profile $\left(t=12000\right)$. (d) Early-time evolution of selected Fourier modes $k=10$, 23, 28, 33, 41, 76, and 121 (the black lowest curve is for $k=10$, the other curves are not significantly different). Inset: average over 50 different runs. (e) Density profile at a late time [$t=160000$, final configuration of (b)]. (f) Same as (c) but for the late-time density profile of (e).

Figure 15

(Color online) (a),(b) Time-averaged profile width for both density (a) and order parameter (b) as a function of noise amplitude in the ordered phase (angular noise, $1024\times 256$ domain, global motion along the major axis, $\rho =2$, and $v_0=0.5$). The dashed vertical blue line marks the order-disorder transition. (c),(d) Typical instantaneous profiles along the long dimension of the system described in (a) and (b) for intermediate noise value [(b) $\eta =0.4$] and in the bandless regime [(c) $\eta =0.15$].

Figure 16

(Color online) Giant density fluctuation and transverse superdiffusion in the bandless ordered phase. (a) Anomalous density fluctuations (see text): $\Delta n$ scales approximately like $n^{0.8}$ (the dashed line has slope 0.8) both in two dimensions (black circles, $L=256$, $\rho =2$, $v_0=0.5$, angular noise amplitude $\eta =0.25$) and in three dimensions (red triangles, $L=64$, $\rho =1/2$ $v_0=0.5$, vectorial noise amplitude $\eta =0.1$, values shifted for clarity). (b) Average doubling time ${\tau }_2$ of the transverse (with respect to mean velocity) interparticle distance ${\delta }_{\perp }$. Black circles: ordered bandless regime ($\rho =4$, angular noise amplitude $\eta =0.2$ in a rectangular box of size $1024\times 256$). The black dashed line marks the expected growth ${\tau }_2\sim {\delta }_{\perp }^{3/2}$. Red squares: same but deep in the disordered phase ($\rho =4$, angular noise amplitude $\eta =1$, $L=512$). The dot-dashed red line shows normal diffusive behavior: ${\tau }_2\sim {\delta }^2$.

Figure 17

(Color online) (a),(b) Cluster size distributions (arbitrary units) for domain sizes $L=32$ (black), 128 (cyan), and 512 (green) from left to right ($d=2$, $\rho =2$, $v_0=0.5$, angular noise). (c),(d) Probability distribution functions of the order parameter $\phi$ (arbitrary units) for the same parameters and system sizes as in (a) and (b). (The most peaked distributions are for the largest size $L=512$.) Left panels (a),(c), $\eta =0.1$, bandless regime; Right panels (b),(d), regime with bands at $\eta =0.4$.

Figure 18

(Color online) Phase ordering from disordered initial conditions ($d=2$, angular noise amplitude $\eta =0.08$, $\rho =1/8$, $v_0=0.5$, system size $L=4096$). Snapshots of the density field coarse grained on a scale $\ell =8$ at times $t=160$, 320, and 640 from left to right.

Figure 19

(Color online) Phase ordering as in Fig. 18 ($L=4096$, $\rho =1/2$, $v_0=0.5$). (a) Two-point density correlation function $C\left(r,t\right)={⟨{\rho }_{\ell }\left(\vec{x},t\right){\rho }_{\ell }\left(\vec{x}+\vec{r},t\right)⟩}_{\vec{x}}$ (coarse grained over a scale $\ell =4$) as a function of distance $r=\left|\vec{r}\right|$ at different time steps: from left to right $t=50$, 50, 100, 200, 400, 800, and 1600. Noise amplitude is $\eta =0.25$; data have been further averaged over $\approx 40$ different realizations. Inset: log scales reveal the intermediate near-algebraic decay and the quasiexponential cutoff. (b) Length scale $\xi$, estimated from the exponential cutoff positions, as a function of time. Empty black circles: $\eta =0.25$ as in (a) (i.e., regime in which bands are observed asymptotically). Red full triangles: $\eta =0.1$ (i.e., in the bandless regime). The dashed black line marks linear growth.

Figure 20

(Color online) Scatter plots of local order parameter vs local density in the ordered phase [angular noise, $\rho =2$, $v_0=0.5$, in a domain of size $1024\times 256$—the same parameters as in Fig. 15a]. The local quantities were measured in boxes of linear size $\ell =8$. Here, the local order is represented by the angle between the orientation ${\Theta }_{\text{local}}$ of the local order parameter and the global direction of motion ${\Theta }_{\text{global}}$. The black solid lines are running averages of the scatter plots. The red solid lines indicate the global density $\rho =2$ (and thus mark the percolation threshold in a two-dimensional square lattice). (a),(b) $\eta =0.1$ and 0.2: in the bandless regime, the ordered plateau starts below $\rho =2$, i.e., ordered regions percolate. (c) Approximately at the limit of existence of bands: the start of the plateau is near $\rho =2$. (d) At higher noise amplitude in the presence of bands.

Figure 21

(Color online) Test of the fluctuation-dissipation relation on the vectorial model with repulsive force ($\rho =2$, $v_0=0.3$, $L=128$). (a) Susceptibility vs time at reduced noise amplitude $\epsilon =1-\eta /{\eta }_t=0.005$, $\left|\vec{h}\right|={10}^{-2}$ (dashed red line), and $\left|\vec{h}\right|={10}^{-3}$ (plain black line). (b) Susceptibility vs correlation at $\left|\vec{h}\right|={10}^{-3}$ and $\epsilon =0.08$, 0.26, and 0.44 from top to bottom. (c) Effective temperature vs noise amplitude. The vertical dashed line marks the transition point.