Emergence of a single cluster in Vicsek's model at very low noise

The classic Vicsek model [Phys. Rev. Lett. 75, 1226 (1995)] is studied in the regime of very low noise intensities, which is shown to be characterized by a cluster [macrocluster (MC)] that contains a macroscopic fraction of the system particles. It is shown that the well-known power-law behavior of the cluster-size distribution loses its cutoff, becoming bimodal at very low noise intensities: A peak develops for larger sizes to settle the emergence of the MC. The average cluster number $m^{*}$ is introduced as a parameter that properly describes this change, i.e., a line in the noise-speed phase portrait can be identified to separates both regimes. Finite-size scaling analysis is performed to show that a phase transition to a macroscopic cluster is taking place. Consistency of the results with the literature is also checked and commented on.

Figure 1

Snapshots of configurations at different noise intensities. Black arrows show the average direction of motion. Lower panels show the CSD. (a) A single macrocluster is revealed. (b) Below and close to ${\eta }_{\mathrm{MC}}$ the MC coexists with many single particles. (c) Above and close to the transition there are no intermediate-size clusters. (d) A roughly spatially homogeneous $\varphi$ distribution arises (see text). Parameters: $v_0=0.1$, $\rho =0.1$, $N=2^{15}$. Clusters are colored according to their sizes $m$; $\varphi >0.98$ for all situations.

Figure 2

Time evolution of the mean number of clusters $M^{*}$ for different combinations of $\rho$ and $v_0$. (a) $\eta =0.0001$ where the macrocluster appears. (b) Clustered regime at $\eta =0.01$ in the homogeneous phase. (c) Clustered regime at $\eta =0.1$ where density waves are observed. Insets: The same information presented for various values of $(\rho ,v_0)$ with the arrows illustrating the direction of increasing values of $\eta$. In all graphs the dashed gray line is a reference with slope $-1$.

Figure 3

Time evolution of $\varphi$ and $m^{*}$. (a) The ferromagnetic order parameter, $\varphi$, stabilizes over $t\simeq {10}^3$. The curves show the same qualitative behavior for different noise intensities. (b) The average cluster size $m^{*}$ changes its behavior at noise intensities close to $\eta \simeq {10}^{-3}$, revealing the emergence of a macrocluster at lower noise intensities. Note the similitude between $\eta ={10}^{-4}$ (red) and $\eta =0$ (black) curves. (c) In the low-density–low-speed regime ($\rho =0.1$–$v_0=0.1$) in a larger system ($N=2^{15}$), the flattening of the bump close to the ODT is most easily observed. (d) Portion of one realization of the time series for $m^{*}$ at times when the MC is losing and reabsorbing particles. The sampling rate is $5\times {10}^4$ time steps.

Figure 4

Cluster-size distributions $P\left(m\right)$ averaged over ${10}^4$ realizations. (a) CSD with linear binning. (b) CSD with logarithmic binning clearly reveals that at very low noise, $\eta ={10}^{-4}$ (red), the MC ($m\simeq N$) exhibits a process of particle gain and loss. As the noise is increased, the well-known power law plus a cutoff behavior emerges. Note that close to the transition, $\eta ={10}^{-3}$ (blue), the CSD becomes nonmonotonic. Straight lines correspond to power-law fittings. Inset: CSD with linear (orange) and logarithmic (red) binning at $\eta ={10}^{-4}$ averaging ${10}^5$ realizations (see text).

Figure 5

Phase portrait $v_0\text{−}\eta$ for $\rho =1$. Dashed line is a rough estimate of the transition line between clustered and macrocluster regimes.

Figure 6

Phase portrait $v_0\text{−}\eta$ of configurations for $\rho =0.1$. It looks similar to $\rho =1$ with a presumably steeper transition. Dashed and dotted lines are estimates of the MCT and ODT lines, respectively.

Figure 7

Transition to the emergence of an MC. (a) The averaged largest cluster size ${\zeta }_1/N$ falls from 1 at nonzero low noise intensity. Fluctuations of ${\zeta }_1$ (b) and values of ${\zeta }_2$ (c) develop peaks close to the critical noise. Lines are log-normal fittings. (d) Fitting of the scaling amplitudes gives the trivial power law $\chi \left({\zeta }_1\right)\sim N$. (e) The peak position ${\eta }_{\max }={\eta }_c+\mathrm{const}\frac{1}{N}$. The extrapolation to $N\to \infty$ gives ${\eta }_c=0.00183\left(8\right)$ and ${\eta }_c=0.0025\left(5\right)$ for the largest and second-largest clusters, respectively.

Figure 8

Cluster-size evolution. The time to form an MC from a specific initial condition drastically changes with system size (continuous lines). This dependency vanishes when random initial conditions are set up (dashed lines).