Phase transitions induced by complex nonlinear noise in a system of self-propelled agents

Influence of anomalous agents on the dynamics of an active system

Swarming behavior in systems of self-propelled particles, whether biological or artificial, has received increased attention in recent years. Here, we show that even a small number of particles with anomalous behavior can change dramatically collective dynamics of the swarming system and can impose unusual behavior and transitions between dynamic states. Our results pave the way to practical approaches and concepts of multiagent dynamics in groups of flocking animals: birds, insects, and fish, i.e., active and living soft matter.

Aggregation of self-propelled particles with sensitivity to local order

We study a system of self-propelled particles (SPPs) in which individual particles are allowed to switch between a fast aligning and a slow nonaligning state depending upon the degree of the alignment in the neighborhood. The switching is modeled using a threshold for the local order parameter. This additional attribute gives rise to a mixed phase, in contrast to the ordered phases found in clean SPP systems. As the threshold is increased from zero, we find the sudden appearance of clusters of nonaligners. Clusters of nonaligners coexist with moving clusters of aligners with continual coalescence and fragmentation. The behavior of the system with respect to the clustering of nonaligners appears to be very different for values of low and high global densities. In the low density regime, for an optimal value of the threshold, the largest cluster of nonaligners grows in size up to a maximum that varies logarithmically with the total number of particles. However, on further increasing the threshold the size decreases. In contrast, for the high density regime, an initial abrupt rise is followed by the appearance of a giant cluster of nonaligners. The latter growth can be characterized as a continuous percolation transition. In addition, we find that the speed differences between aligners and nonaligners is necessary for the segregation of aligners and nonaligners.

Effect of particle fraction on phase transitions in an active-passive particles system

We study phase transition in a binary system of monodisperse active and passive particles. The particles are initially randomly positioned inside a fixed boundary square enclosure. The active particles can move with their self-propulsion force. Whereas, the passive particles do not have any self-propulsion force, and they move by the spatial interactions with other particles. An alignment force in our discrete element model causes the emergence of collective milling motion. Without this alignment interaction, the particle system remains in a disordered phase. Whereas, the ordered milling phase is attained after achieving a minimum coordination among neighboring particles. The phase transition from disordered to ordered depends upon the relative effect of self-propulsion and the alignment, initial states of the particles, noise level, and the fraction of the active particles present in the system. The phase transition we observed is of first-order nature.

Two types of exclusion interactions for self-propelled objects and collective motion induced by their combination

Exclusive interactions between self-driven objects may play crucial roles in their collective behavior, e.g., in collective migration of living cells. Here, such collective behavior is studied based on a simple but sufficient model taking account the exclusion effects, which incorporate the following two distinct kinds of exclusion interactions in two dimensions: The first is the mechanical exclusion wherein two objects mechanically repel each other when they overlap. The second is the scattering exclusion, wherein the directions along which each object tries to move are modulated to avoid overlapping. We propose a theoretical model based on two principles: (1) Each object maintains its own polarity with a fixed strength and attempts to move into the polarity direction and (2) objects interact with each other through the abovementioned exclusions. Based on this model, we look at the difference of consequences and combinatory effects of these two kinds of exclusions. Furthermore, we calculate the polar order of polarity directions without an external directional bias. Our results suggest that the combination of these two kinds of exclusions leads to effectively inelastic scattering of two objects, which eventually gives rise to global polar ordering. We also find that the traveling band can arise by this mechanism of alignment at the intermediate density, as generally seen in collective motion with polar alignment and investigated in various earlier works. Characteristics of transitions among disordered, traveling band, and homogeneously ordered states of the presented model are investigated, and their similarities and differences with those given by the explicit alignment interaction are discussed.

Delay in the dispersal of flocks moving in unbounded space using long-range interactions

Since the pioneering work by Vicsek and his collaborators on the motion of self-propelled particles, most of the subsequent studies have focused on the onset of ordered states through a phase transition driven by particle density and noise. Usually, the particles in these systems are placed within periodic boundary conditions and interact via short-range velocity alignment forces. However, when the periodic boundaries are eliminated, letting the particles move in open space, the system is not able to organize into a coherently moving group since even small amounts of noise cause the flock to break apart. While the phase transition has been thoroughly studied, the conditions to keep the flock cohesive in open space are still poorly understood. Here we extend the Vicsek model of collective motion by introducing long-range alignment interactions between the particles. We show that just a small number of these interactions is enough for the system to build up long lasting ordered states of collective motion in open space and in the presence of noise. This finding was verified for other models in addition to the Vicsek one, suggesting its generality and revealing the importance that long-range interactions can have for the cohesion of the flock.

Collective Motion of Swarming Agents Evolving on a Sphere Manifold: A Fundamental Framework and Characterization

Collective motion of self-propelled agents has attracted much attention in vast disciplines. However, almost all investigations focus on such agents evolving in the Euclidean space, with rare concern of swarms on non-Euclidean manifolds. Here we present a novel and fundamental framework for agents evolving on a sphere manifold, with which a variety of concrete cooperative-rules of agents can be designed separately and integrated easily into the framework, which may perhaps pave a way for considering general spherical collective motion (SCM) of a swarm. As an example, one concrete cooperative-rule, i.e., the spherical direction-alignment (SDA), is provided, which corresponds to the usual and popular direction-alignment rule in the Euclidean space. The SCM of the agents with the SDA has many unique statistical properties and phase-transitions that are unexpected in the counterpart models evolving in the Euclidean space, which unveils that the topology of the sphere has an important impact on swarming emergence.

Emergence of Collective Motion in a Model of Interacting Brownian Particles

By studying a system of Brownian particles that interact among themselves only through a local velocity-alignment force that does not affect their speed, we show that self-propulsion is not a necessary feature for the flocking transition to take place as long as underdamped particle dynamics can be guaranteed. Moreover, the system transits from stationary phases close to thermal equilibrium, with no net flux of particles, to far-from-equilibrium ones exhibiting collective motion, phase coexistence, long-range order, and giant number fluctuations, features typically associated with ordered phases of models where self-propelled particles with overdamped dynamics are considered.

Active matter beyond mean-field: ring-kinetic theory for self-propelled particles

Recently, Hanke et al. [Phys. Rev. E 88, 052309 (2013)] showed that mean-field kinetic theory fails to describe collective motion in soft active colloids and that correlations must not be neglected. Correlation effects are also expected to be essential in systems of biofilaments driven by molecular motors and in swarms of midges. To obtain correlations in an active matter system from first principles, we derive a ring-kinetic theory for Vicsek-style models of self-propelled agents from the exact N-particle evolution equation in phase space. The theory goes beyond mean-field and does not rely on Boltzmann's approximation of molecular chaos. It can handle precollisional correlations and cluster formation, which are both important to understand the phase transition to collective motion. We propose a diagrammatic technique to perform a small-density expansion of the collision operator and derive the first two equations of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy. An algorithm is presented that numerically solves the evolution equation for the two-particle correlations on a lattice. Agent-based simulations are performed and informative quantities such as orientational and density correlation functions are compared with those obtained by ring-kinetic theory. Excellent quantitative agreement between simulations and theory is found at not-too-small noises and mean free paths. This shows that there are parameter ranges in Vicsek-like models where the correlated closure of the BBGKY hierarchy gives correct and nontrivial results. We calculate the dependence of the orientational correlations on distance in the disordered phase and find that it seems to be consistent with a power law with an exponent around -1.8, followed by an exponential decay. General limitations of the kinetic theory and its numerical solution are discussed.

Consequences of animal interactions on their dynamics: emergence of home ranges and territoriality

Animal spacing has important implications for population abundance, species demography and the environment. Mechanisms underlying spatial segregation have their roots in the characteristics of the animals, their mutual interaction and their response, collective as well as individual, to environmental variables. This review describes how the combination of these factors shapes the patterns we observe and presents a practical, usable framework for the analysis of movement data in confined spaces. The basis of the framework is the theory of interacting random walks and the mathematical description of out-of-equilibrium systems. Although our focus is on modelling and interpreting animal home ranges and territories in vertebrates, we believe further studies on invertebrates may also help to answer questions and resolve unanswered puzzles that are still inaccessible to experimental investigation in vertebrate species.

Invasion-wave-induced first-order phase transition in systems of active particles

An instability near the transition to collective motion of self-propelled particles is studied numerically by Enskog-like kinetic theory. While hydrodynamics breaks down, the kinetic approach leads to steep solitonlike waves. These supersonic waves show hysteresis and lead to an abrupt jump of the global order parameter if the noise level is changed. Thus they provide a mean-field mechanism to change the second-order character of the phase transition to first order. The shape of the wave is shown to follow a scaling law and to quantitatively agree with agent-based simulations.

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.

Singular perturbation analysis of a reduced model for collective motion: a renormalization group approach

In a system of noisy self-propelled particles with interactions that favor directional alignment, collective motion will appear if the density of particles is beyond a critical density. Starting with a reduced model for collective motion, we determine how the critical density depends on the form of the initial perturbation. Specifically, we employ a renormalization-group improved perturbative method to analyze the model equations and show analytically, up to first order in the perturbation parameter, how the critical density is modified by the strength of the initial angular perturbation in the system.

Collective behavior of inanimate boats

We propose an inanimate system composed of camphor boats in an annular water channel in order to understand the collective motions. The boats move on the water surface spontaneously and interact with one another through the concentration of the camphor molecules on the water. We observed several modes of collective motion, e.g., behaviors analogous to traffic flow or an ant trail. Our system provides a convenient experimental setup for the investigation of a variety of collective motions.

Fluctuation-induced collective motion: a single-particle density analysis

In a system of noisy self-propelled particles with interactions that favor directional alignment, collective motion will appear if the density of particles increases beyond a certain threshold. In this paper, we argue that such a threshold may depend also on the profiles of the perturbation in the particle directions. Specifically, we perform mean-field, linear stability, perturbative, and numerical analyses on an approximated form of the Fokker-Planck equation describing the system. We find that if an angular perturbation to an initially homogeneous system is large in magnitude and highly localized in space, it will be amplified and thus serves as an indication of the onset of collective motion. Our results also demonstrate that high particle speed promotes collective motion.

Nature of the order-disorder transition in the Vicsek model for the collective motion of self-propelled particles

One of the most popular approaches to the study of the collective behavior of self-driven individuals is the well-known Vicsek model (VM) [T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, Phys. Rev. Lett. 75, 1226 (1995)]. In the VM one has that each individual tends to adopt the direction of motion of its neighbors with the perturbation of some noise. For low enough noise the individuals move in an ordered fashion with net transport of mass; however, when the noise is increased, one observes disordered motion in a gaslike scenario. The nature of the order-disorder transition, i.e., first-versus second-order, has originated an ongoing controversy. Here, we analyze the most used variants of the VM unambiguously establishing those that lead either to first- or second-order behavior. By requesting the invariance of the order of the transition upon rotation of the observational frame, we easily identify artifacts due to the interplay between finite-size and boundary conditions, which had erroneously led some authors to observe first-order transitionlike behavior.