We demonstrate that the clustering statistics and the corresponding phase transition to non-equilibrium clustering found in many experiments and simulation studies with self-propelled particles (SPPs) with alignment can be obtained from a simple kinetic model. The key elements of this approach are the scaling of the cluster cross-section with the cluster mass -- characterized by an exponent $α$ -- and the scaling of the cluster perimeter with the cluster mass -- described by an exponent $β$. The analysis of the kinetic approach reveals that the SPPs exhibit two phases: i) an individual phase, where the cluster size distribution (CSD) is dominated by an exponential tail that defines a characteristic cluster size, and ii) a collective phase characterized by the presence of non-monotonic CSD with a local maximum at large cluster sizes. At the transition between these two phases the CSD is well described by a power-law with a critical exponent $γ$, which is a function of $α$ and $β$ only. The critical exponent is found to be in the range $0.8 < γ< 1.5$ in line with observations in experiments and simulations.
We demonstrate that a system of self-propelled particles (SPP) exhibits spontaneous symmetry breaking and self-organization in one dimension, in contrast with previous analytical predictions. To explain this surprising result we derive a new continuum theory that can account for the development of the symmetry broken state and belongs to the same universality class as the discrete SPP model.
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 t…
Motivated by aggregation phenomena in gliding bacteria, we study collective motion in a twodimensional model of active, self-propelled rods interacting through volume exclusion. In simulations with individual particles, we find that particle clustering is facilitated by a sufficiently large packing fraction (eta) or length-to-width ratio (kappa). The transition to clustering in simulations is well captured by a mean-field model for the cluster size distribution, which predicts that the transition values kappa_c of the aspect ratio for a fixed packing fraction is given by kappa_c = C/eta - 1 wh…
A ring-kinetic theory for Vicsek-style models of self-propelled agents is derived 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 pre-collisional correlations and cluster formation which both seem 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 BBGKY-hierarchy. An algorithm is presented that numerically solves …
We introduce a kinetic Monte-Carlo model for self-propelled hard disks to capture with minimal ingredients the interplay between thermal fluctuations, excluded volume and self-propulsion in large assemblies of active particles. We analyze in detail the resulting (density, self-propulsion) nonequilibrium phase diagram over a broad range of parameters. We find that purely repulsive hard disks spontaneously aggregate into fractal clusters as self-propulsion is increased, and rationalize the evolution of the average cluster size by developing a kinetic model of reversible aggregation. As density i…
We analyse the collective dynamics of self-propelled particles in the large density regime where passive particles undergo a kinetic arrest to an amorphous glassy state. We capture the competition between self-propulsion and crowding effects using a two-dimensional model of self-propelled hard disks, which we study using Monte-Carlo simulations. Although the activity drives the system far from equilibrium, self-propelled particles undergo a kinetic arrest, which we characterize in detail and compare with its equilibrium counterpart. In particular, the critical density for dynamic arrest contin…
A model of self-driven particles similar to the Vicsek model [Phys. Rev. Lett. 75 (1995) 1226] but with metric-free interactions is studied by means of a novel Enskog-type kinetic theory. In this model, N particles of constant speed v0 try to align their travel directions with the average direction of a fixed number of closest neighbors. At strong alignment a global flocking state forms. The alignment is defined by a stochastic rule, not by a Hamiltonian. The corresponding interactions are of genuine multi-body nature. The theory is based on a Master equation in 3N-dimensional phase space, whi…
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 systems 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 decreasi…
Recent studies of the phase diagram for spherical, purely repulsive, active particles established the existence of a transition from a liquid-like to a solid-like phase analogous to the one observed in colloidal systems at thermal equilibrium, in particular in two dimensions an intermediate hexatic phase is observed. Here, we present evidence that the active dense phases (solid, hexatic and liquid) exhibit interesting dynamical anomalies. First, we unveil the growth - with density and activity - of ordered domains where the particles' velocities align in parallel or vortex-like patches. Second…
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 coalesce…
Starting from a microscopic model of self-propelled hard spheres we use tools of non-equilibrium statistical mechanics and the kinetic theory of hard spheres to derive a Smoluchowski equation for interacting Active Brownian particles. We illustrate the utility of the statistical mechanics framework developed with two applications. First, we derive the steady state pressure of the hard sphere active fluid in terms of the microscopic parameters and second, we identify the critical density for the onset of motility-induced phase separation in this system. We show that both these quantities agree …
Inspired by the Turing mechanism for pattern formation, we propose a simple self-propelled particle model with short-ranged alignment and anti-alignment at larger distances. It is able to produce orientationally ordered states, periodic vortex patterns as well as meso-scale turbulence. The latter phase resembles observations in dense bacterial suspensions. 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 such pattern formation as well as spatially homogeneous orientationa…
Collective behavior of self-propelled particles is observed on a microscale for swimmers such as sperm and bacteria as well as for protein filaments in motility assays. The properties of such systems depend both on their dimensionality and the interactions between their particles. We introduce a model for self-propelled rods in two dimensions that interact via a separation-shifted Lennard-Jones potential. Due to the finite potential barrier, the rods are able to cross. This model allows us to efficiently simulate systems of self-propelled rods that effectively move in two dimensions but can oc…
A number of novel experimental and theoretical results have recently been obtained on active soft matter, demonstrating the various interesting universal and anomalous features of this kind of driven systems. Here we consider a fundamental but still unexplored aspect of the patterns arising in the system of actively moving units, i.e., their segregation taking place when two kinds of them with different adhesive properties are present. The process of segregation is studied by a model made of self-propelled particles such that the particles have a tendency to adhere only to those which are of t…
Kinetic and hydrodynamic theories are widely employed for describing the collective behaviour of active matter systems. At the fluctuating level, these have been obtained from explicit coarse-graining procedures in the limit where each particle interacts weakly with many others, so that the total forces and torques exerted on each of them is of order unity at all times. Such limit is however not relevant for dilute systems that mostly interact via alignment; there, collisions are rare and make the self-propulsion direction to change abruptly. We derive a fluctuating kinetic theory, and the cor…
Recently, an Enskog-type kinetic theory for Vicsek-type models for self-propelled particles has been proposed [T. Ihle, Phys. Rev. E 83, 030901 (2011)]. This theory is based on an exact equation for a Markov chain in phase space and is not limited to small density. Previously, the hydrodynamic equations were derived from this theory and its transport coefficients were given in terms of infinite series. Here, I show that the transport coefficients take a simple form in the large density limit. This allows me to analytically evaluate the well-known density instability of the polarly ordered phas…
The motion of several self-propelled boats in a narrow channel displays spontaneous pattern formation and kinetic phase transitions. In contrast with previous studies on self-propelled particles, this model does not require stochastic fluctuations and it is experimentally accessible. By varying the viscosity in the system, it is possible to form either a stationary state, correlated or uncorrelated oscillations, or unidirectional flow. Here, we describe and analyze these self organized patterns and their transitions.
We study self-propelled particles with velocity reversal interacting by uniaxial (nematic) alignment within a coarse-grained hydrodynamic theory. Combining analytical and numerical continuation techniques, we show that the physics of this active system is essentially controlled by the reversal frequency. In particular, we find that elongated, high-density, ordered patterns, called bands, emerge via subcritical bifurcations from spatially homogeneous states. Our analysis reveals further that the interaction of bands is weakly attractive and, consequently, bands fuse upon collision in analogy wi…
An Enskog-like kinetic equation for self-propelled particles is solved numerically. I study a density instability near the transition to collective motion and find that while hydrodynamics breaks down, the kinetic approach leads to soliton-like supersonic waves with steep leading kinks and Knudsen numbers of order one. These waves show hysteresis, modify the transition threshold and lead to an abrupt jump of the global order parameter if the noise level is changed. Thus they provide a mechanism to change the second-order character of the phase transition to first order.
Kinetic theories constitute one of the most promising tools to decipher the characteristic spatio-temporal 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 …
We consider the solid or hexatic non-equilibrium phases of an interacting two-dimensional system of Active Brownian Particles at high density and investigate numerically and theoretically the properties of the velocity distribution function and the associated kinetic temperature. We obtain approximate analytical predictions for the shape of the velocity distribution and find a transition from a Mexican-hat-like to a Gaussian-like distribution as the persistence time of the active force changes from the small to the large persistence regime. Through a detailed numerical and theoretical analysis…
The emergence of clustering and coarsening in crowded ensembles of self-propelled agents is studied using a lattice model in one-dimension. The persistent exclusion process, where particles move at directions that change randomly at a low tumble rate $α$, is extended allowing sites to be occupied by more than one particle, with a maximum $n_\text{max}$ per site. Three phases are distinguished. For $n_\text{max}=1$ a gas of clusters form, with sizes distributed exponentially and no coarsening takes place. For $n_\text{max}\geq 3$ and small values of $α$, coarsening takes place and few large clu…
In this work we derive and analyze coarse-grained descriptions of self-propelled particles with selective attraction-repulsion interaction, where individuals may respond differently to their neighbours depending on their relative state of motion (approach versus movement away). Based on the formulation of a nonlinear Fokker-Planck equation, we derive a kinetic description of the system dynamics in terms of equations for the Fourier modes of a one-particle density function. This approach allows effective numerical investigation of the stability of possible solutions of the system. The detailed …
Active matter research focuses on the emergent behavior among interacting self-propelled particles. Unification of seemingly disconnected paradigms -- active phase-separation of repulsive discs and collective motion of self-propelled rods -- is a major challenge in contemporary active matter. Inspired by the quanto-mechanical wave-particle duality, we develop an approach based on the representation of active particles by smoothed continuum fields. On the basis of the collision kinetics, we demonstrate analytically and numerically how nonequilibrium stresses acting among self-driven, anisotropi…
Suspensions of purely repulsive but self-propelled Brownian particles might undergo phase separation, a phenomenon that strongly resembles the phase separation of passive particles with attractions. Here we employ computer simulations to study the nucleation kinetics and the microscopic pathway active Brownian disks take in two dimensions when quenched from the homogeneous suspension to propulsion speeds beyond the binodal. We find the same qualitative behavior for the nucleation rate as a function of density as for a passive suspension undergoing liquid-vapor separation, suggesting that the s…
Motility-induced phase separation (MIPS) arises generically in fluids of self-propelled particles when interactions lead to a kinetic slowdown at high densities. Starting from a continuum description of scalar active matter, akin to a generalized Cahn-Hilliard equation, we give a general prescription for the mean densities of coexisting phases in flux-free steady states that amounts, at a hydrodynamics scale, to extremizing an effective free energy. We illustrate our approach on two well-known models: self-propelled particles interacting either through a density-dependent propulsion speed or v…
Phase behaviors and dynamics of active particle systems in double-well potential
College of Physics, Changchun Normal University（长春师范大学物理学院）；Complex Systems Division, Beijing Computational Science Research Center（北京计算科学研究中心复杂系统部门）；School of Mathematics and Physics, Anqing Normal University（安庆师范大学数理学院）
Diffusion and clustering of lattice vacancies in silicon as a function of temperature, concentration, and interaction range are investigated by Kinetic Lattice Monte Carlo simulations. It is found that higher temperatures lead to larger clusters with shorter lifetimes on average, which grow by attracting free vacancies, while clusters at lower temperatures grow by aggregation of smaller clusters. Long interaction ranges produce enhanced diffusivity and fewer clusters. Greater vacancy concentrations lead to more clusters, with fewer free vacancies, but the size of the clusters is largely indepe…
Self-propelled particles include both self-phoretic synthetic colloids and various micro-organisms. By continually consuming energy, they bypass the laws of equilibrium thermodynamics. These laws enforce the Boltzmann distribution in thermal equilibrium: the steady state is then independent of kinetic parameters. In contrast, self-propelled particles tend to accumulate where they move more slowly. They may also slow down at high density, for either biochemical or steric reasons. This creates positive feedback which can lead to motility-induced phase separation (MIPS) between dense and dilute f…
We study non-equilibrium phases for interacting two-dimensional self-propelled particles with isotropic pair-wise interactions using a persistent kinetic Monte Carlo (MC) approach. We establish the quantitative phase diagram, including the motility-induced phase separation (MIPS) that is a commonly observed collective phenomena in active matter. In addition, we demonstrate for several different potential forms the presence of two-step melting, with an intermediate hexatic phase, in regions far from equilibrium. Increased activity can melt a two-dimensional solid and the melting lines remain di…
In this paper, we explore experimentally the phase behavior of a dense active suspension of self- propelled colloids. In addition to a solid-like and a gas-like phase observed for high and low densities, a novel cluster phase is reported at intermediate densities. This takes the form of a stationary assembly of dense aggregates, with an average size which grows with activity as a linear function of the self-propelling velocity. While different possible scenarii can be considered to account for these observations - such as a generic velocity weakening instability recently put forward -, we show…
Some models of clustering processes are formulated and analytically solved employing generating functions methods. Those models include events which result from combined action of the coagulation and fragmentation processes. Fragmentation processes of two kinds, so-called similar and arbitrary, ones, are brought forward, and the explicit forms of their solutions are produced. This implies some possibility of existence of different aggregation mechanisms for clusters creation differing in their inner structure. All the models are based on "the three-level bunch" scheme of interaction between th…
Kinetic Monte Carlo approach is developed to study aspects of sintering of dispersed nanoparticles of bimodal size distributions. We explore mechanisms of neck development when sintering is initiated at elevated temperatures for nanosize crystalline surfaces of particles of different sizes. Specifically, the role of smaller particles fitting between larger particles, on the sintering of the latter is considered. Formation of stable necks bridging particles at the nanoscale was found to be governed by layering or clustering mechanisms at the facing surfaces, with clustering leading to a much fa…
Dynamical properties of a Vicsek-like gas of self-propelled particles are investigated by means of kinetic theory and agent based simulations. While memory effects have been observed in disordered systems, we show that they also occur in active matter systems. In particular, we find that the system exhibits a giant Kovacs-like memory effect that is much larger than predicted by a generic linear theory. Based on a separation of time scales we develop a nonlinear theory to explain this effect. We apply this theory to driven granular gases and propose further applications to spin glasses.
The behavior of a bidisperse inelastic gas vertically shaken in a compartmentalized container is investigated using two different approaches: the first is a mean-field dynamical model, which treats the number of particles in the two compartments and the associated kinetic temperatures in a self-consistent fashion; the second is an event-driven numerical simulation. Both approaches reveal a non-stationary regime, which has no counterpart in the case of monodisperse granular gases. Specifically, when the mass difference between the two species exceeds a certain threshold the populations display …
The hypothesis of molecular chaos plays the central role in kinetic theory, which provides a closure leading to the Boltzmann equation for quantitative description of classic fluids. Yet how to properly extend it to active systems is still an open question in nonequilibrium physics. Combining experiment, simulation, and theory, we investigate the emergent collective behaviors of self-propelled particles that exhibit collision avoidance, a moving strategy commonly adopted in natural and engineering active systems. This dense active system shows unusual phase dynamics strongly regulated by many-…
We numerically study the escape kinetics of a self-propelled Janus particle, carrying a cargo, from a meta-stable state. We assume that the cargo is attached to the Janus particle by a flexible harmonic spring. We take into account the effect of velocity field, created in the fluid due to movements of dimer's components, by considering space-dependent diffusion tensor (Oseen tensor). Our simulation results show that the synchronization between barrier crossing events and rotational relaxation process can enhance escape rate to a large extent. Also, the load carrying capability of a Janus parti…
Via event-driven molecular dynamics simulations we study kinetics of clustering in assemblies of inelastic particles in various space dimensions. We consider two models, viz., the ballistic aggregation model (BAM) and the freely cooling granular gas model (GGM), for each of which we quantify the time dependence of kinetic energy and average mass of clusters (that form due to inelastic collisions). These quantities, for both the models, exhibit power-law behavior, at least in the long time limit. For the BAM, corresponding exponents exhibit strong dimension dependence and follow a hyperscaling …
Equilibrium and non-equilibrium properties of active matter systems
Indian Association for the Cultivation of Science（印度科学文化促进协会）；School of Mathematical & Computational Sciences, Indian Association for the Cultivation of Science（印度科学文化促进协会数学与计算科学学院）
In the last 50 years, equilibrium density functional theory (DFT) has been proven to be a powerful, versatile and predictive approach for the statics and structure of classical particles. This theory can be extended to the nonequilibrium dynamics of completely overdamped Brownian colloidal particles towards so-called dynamical density functional theory (DDFT). The success of DDFT makes it a promising candidate for a first-principle description of active matter. In this lecture, we shall first recapitulate classical DDFT for passive colloidal particles typically described by Smoluchowski equati…
We study the phase behavior of polar Active Brownian Particles moving in two-spatial dimensions and interacting through volume exclusion and velocity alignment. We combine particle-based simulations of the microscopic model with a simple mean-field kinetic model to understand the impact of velocity alignment on the motility-induced phase separation of self-propelled disks. We show that, as the alignment strength is increased, approaching the onset of collective motion from below, orientational correlations grow, rendering the diffusive reorientation dynamics slower. As a consequence, the tende…
We have defined a new type of clustering scheme preserving the connectivity of the nodes in network ignored by the conventional Migdal-Kadanoff bond moving process. Our new clustering scheme performs much better for correlation length and dynamical critical exponents in high dimensions, where the conventional Migdal-Kadanoff bond moving scheme breaks down. In two and three dimensions we find the dynamical critical exponents for the kinetic Ising Model to be z=2.13 and z=2.09, respectively at pure Ising fixed point. These values are in very good agreement with recent Monte Carlo results. We inv…
We study a simple lattice model of shear-induced clustering in two dimensions in which clusters of particles aggregate under an imposed shear flow and fragment stochastically. Two non-equilibrium steady states are identified: an unjammed state and a jammed state characterised by a system-spanning cluster. A discontinuous jamming transition with strong hysteresis occurs as the shear rate is increased or fragmentation rate decreased. We study the kinetics of jamming and measure power law cluster size distributions. We also consider some general simulation issues including the role of Galilean in…
