Criticality and the onset of ordering in the standard Vicsek model

Finite-size scaling in kinetics of phase separation in certain models of aligning active particles

To study the kinetics of phase separation in active matter systems, we consider models that impose a Vicsek-type self-propulsion rule on otherwise passive particles interacting via the Lennard-Jones potential. Two types of kinetics are of interest: one conserves the total momentum of all the constituents and the other does not. We carry out molecular dynamics simulations to obtain results on structural, growth, and aging properties. Results from our studies, with various finite boxes, show that there exist scalings with respect to the system sizes, in both the latter quantities, as in the standard passive cases. We have exploited this scaling picture to accurately estimate the corresponding exponents, in the thermodynamically large system size limit, for power-law time dependences. It is shown that certain analytical functions describe the behavior of these quantities quite accurately, including the finite-size limits. Our results demonstrate that even though the conservation of velocity has at best weak effects on the dynamics of evolution in the thermodynamic limit, the finite-size behavior is strongly influenced by the presence (or the absence) of it.

Collective motion of chiral particles in complex noise environments

Collective motion of chiral particles in complex noise environments is investigated based on the Vicsek model. In the model, we added chirality, along with complex noise, affecting particles clustering motion. Particles can only avoid noise interference in a specific channel, and this consideration is more realistic due to the complexity of the environment. Via simulations, we find that the channel proportion, p, critically influences chiral particle synchronization. Specifically, we observe a disorder-order transition at critical [Formula: see text], only when [Formula: see text], the system can achieve global synchronization. Combined with our definition of spatial distribution parameter and observation of the model, the reason is that particles begin to escape from the noise region under the influence of complex noise. In addition, the value of [Formula: see text] increases linearly with velocity, while it decreases monotonically with the increase in chirality and interaction radius. Interestingly, an appropriate noise amplitude minimizes [Formula: see text]. Our findings may inspire novel strategies to manipulate self-propelled particles of distinct chirality to achieve desired spatial migration and global synchronization.

Perspectives on a Few Puzzles in Phase Transformations: When Should the Farthest Reach the Earliest?

We briefly review the facts concerning two important aspects of phase transitions, namely, critical and coarsening phenomena. A discussion of the universal features, highlighting the current challenges, is provided. Following this, we elaborate on a topic of much recent interest, viz., the Mpemba effect, a puzzle that found mention even in the works of Aristotle. After a description of the debated case of faster freezing of a hotter sample of liquid water, into ice, than a colder one, when quenched to the same subzero temperature, we discuss more modern interest. There one asks, should a hotter body of a material equilibrate faster than a colder one when quenched to a common lower temperature? Within this broad scenario, we focus on magnetic systems. A surprising observation of the effect during the para- to ferromagnetic transition, in a simple model system, viz., the nearest-neighbor Ising model, without any built-in frustration, is described. Some associated future directions are pointed out. A discussion is provided by considering the effect as a kinetic outcome in the background of critical phenomena. A picture is drawn by putting emphasis on the role of spatial correlations in the initial configurations alongside discussing the importance of frustration and metastability in evolution from one state to another. In connection with dynamical freezing, concerning metastability, we have introduced the complex Ginzburg-Landau equation that has relevance in phase transitions, chemical oscillations, and elsewhere. For this model and a few other cases also, we have described how a lack of order or correlation in certain parameters can lead to quicker evolution.

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.

Swarming Transition in Super-Diffusive Self-Propelled Particles

A super-diffusive Vicsek model is introduced in this paper that incorporates Levy flights with exponent α. The inclusion of this feature leads to an increase in the fluctuations of the order parameter, ultimately resulting in the disorder phase becoming more dominant as α increases. The study finds that for α values close to two, the order-disorder transition is of the first order, while for small enough values of α, it shows degrees of similarities with the second-order phase transitions. The article formulates a mean field theory based on the growth of the swarmed clusters that accounts for the decrease in the transition point as α increases. The simulation results show that the order parameter exponent β, correlation length exponent ν, and susceptibility exponent γ remain constant when α is altered, satisfying a hyperscaling relation. The same happens for the mass fractal dimension, information dimension, and correlation dimension when α is far from two. The study reveals that the fractal dimension of the external perimeter of connected self-similar clusters conforms to the fractal dimension of Fortuin-Kasteleyn clusters of the two-dimensional Q=2 Potts (Ising) model. The critical exponents linked to the distribution function of global observables vary when α changes.

A physics perspective on collective animal behavior

The beautiful dynamic patterns and coordinated motion displayed by groups of social animals are a beautiful example of self-organization in natural farfrom-equilibrium systems. Recent advances in active-matter physics have enticed physicists to begin to consider how their results can be extended from microscale physical or biological systems to groups of real, macroscopic animals. At the same time, advances in measurement technology have led to the increasing availability of high-quality empirical data for the behavior of animal groups both in the laboratory and in the wild. In this review, I survey this available data and the ways that it has been analyzed. I then describe how physicists have approached synthesizing, modeling, and interpreting this information, both at the level of individual animals and at the group scale. In particular, I focus on the kinds of analogies that physicists have made between animal groups and more traditional areas of physics.

Phase transitions in biology: from bird flocks to population dynamics

Phase transitions are an important and extensively studied concept in physics. The insights derived from understanding phase transitions in physics have recently and successfully been applied to a number of different phenomena in biological systems. Here, we provide a brief review of phase transitions and their role in explaining biological processes ranging from collective behaviour in animal flocks to neuronal firing. We also highlight a new and exciting area where phase transition theory is particularly applicable: population collapse and extinction. We discuss how phase transition theory can give insight into a range of extinction events such as population decline due to climate change or microbial responses to stressors such as antibiotic treatment.

How do clusters in phase-separating active matter systems grow? A study for Vicsek activity in systems undergoing vapor-solid transition

Via molecular dynamics simulations, we have studied the kinetics of vapor-"solid" phase transition in an active matter model in which self-propulsion is introduced via the well-known Vicsek rule. The overall density of the particles is chosen in such a way that the evolution morphology consists of disconnected clusters that are defined as regions of high density of particles. Our focus has been on understanding the influence of the above-mentioned self-propulsion on structure and growth of these clusters by comparing the results with those for the passive limit of the model that also exhibits vapor-"solid" transition. While in the passive case growth occurs due to a standard diffusive mechanism, the Vicsek activity leads to very rapid growth, via a process that is practically equivalent to the ballistic aggregation mechanism. The emerging growth law in the latter case has been accurately estimated and explained by invoking information on velocity and structural aspects of the clusters into a relevant theory. Some of these results are also discussed with reference to a model for active Brownian particles.

Dynamical self-assembly of dipolar active Brownian particles in two dimensions

Based on Brownian Dynamics (BD) simulations, we study the dynamical self-assembly of active Brownian particles with dipole-dipole interactions, stemming from a permanent point dipole at the particle center. The propulsion direction of each particle is chosen to be parallel to its dipole moment. We explore a wide range of motilities and dipolar coupling strengths and characterize the corresponding behavior based on several order parameters. At low densities and low motilities, the most important structural phenomenon is the aggregation of the dipolar particles into chains. Upon increasing the particle motility, these chain-like structures break, and the system transforms into a weakly correlated isotropic fluid. At high densities, we observe that the motility-induced phase separation is strongly suppressed by the dipolar coupling. Once the dipolar coupling dominates the thermal energy, the phase separation disappears, and the system rather displays a flocking state, where particles form giant clusters and move collective along one direction. We provide arguments for the emergence of the flocking behavior, which is absent in the passive dipolar system.

Information flow in finite flocks

We explore information flow in finite active matter flocks by simulating the canonical Vicsek model and estimating the flow of information as a function of noise (the variability in the extent to which each animal aligns with its neighbours). We show that the global transfer entropy for finite flocks not only fails to peak near the phase transition, as demonstrated for the canonical 2D Ising model, but remains constant from the transition throughout the entire ordered regime to very low noise values. This provides a foundation for future study regarding information flow in more complex models and real-world flocking data.

Coarsening dynamics in the Vicsek model of active matter

We study the flocking model introduced by Vicsek et al. (Phys. Rev. Lett. 75, 1226 (1995)) in the "coarsening" regime. At standard self-propulsion speeds, we find two distinct growth laws for the coupled density and velocity fields. The characteristic length scale of the density domains grows as [Formula: see text] (with [Formula: see text] , while the velocity length scale grows much faster, viz., [Formula: see text] (with [Formula: see text] . The spatial fluctuations in the density and velocity fields are studied by calculating the two-point correlation function and the structure factor, which show deviations from the well-known Porod's law. This is a natural consequence of scattering from irregular morphologies that dynamically arise in the system. At large values of the scaled wave vector, the scaled structure factors for the density and velocity fields decay with powers -2.6 and -1.52 , respectively.

Pattern, growth, and aging in aggregation kinetics of a Vicsek-like active matter model

Via molecular dynamics simulations, we study kinetics in a Vicsek-like phase-separating active matter model. Quantitative results, for isotropic bicontinuous pattern, are presented on the structure, growth, and aging. These are obtained via the two-point equal-time density-density correlation function, the average domain length, and the two-time density autocorrelation function. Both the correlation functions exhibit basic scaling properties, implying self-similarity in the pattern dynamics, for which the average domain size exhibits a power-law growth in time. The equal-time correlation has a short distance behavior that provides reasonable agreement between the corresponding structure factor tail and the Porod law. The autocorrelation decay is a power-law in the average domain size. Apart from these basic similarities, the overall quantitative behavior of the above-mentioned observables is found to be vastly different from those of the corresponding passive limit of the model which also undergoes phase separation. The functional forms of these have been quantified. An exceptionally rapid growth in the active system occurs due to fast coherent motion of the particles, mean-squared-displacements of which exhibit multiple scaling regimes, including a long time ballistic one.

Flocking with discrete symmetry: The two-dimensional active Ising model

We study in detail the active Ising model, a stochastic lattice gas where collective motion emerges from the spontaneous breaking of a discrete symmetry. On a two-dimensional lattice, active particles undergo a diffusion biased in one of two possible directions (left and right) and align ferromagnetically their direction of motion, hence yielding a minimal flocking model with discrete rotational symmetry. We show that the transition to collective motion amounts in this model to a bona fide liquid-gas phase transition in the canonical ensemble. The phase diagram in the density-velocity parameter plane has a critical point at zero velocity which belongs to the Ising universality class. In the density-temperature "canonical" ensemble, the usual critical point of the equilibrium liquid-gas transition is sent to infinite density because the different symmetries between liquid and gas phases preclude a supercritical region. We build a continuum theory which reproduces qualitatively the behavior of the microscopic model. In particular, we predict analytically the shapes of the phase diagrams in the vicinity of the critical points, the binodal and spinodal densities at coexistence, and the speeds and shapes of the phase-separated profiles.

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.

Collective behaviour without collective order in wild swarms of midges

Collective behaviour is a widespread phenomenon in biology, cutting through a huge span of scales, from cell colonies up to bird flocks and fish schools. The most prominent trait of collective behaviour is the emergence of global order: individuals synchronize their states, giving the stunning impression that the group behaves as one. In many biological systems, though, it is unclear whether global order is present. A paradigmatic case is that of insect swarms, whose erratic movements seem to suggest that group formation is a mere epiphenomenon of the independent interaction of each individual with an external landmark. In these cases, whether or not the group behaves truly collectively is debated. Here, we experimentally study swarms of midges in the field and measure how much the change of direction of one midge affects that of other individuals. We discover that, despite the lack of collective order, swarms display very strong correlations, totally incompatible with models of non-interacting particles. We find that correlation increases sharply with the swarm's density, indicating that the interaction between midges is based on a metric perception mechanism. By means of numerical simulations we demonstrate that such growing correlation is typical of a system close to an ordering transition. Our findings suggest that correlation, rather than order, is the true hallmark of collective behaviour in biological systems.

Our perception of collective behaviour in biological systems is closely associated to the emergence of order on a group scale. For example, birds within a flock align their directions of motion, giving the stunning impression that the group is just one organism. Large swarms of midges, mosquitoes and flies, however, look quite chaotic and do not exhibit any group ordering. It is therefore unclear whether these systems are true instances of collective behaviour. Here we perform the three dimensional tracking of large swarms of midges in the field and find that swarms display strong collective behaviour despite the absence of collective order. In fact, we discover that the capability of swarms to collectively respond to perturbations is surprisingly large, comparable to that of highly ordered groups of vertebrates.