Systems Of Self-propelled Particles Research Articles

Effect of the orientational relaxation on the collective motion of patterns formed by self-propelled particles

We investigate the collective behavior of self-propelled particles (SPPs) undergoing competitive processes of pattern formation and rotational relaxation of their self-propulsion velocities. In full accordance with previous work, we observe transitions between different steady states of the SPPs caused by the intricate interplay among the involved effects of pattern formation, orientational order, and coupling between the SPP density and orientation fields. Based on rigorous analytical and numerical calculations, we prove that the rate of the orientational relaxation of the SPP velocity field is the main factor determining the steady states of the SPP system. Further, we determine the boundaries between domains in the parameter plane that delineate qualitatively different resting and moving states. In addition, we analytically calculate the collective velocity of the SPPs and show that it perfectly agrees with our numerical results. We quantitatively demonstrate that does not vanish upon approaching the transition boundary between the moving pattern and homogeneous steady states.

Superdiffusion in dispersions of active colloids driven by an external field and their sedimentation equilibrium.

The diffusive behaviors of active colloids with run-and-tumble movement are explored by dissipative particle dynamics simulations for self-propelled particles (force dipole) and external field-driven particles (point force). The self-diffusion of tracers (solvent) is investigated as well. The influences of the active force, run time, and concentration associated with active particles are studied. For the system of self-propelled particles, the normal diffusion is observed for both active particles and tracers. The diffusivity of the former is significantly greater than that of the latter. For the system of field-driven particles, the superdiffusion is seen for both active particles and tracers. In contrast, it is found that the anomalous diffusion exponent of the former is slightly less than that of the latter. The anomalous diffusion is caused by the many-body, long-range hydrodynamic interactions. In spite of the superdiffusion, the sedimentation equilibrium of field-driven particles can be acquired and the density profile is still exponentially decayed. The sedimentation length of field-driven particles is always greater than that of self-propelled particles.

Non-universality of critical exponents of a self-propelled particles system

The purpose of this paper is the study of the universality of the critical exponents β = 0.45 and δ = 0.35, detected in the Viscek model, under the effect of three zones (repulsion zone, orientation zone and attraction zone). Our aims are to understand more the behavior of non-equilibrium multi-agent system by extending the simple model proposed by Viscek et al. earlier for one zone. In our study, the particles corresponding to agents interact with their neighbors, by choosing at each time step a velocity va depending on their direction and perturbed by an external noise η. We demonstrate, by fixing the radius R1, R3 and by varying the radius R2 a phase transition can be obtained from disordered movement of individuals to highly align collective motion. The varied zone of orientation will be important for this research. Our findings confirm that the exponents β and δ depend on the size of orientation strongly indicate the non-universality of these critical exponents. This investigation is performed over different situations via a numerical simulation.

Theory for the dynamics of dense systems of athermal self-propelled particles.

We present a derivation of a recently proposed theory for the time dependence of density fluctuations in stationary states of strongly interacting, athermal, self-propelled particles. The derivation consists of two steps. First, we start from the equation of motion for the joint distribution of particles' positions and self-propulsions and we integrate out the self-propulsions. In this way we derive an approximate, many-particle evolution equation for the probability distribution of the particles' positions. Second, we use this evolution equation to describe the time dependence of steady-state density correlations. We derive a memory function representation of the density correlation function and then we use a factorization approximation to obtain an approximate expression for the memory function. In the final equation of motion for the density correlation function the nonequilibrium character of the active system manifests itself through the presence of a new steady-state correlation function that quantifies spatial correlations of the velocities of the particles. This correlation function enters into the frequency term, and thus it describes the dependence of the short-time dynamics on the properties of the self-propulsions. More importantly, the correlation function of particles' velocities enters into the vertex of the memory function and through the vertex it modifies the long-time glassy dynamics.

Collective motion of bacteria in two dimensions

Collective motion can be observed in biological systems over a wide range of length scales, from large animals to bacteria. Collective motion is thought to confer an advantage for defense and adaptation. A central question in the study of biological collective motion is how the traits of individuals give rise to the emergent behavior at population level. This question is relevant to the dynamics of general self‐propelled particle systems, biological self‐organization, and active fluids. Bacteria provide a tractable system to address this question, because bacteria are simple and their behavior is relatively easy to control. In this mini review we will focus on a special form of bacterial collective motion, i.e., bacterial swarming in two dimensions. We will introduce some organization principles known in bacterial swarming and discuss potential means of controlling its dynamics. The simplicity and controllability of 2D bacterial behavior during swarming would allow experimental examination of theory predictions on general collective motion.

Mathematics, complexity and multiscale features of large systems of self-propelled particles

This issue is devoted to complex systems in life sciences. Some perspective ideas on possible objectives of future research are extracted from the contents of this issue and brought to the reader’s attention. The final ambitious aim is the development of a mathematical theory for complex living systems.

Self-propelled hard disks: implicit alignment and transition to collective motion

We formulate a model of self-propelled hard disks whose dynamics is governed by mutually coupled vectors for velocity and body orientation. Numerical integration at low densities reveals that the expected transition from isotropic to aligned collective motion is present. However, the transition at the Landau mean-field level is strongly first-order, while it is continuous in the Vicsek model. We show that this difference is rooted in the completely opposite effect that individual scattering events have on alignment. We argue that such differences will generically hold for systems of self-propelled particles with repulsive short-ranged interactions. We obtain these results by matching the numerical results to the framework of Boltzmann theory, based on the statistics of binary scattering properties, always assuming that the system is small enough to stay homogeneous. We further show that the presence of noise in the dynamics can change the nature of the transition from discontinuous to continuous.

Critical exponents of a self-propelled particles system

The Vicsek model of self-propelled particles is an important tool in the study of the collective motion of live organisms. The model consists of particles that move with a constant velocity and adopt, in a region called the zone of repulsion, the average motion direction of their neighbors disturbed by an external noise. A second-order phase transition from a disordered state, with motion in random directions, to an ordered motion state was observed. In this work, we have estimated, using finite-size scaling arguments, the critical exponents β, γ and ν of the original Vicsek model as a function of parameters important to the model, such as the orientation radius size, density, and velocity modulus. Our results show that the critical exponents depend greatly on these parameters.

Effects of Two Distinct Types of Noise on the Directional Alignment of a Self-Propelled Particles System

An important tool for describing the collective motion of autonomous particles is based on the concept of self-propelled particles, where each particle interacts locally with its neighbors, subject to the interference of some noise. There are two ways to introduce noise into the system: either during (extrinsic noise) or after (intrinsic noise) the interactions have occurred. In this work, we present a study of the effects of these two types of noise on the directional order of self-propelled particles. Our numerical investigations show that the directional alignment degree is strongly influenced by the way in which one introduces the noise.

Effect of vision angle on the phase transition in flocking behavior of animal groups.

The nature of the phase transition in a system of self-propelling particles has been extensively studied during the past few decades. A theoretical model was proposed by [T. Vicsek et al. Phys. Rev. Lett. 75, 1226 (1995)PRLTAO0031-900710.1103/PhysRevLett.75.1226] with a simple rule for updating the direction of motion of each particle. Based on the model of Vicsek etal., in this paper, we consider a group of animals as particles moving freely in a two-dimensional space. Due to the fact that the viewable area of animals depends on the species, we consider the motion of each individual within an angle φ=ϕ/2 (ϕ is called the angle of view) of a circle centered at its position of radius R. We obtained a phase diagram in the space (φ,η_{c}) with η_{c} being the critical noise. We show that the phase transition exists only in the case of a wide view's angle φ≥0.5π. The flocking of animals is a universal behavior of the species of prey but not the one of the predator. Our simulation results are in good agreement with experimental observation [C. Beccoa et al., Physica A 367, 487 (2006)PHYADX0378-437110.1016/j.physa.2005.11.041].

Statistical physics of self-propelled particles

Self-propelled particles (SPP) are a modern topic of interdisciplinary research at the frontier between physics and biology. These entities take up energy from their environment and convert (a part of) it into motion. This ability to move autonomously enables them to generate a new variety of singular properties and of collective dynamical behaviors. Autonomous motion and the behavioral patterns associated to SPPs are inherent to many living objects but also to some engineered particles and constructs. Currently, various aspects of the physics of self-propelling particles are in the focus of scientific research. They encompass, among others, the mechanisms that bring about autonomous motion, the description of the types of motion, as well as the role played by fluctuations in the emergence of novel behaviors. Currently, the research on SPP points to three major directions, namely (i) the motion and transport of individual self-propelled objects, (ii) the motion of active particles in external and/or self-generated fields, and (iii) the collective behavior of a group of particles that act upon each other via binary interactions. Recently, an increasing number of scientific studies on SPP is observed with a main focus on the determination of the statistical properties of the motion. This is the case in studies of contemplative motion, i.e., motion where the particle is free to move without any confinements or gradients acting upon them. One of the main objectives of such studies is to assess and understand the role of fluctuations in the motion of SPPs. The statistics of such free migration may provide useful information on the efficiency of the different types of motion. Many systems, however, move either in an external gradient field, or even produce their own gradient field that, in turn, affects the motion of the SPPs. Such types of interplay may be found in biological systems where a cell or organism relays a chemoattractant, pheromone, or the like, but also in non-living systems, where the swimming of a particle may induce hydrodynamic fields that affect the swimming of the other particles. Such interactions may lead to motions, the statistics of which differ considerably from those found for free migration. Finally, systems of SPPs that are coupled to each other by binary interactions may generate interesting patterns of group dynamics, for instance, as seen in herds of sheep, schools of fish, or flocks of birds. Such collective motion may provide huge advantages for the members of the group, which adds to the scientific interest in such

Active particles in heterogeneous media display new physics

We present a detailed study of the large-scale collective properties of self-propelled particles (SPPs) moving in two-dimensional heterogeneous space. The impact of spatial heterogeneities on the ordered, collectively moving phase is investigated. We show that for strong enough spatial heterogeneity, the well-documented high-density, high-ordered propagating bands that emerge in homogeneous space disappear. Moreover, the ordered phase does not exhibit long-range order, as occurs in homogeneous systems, but rather quasi-long range order: i.e. the SPP system becomes disordered in the thermodynamical limit. For finite size systems, we find that there is an optimal noise value that maximizes order. Interestingly, the system becomes disordered in two limits, for high noise values as well as for vanishing noise. This remarkable finding strongly suggests the existence of two critical points, instead of only one, associated to the collective motion transition. Density fluctuations are consistent with these observations, being higher and anomalously strong at the optimal noise, and decreasing and crossing over to normal for high and low noise values. Collective properties are investigated in static as well as dynamic heterogeneous environments, and by changing the symmetry of the velocity alignment mechanism of the SPPs.

Particle-based simulations of self-motile suspensions

A simple model for simulating flows of active suspensions is investigated. The approach is based on dissipative particle dynamics. While the model is potentially applicable to a wide range of self-propelled particle systems, the specific class of self-motile bacterial suspensions is considered as a modeling scenario. To mimic the rod-like geometry of a bacterium, two dissipative particle dynamics particles are connected by a stiff harmonic spring to form an aggregate dissipative particle dynamics molecule. Bacterial motility is modeled through a constant self-propulsion force applied along the axis of each such aggregate molecule. The model accounts for hydrodynamic interactions between self-propelled agents through the pairwise dissipative interactions conventional to dissipative particle dynamics. Numerical simulations are performed using a customized version of the open-source software package LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) software package. Detailed studies of the influence of agent concentration, pairwise dissipative interactions, and Stokes friction on the statistics of the system are provided. The simulations are used to explore the influence of hydrodynamic interactions in active suspensions. For high agent concentrations in combination with dominating pairwise dissipative forces, strongly correlated motion patterns and a fluid-like spectral distributions of kinetic energy are found. In contrast, systems dominated by Stokes friction exhibit weaker spatial correlations of the velocity field. These results indicate that hydrodynamic interactions may play an important role in the formation of spatially extended structures in active suspensions.

Focusing by blocking: Repeatedly generating central density peaks in self-propelled particle systems by exploiting diffusive processes

Over the past few years the displacement statistics of self-propelled particles has been intensely studied, revealing their long-time diffusive behavior. Here, we demonstrate that a concerted combination of boundary conditions and switching on and off the self-propelling drive can generate and afterwards arbitrarily often restore a non-stationary centered peak in their spatial distribution. This corresponds to a partial reversibility of their statistical behavior, in opposition to the above-mentioned long-time diffusive nature. Interestingly, it is a diffusive process that mediates and makes possible this procedure. It should be straightforward to verify our predictions in a real experimental system.

Effective strategies of collective evacuation from an enclosed space

On the basis of fundamental principles of the Vicsek model and the leader–follower model, we develop an extended evacuation model of self-propelled particles system considering movable exits, and then propose effective strategies of self-organization evacuating from an enclosed space. It is found that placing exits in the corner is an effective strategy for evacuation via simulations. Furthermore, increasing the intensity of exit sign takes only effect in some extent. In addition, multi exits will make the evacuation more slowly. In general, one corner exit is the best choice for collective evacuation. Our results provide new insights into designing a safe passage in some enclosed places, such as the cinema and conference halls.

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.

Vortex arrays and mesoscale turbulence of self-propelled particles.

Inspired by the Turing mechanism for pattern formation, we propose a simple self-propelled particle model with short-range alignment and antialignment at larger distances. It is able to produce orientationally ordered states, periodic vortex patterns, and mesoscale turbulence, which resembles observations in dense suspensions of swimming bacteria. The model allows a systematic derivation and analysis of a kinetic theory as well as hydrodynamic equations for density and momentum fields. A phase diagram with regions of pattern formation as well as orientational order is obtained from a linear stability analysis of these continuum equations. Microscopic Langevin simulations of self-propelled particles are in agreement with these findings.

Numerical Treatment of the Boltzmann Equation for Self-Propelled Particle Systems

Kinetic theories constitute one of the most promising tools to decipher the characteristic spatiotemporal dynamics in systems of actively propelled particles. In this context, the Boltzmann equation plays a pivotal role, since it provides a natural translation between a particle-level description of the system’s dynamics and the corresponding hydrodynamic fields. Yet, the intricate mathematical structure of the Boltzmann equation substantially limits the progress toward a full understanding of this equation by solely analytical means. Here, we propose a general framework to numerically solve the Boltzmann equation for self-propelled particle systems in two spatial dimensions and with arbitrary boundary conditions. We discuss potential applications of this numerical framework to active matter systems and use the algorithm to give a detailed analysis to a model system of self-propelled particles with polar interactions. In accordance with previous studies, we find that spatially homogeneous isotropic and broken-symmetry states populate two distinct regions in parameter space, which are separated by a narrow region of spatially inhomogeneous, density-segregated moving patterns. We find clear evidence that these three regions in parameter space are connected by first-order phase transitions and that the transition between the spatially homogeneous isotropic and polar ordered phases bears striking similarities to liquid-gas phase transitions in equilibrium systems. Within the density-segregated parameter regime, we find a novel stable limit-cycle solution of the Boltzmann equation, which consists of parallel lanes of polar clusters moving in opposite directions, so as to render the overall symmetry of the system’s ordered state nematic, despite purely polar interactions on the level of single particles.5 MoreReceived 2 April 2014DOI:https://doi.org/10.1103/PhysRevX.4.041030This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical Society

Collision dynamics of traveling bands in systems of deformable self-propelled particles.

We study the soliton-like character of traveling bands in systems of interacting deformable self-propelled particles in two dimensions. The collision dynamics of the model in which migration velocity increases with increasing local density is investigated numerically by changing the relaxation rate of deformations. The bands are unstable upon head-on collisions for larger relaxation rates. This clearly indicates that deformability plays a crucial role of the soliton-like behavior.

The hydrodynamic description for the system of self-propelled particles: Ideal Viscek fluid

We use the method of the microscopic phase density to get the kinetic equation for the system of self-propelled particles with Vicsek-like alignment rule. The hydrodynamic equations are derived for the ordered phase taking into account the mean-field force only. The equation for the hydrodynamic velocity plays the role of the Euler equation for the self-propelled Vicsek fluid. The hydrodynamics of such ideal self-propelled fluid demonstrates the dynamical transition from disordered initial state to the completely ordered motion. To take the noise into account we consider how the framework of the local equilibrium approximation affects the hydrodynamic equations and the viscous tensor and show that in such approximation the shear viscosity vanishes.