The Vicsek Model

Module 2: Synchronization and Collective Behaviour

Reading

Compulsory: Vicsek et al. (1995).
Recommended: Couzin et al. (2002); Helbing and Molnár (1995).

The Kuramoto model synchronizes phases in time. The Vicsek model aligns headings in space. It describes $N$ self-propelled particles that move at constant speed and copy the average heading of their neighbours, with some noise (Vicsek et al. 1995). There is no leader and no global signal, yet the whole group can move in one direction.

Rules

Each particle $i$ has a position $\mathbf{x}_i(t)$ in a square box of side $L$ with periodic boundaries, and a heading $\theta_i(t)$. At every time step $\Delta t$ all particles update at the same time:

\[ \theta_i(t + \Delta t) = \langle \theta(t) \rangle_{r,i} + \Delta\theta_i, \tag{1}\]

\[ \mathbf{x}_i(t + \Delta t) = \mathbf{x}_i(t) + v_0 \Delta t \begin{bmatrix} \cos\theta_i(t + \Delta t) \\ \sin\theta_i(t + \Delta t) \end{bmatrix}. \tag{2}\]

$\langle \theta \rangle_{r,i}$ is the mean direction of all particles within distance $r$ of particle $i$, including $i$ itself. It is computed as $\arctan\big(\langle\sin\theta\rangle / \langle\cos\theta\rangle\big)$, using the quadrant of the vector $(\langle\cos\theta\rangle, \langle\sin\theta\rangle)$.
$\Delta\theta_i$ is drawn uniformly from $[-\eta/2, \eta/2]$. The noise amplitude $\eta$ runs from $0$ (no noise) to $2\pi$ (a random heading every step).
The speed $v_0$ is the same for all particles.

Vicsek et al. (1995) used $r = 1$, $\Delta t = 1$ and $v_0 = 0.03$. The two control parameters are the noise $\eta$ and the density $\rho = N/L^2$.

Order parameter

Collective motion is measured by the normalized average velocity

\[ v_a = \frac{1}{N v_0}\left|\sum_{i=1}^N \mathbf{v}_i\right| = \left|\frac{1}{N}\sum_{i=1}^N e^{i\theta_i}\right|, \tag{3}\]

which is the Kuramoto order parameter applied to headings. $v_a \approx 1$ when all particles move the same way and $v_a \approx 0$ when headings are random.

Kinetic phase transition

Vicsek et al. (1995) found that $v_a$ falls from near $1$ to near $0$ as the noise increases, and rises as the density increases. Near the transition they reported the scaling laws

\[ v_a \sim \big(\eta_c(\rho) - \eta\big)^{\beta}, \qquad v_a \sim \big(\rho - \rho_c(\eta)\big)^{\delta}, \]

with $\beta = 0.45 \pm 0.07$ and $\delta = 0.35 \pm 0.06$. They interpreted it as a continuous transition. Later simulations of much larger systems by Grégoire and Chaté (2004) and Chaté et al. (2008) found that the onset of collective motion is discontinuous. Near the threshold, the ordered phase organizes into dense travelling bands. Small simulations like the ones in this course cannot distinguish the two scenarios.

Figure 1: Order parameter against noise at density $\rho = 4$ for $N = 40$ and $N = 100$, one realization each, generated with `amlab.collective_motion.vicsek`. Compare with Figure 2(a) of Vicsek et al. (1995).

Exercise

In Figure 1, $v_a$ stays around $0.1$ to $0.2$ at the largest noise, instead of going to zero. Why? Use what you know from the Kuramoto model.

Solution

At high noise the headings are close to independent and uniform. Then $v_a$ is the modulus of the mean of $N$ random unit vectors, so $\mathbb{E}[v_a^2] = 1/N$ (The Order Parameter). For $N = 40$ and $N = 100$ this gives typical values of $0.16$ and $0.1$, which is what the figure shows.

Exercise

Compare the Vicsek model with the mean-field Kuramoto model. Name one analogue of the coupling strength $K$ and one analogue of the frequency spread $\sigma$.

Solution

Alignment plays the role of coupling. Its effective strength grows with the number of neighbours inside radius $r$, so the density $\rho$ (or $r$ itself) acts like $K$. The noise $\eta$ plays the role of the disorder that opposes alignment, like $\sigma$. One important difference: in the Vicsek model the neighbours change as particles move, so the interaction network is itself dynamic.

References

Chaté, Hugues, Francesco Ginelli, Guillaume Grégoire, and Franck Raynaud. 2008. “Collective Motion of Self-Propelled Particles Interacting Without Cohesion.” Physical Review E 77 (4): 046113. https://doi.org/10.1103/PhysRevE.77.046113.

Couzin, Iain D, Jens Krause, Richard James, Graeme D Ruxton, and Nigel R Franks. 2002. “Collective Memory and Spatial Sorting in Animal Groups.” Journal of Theoretical Biology 218 (1): 1–11.

Grégoire, Guillaume, and Hugues Chaté. 2004. “Onset of Collective and Cohesive Motion.” Physical Review Letters 92 (2): 025702. https://doi.org/10.1103/PhysRevLett.92.025702.

Helbing, Dirk, and Péter Molnár. 1995. “Social Force Model for Pedestrian Dynamics.” Physical Review E 51 (5): 4282–86. https://doi.org/10.1103/PhysRevE.51.4282.

Vicsek, Tamás, András Czirók, Eshel Ben-Jacob, Inon Cohen, and Ofer Shochet. 1995. “Novel Type of Phase Transition in a System of Self-Driven Particles.” Physical Review Letters 75 (6): 1226.

--- title: "The Vicsek Model" subtitle: "Module 2: Synchronization and Collective Behaviour" format: html --- ::: {.callout-note} ## Reading - **Compulsory:** @vicsek1995novel. - **Recommended:** @couzin2002collective; @helbing1995social. ::: The Kuramoto model synchronizes phases in time. The Vicsek model aligns headings in space. It describes $N$ self-propelled particles that move at constant speed and copy the average heading of their neighbours, with some noise [@vicsek1995novel]. There is no leader and no global signal, yet the whole group can move in one direction. ## Rules Each particle $i$ has a position $\mathbf{x}_i(t)$ in a square box of side $L$ with periodic boundaries, and a heading $\theta_i(t)$. At every time step $\Delta t$ all particles update at the same time: $$ \theta_i(t + \Delta t) = \langle \theta(t) \rangle_{r,i} + \Delta\theta_i, $$ {#eq-vicsek-theta-rule} $$ \mathbf{x}_i(t + \Delta t) = \mathbf{x}_i(t) + v_0 \Delta t \begin{bmatrix} \cos\theta_i(t + \Delta t) \\ \sin\theta_i(t + \Delta t) \end{bmatrix}. $$ {#eq-vicsek-x-rule} - $\langle \theta \rangle_{r,i}$ is the mean direction of all particles within distance $r$ of particle $i$, including $i$ itself. It is computed as $\arctan\big(\langle\sin\theta\rangle / \langle\cos\theta\rangle\big)$, using the quadrant of the vector $(\langle\cos\theta\rangle, \langle\sin\theta\rangle)$. - $\Delta\theta_i$ is drawn uniformly from $[-\eta/2, \eta/2]$. The noise amplitude $\eta$ runs from $0$ (no noise) to $2\pi$ (a random heading every step). - The speed $v_0$ is the same for all particles. @vicsek1995novel used $r = 1$, $\Delta t = 1$ and $v_0 = 0.03$. The two control parameters are the noise $\eta$ and the density $\rho = N/L^2$. ## Order parameter Collective motion is measured by the normalized average velocity $$ v_a = \frac{1}{N v_0}\left|\sum_{i=1}^N \mathbf{v}_i\right| = \left|\frac{1}{N}\sum_{i=1}^N e^{i\theta_i}\right|, $$ {#eq-vicsek-va} which is the Kuramoto order parameter applied to headings. $v_a \approx 1$ when all particles move the same way and $v_a \approx 0$ when headings are random. ## Kinetic phase transition @vicsek1995novel found that $v_a$ falls from near $1$ to near $0$ as the noise increases, and rises as the density increases. Near the transition they reported the scaling laws $$ v_a \sim \big(\eta_c(\rho) - \eta\big)^{\beta}, \qquad v_a \sim \big(\rho - \rho_c(\eta)\big)^{\delta}, $$ with $\beta = 0.45 \pm 0.07$ and $\delta = 0.35 \pm 0.06$. They interpreted it as a continuous transition. Later simulations of much larger systems by @gregoire2004onset and @chate2008collective found that the onset of collective motion is discontinuous. Near the threshold, the ordered phase organizes into dense travelling bands. Small simulations like the ones in this course cannot distinguish the two scenarios. ![Order parameter against noise at density $\rho = 4$ for $N = 40$ and $N = 100$, one realization each, generated with `amlab.collective_motion.vicsek`. Compare with Figure 2(a) of @vicsek1995novel.](../../img/s04_order_parameter_vs_noise.png){#fig-vicsek-noise width=80%} ::: {.callout-note} ## Exercise In @fig-vicsek-noise, $v_a$ stays around $0.1$ to $0.2$ at the largest noise, instead of going to zero. Why? Use what you know from the Kuramoto model. ::: ::: {.callout-tip collapse="true"} ## Solution At high noise the headings are close to independent and uniform. Then $v_a$ is the modulus of the mean of $N$ random unit vectors, so $\mathbb{E}[v_a^2] = 1/N$ ([The Order Parameter](order-parameter.qmd)). For $N = 40$ and $N = 100$ this gives typical values of $0.16$ and $0.1$, which is what the figure shows. ::: ::: {.callout-note} ## Exercise Compare the Vicsek model with the mean-field Kuramoto model. Name one analogue of the coupling strength $K$ and one analogue of the frequency spread $\sigma$. ::: ::: {.callout-tip collapse="true"} ## Solution Alignment plays the role of coupling. Its effective strength grows with the number of neighbours inside radius $r$, so the density $\rho$ (or $r$ itself) acts like $K$. The noise $\eta$ plays the role of the disorder that opposes alignment, like $\sigma$. One important difference: in the Vicsek model the neighbours change as particles move, so the interaction network is itself dynamic. :::