Standard Vicsek Model Research Articles

Motional consensus of self-propelled particles

The motional consensus of self-propelled particles is studied in both noise-free cases and cases with noise by the standard Vicsek model. In the absence of noise, we propose a simple method, using grid-based technique and defining the normalized variance of the ratio of the number of particles locally to globally, to quantitatively study the movement pattern of the system by the spatial distribution of the particles and the degree of aggregation of particles. It is found that the weaker correlation of velocity leads to larger degree of aggregation of the particles. In the cases with noise, we quantify the competition between velocity alignment and noise by considering the difference of the variety of order parameter result from the velocity alignment and noise. The variation of the effect of noise on motional consensus is non-monotonic for the change of the probability distribution of noise from uniform to non-uniform. Our results may be useful and encourage further efforts in exploring the basic principles of collective motion.

Dry Active Matter Exhibits a Self-Organized Cross Sea Phase.

The Vicsek model of self-propelled particles is known in three different phases: a polar ordered homogeneous phase, also called the Toner-Tu phase, a phase of polar ordered regularly arranged high density bands with surrounding low density regions without polar order, and a homogeneous phase without polar order. Here, we show that the standard Vicsek model has a fourth phase for large system sizes: a polar ordered cross sea phase. We demonstrate that the cross sea phase is not just a superposition of two waves, but it is an independent complex pattern with an inherently selected crossing angle.

ACCELERATING THE EMERGENCE OF ORDER IN SWARMING SYSTEMS

Our ability to understand and control the emergence of order in swarming systems is a fundamental challenge in contemporary science. The standard Vicsek model (SVM) — a minimal model for swarming systems of self-propelled particles — describes a large population of agents reaching global alignment without the need of central control. Yet, the emergence of order in this model takes time and is not robust to noise. In many real-world scenarios, we need a decentralized protocol to guide a swarming system (e.g., unmanned vehicles or nanorobots) to reach an ordered state in a prompt and noise-robust manner. Here, we find that introducing a simple adaptive rule based on the heading differences of neighboring particles in the Vicsek model can effectively speed up their global alignment, mitigate the disturbance of noise to alignment, and maintain a robust alignment under predation. This simple adaptive model of swarming systems could offer new insights in understanding the prompt and flexible formation of animals and help us design better protocols to achieve fast and robust alignment for multi-agent systems.

Effects of Heterogeneous Social Interactions on Flocking Dynamics.

Social relationships characterize the interactions that occur within social species and may have an important impact on collective animal motion. Here, we consider a variation of the standard Vicsek model for collective motion in which interactions are mediated by an empirically motivated scale-free topology that represents a heterogeneous pattern of social contacts. We observe that the degree of order of the model is strongly affected by network heterogeneity: more heterogeneous networks show a more resilient ordered state, while less heterogeneity leads to a more fragile ordered state that can be destroyed by sufficient external noise. Our results challenge the previously accepted equivalence between the static Vicsek model and the equilibrium XY model on the network of connections, and point towards a possible equivalence with models exhibiting a different symmetry.

Chapman–Enskog expansion for the Vicsek model of self-propelled particles

Using the standard Vicsek model, I show how the macroscopic transport equations can be systematically derived from microscopic collision rules. The approach starts with the exact evolution equation for the N-particle probability distribution and, after making the mean-field assumption of molecular chaos, leads to a multi-particle Enskog-type equation. This equation is treated by a non-standard Chapman–Enskog expansion to extract the macroscopic behavior. The expansion includes terms up to third order in a formal expansion parameter ϵ, and involves a fast time scale. A self-consistent closure of the moment equations is presented that leads to a continuity equation for the particle density and a Navier–Stokes-like equation for the momentum density. Expressions for all transport coefficients in these macroscopic equations are given explicitly in terms of microscopic parameters of the model. The transport coefficients depend on specific angular integrals which are evaluated asymptotically in the limit of infinitely many collision partners, using an analogy to a random walk. The consistency of the Chapman–Enskog approach is checked by an independent calculation of the shear viscosity using a Green–Kubo relation.

Model flocks in a steady vortical flow.

We modify the standard Vicsek model to clearly distinguish between intrinsic noise due to imperfect alignment between organisms and extrinsic noise due to fluid motion. We then consider the effect of a steady vortical flow, the Taylor-Green vortex, on the dynamics of the flock, for various flow speeds, with a fixed intrinsic particle speed. We pay particular attention to the morphology of the flow, and quantify its filamentarity. Strikingly, above a critical flow speed there is a pronounced increase in the filamentarity of the flock, when compared to the zero-flow case. This is due to the fact that particles appear confined to areas of low vorticity; a familiar phenomena, commonly seen in the clustering of inertial particles in vortical flows. Hence, the cooperative motion of the particles gives them an effective inertia, which is seen to have a profound effect on the morphology of the flock, in the presence of external fluid motion. Finally, we investigate the angle between the flow and the particles direction of movement and find it follows a power-law distribution.

Tricritical points in a Vicsek model of self-propelled particles with bounded confidence.

We study the orientational ordering in systems of self-propelled particles with selective interactions. To introduce the selectivity we augment the standard Vicsek model with a bounded-confidence collision rule: a given particle only aligns to neighbors who have directions quite similar to its own. Neighbors whose directions deviate more than a fixed restriction angle α are ignored. The collective dynamics of this system is studied by agent-based simulations and kinetic mean-field theory. We demonstrate that the reduction of the restriction angle leads to a critical noise amplitude decreasing monotonically with that angle, turning into a power law with exponent 3/2 for small angles. Moreover, for small system sizes we show that upon decreasing the restriction angle, the kind of the transition to polar collective motion changes from continuous to discontinuous. Thus, an apparent tricritical point with different scaling laws is identified and calculated analytically. We investigate the shifting and vanishing of this point due to the formation of density bands as the system size is increased. Agent-based simulations in small systems with large particle velocities show excellent agreement with the kinetic theory predictions. We also find that at very small interaction angles, the polar ordered phase becomes unstable with respect to the apolar phase. We derive analytical expressions for the dependence of the threshold noise on the restriction angle. We show that the mean-field kinetic theory also permits stationary nematic states below a restriction angle of 0.681π. We calculate the critical noise, at which the disordered state bifurcates to a nematic state, and find that it is always smaller than the threshold noise for the transition from disorder to polar order. The disordered-nematic transition features two tricritical points: At low and high restriction angle, the transition is discontinuous but continuous at intermediate α. We generalize our results to systems that show fragmentation into more than two groups and obtain scaling laws for the transition lines and the corresponding tricritical points. A numerical method to evaluate the nonlinear Fredholm integral equation for the stationary distribution function is also presented. This method is shown to give excellent agreement with agent-based simulations, even in strongly ordered systems at noise values close to zero.

Discussion on Ohta et al., “Traveling bands in self-propelled soft particles”

A discussion on the contribution of Ohta and Yamanaka in this special issue, supplemented by new agent-based simulations of band collisions within the standard Vicsek-model.

Complex Network Structure of Flocks in the Standard Vicsek Model

In flocking models, the collective motion of self-driven individuals leads to the formation of complex spatiotemporal patterns. The Standard Vicsek Model (SVM) considers individuals that tend to adopt the direction of movement of their neighbors under the influence of noise. By performing an extensive complex network characterization of the structure of SVM flocks, we show that flocks are highly clustered, assortative, and non-hierarchical networks with short-tailed degree distributions. Moreover, we also find that the SVM dynamics leads to the formation of complex structures with an effective dimension higher than that of the space where the actual displacements take place. Furthermore, we show that these structures are capable of sustaining mean-field-like orientationally ordered states when the displacements are suppressed, thus suggesting a linkage between the onset of order and the enhanced dimensionality of SVM flocks.

Gregarious versus individualistic behavior in Vicsek swarms and the onset of first-order phase transitions

The standard Vicsek model (SVM) is a minimal non-equilibrium model of self-propelled particles that appears to capture the essential ingredients of critical flocking phenomena. In the SVM, particles tend to align with each other and form ordered flocks of collective motion; however, perturbations controlled by a noise term lead to a noise-driven continuous order–disorder phase transition. In this work, we extend the SVM by introducing a parameter α that allows particles to be individualistic instead of gregarious, i.e. to choose a direction of motion independently of their neighbors. By focusing on the small-noise regime, we show that a relatively small probability of individualistic motion (around 10%) is sufficient to drive the system from a Vicsek-like ordered phase to a disordered phase. Despite the fact that the α-extended model preserves the O(n) symmetry and the interaction range, as well as the dimensionality of the underlying SVM, this novel phase transition is found to be discontinuous (first order), an intriguing manifestation of the richness of the non-equilibrium flocking/swarming phenomenon.

Criticality and the onset of ordering in the standard Vicsek model

Experimental observations of animal collective behaviour have shown stunning evidence for the emergence of large-scale cooperative phenomena resembling phase transitions in physical systems. Indeed, quantitative studies have found scale-free correlations and critical behaviour consistent with the occurrence of continuous, second-order phase transitions. The standard Vicsek model (SVM), a minimal model of self-propelled particles in which their tendency to align with each other competes with perturbations controlled by a noise term, appears to capture the essential ingredients of critical flocking phenomena. In this paper, we review recent finite-size scaling and dynamical studies of the SVM, which present a full characterization of the continuous phase transition through dynamical and critical exponents. We also present a complex network analysis of SVM flocks and discuss the onset of ordering in connection with XY-like spin models.

Accelerating consensus of self-driven swarm via adaptive speed

In recent years, the well-developed Vicsek model has attracted more and more attention. Unfortunately, in-depth research on its convergence speed is not yet completed. In this paper, we investigate some key factors governing the convergence speed of the Vicsek model with the assistance of extensive numerical simulations. A significant phenomenon surfaces that the convergence time scales obeys a power law with r 2 ln N , with r and N being the horizon radius and the number of particles, respectively. To further accelerate the convergence procedure, we propose a kind of improved Vicsek model with self-driven particles governed by variational speeds, which can remarkably shorten the convergence time of the standard Vicsek model.