From phase to microphase separation in flocking models: the essential role of nonequilibrium fluctuations

Motility-Induced Microphase and Macrophase Separation in a Two-Dimensional Active Brownian Particle System

As a result of nonequilibrium forces, purely repulsive self-propelled particles undergo macrophase separation between a dense and a dilute phase. We present a thorough study of the ordering kinetics of such motility-induced phase separation (MIPS) in active Brownian particles in two dimensions, and we show that it is generically accompanied by microphase separation. The growth of the dense phase follows a law akin to the one of liquid-gas phase separation. However, it is made of a mosaic of hexatic microdomains whose size does not coarsen indefinitely, leaving behind a network of extended topological defects from which microscopic dilute bubbles arise. The characteristic length of these finite-size structures increases with activity, independently of the choice of initial conditions.

Adaptive stochastic continuation with a modified lifting procedure applied to complex systems

Many complex systems occurring in the natural or social sciences or economics are frequently described on a microscopic level, e.g., by lattice- or agent-based models. To analyze the states of such systems and their bifurcation structure on the level of macroscopic observables, one has to rely on equation-free methods like stochastic continuation. Here we investigate how to improve stochastic continuation techniques by adaptively choosing the parameters of the algorithm. This allows one to obtain bifurcation diagrams quite accurately, especially near bifurcation points. We introduce lifting techniques which generate microscopic states with a naturally grown structure, which can be crucial for a reliable evaluation of macroscopic quantities. We show how to calculate fixed points of fluctuating functions by employing suitable linear fits. This procedure offers a simple measure of the statistical error. We demonstrate these improvements by applying the approach in analyses of (i) the Ising model in two dimensions, (ii) an active Ising model, and (iii) a stochastic Swift-Hohenberg model. We conclude by discussing the abilities and remaining problems of the technique.

Flocking with a q-fold discrete symmetry: Band-to-lane transition in the active Potts model

We study the q-state active Potts model (APM) on a two-dimensional lattice in which self-propelled particles have q internal states corresponding to the q directions of motion. A local alignment rule inspired by the ferromagnetic q-state Potts model and self-propulsion via biased diffusion according to the internal particle states elicits collective motion at high densities and low noise. We formulate a coarse-grained hydrodynamic theory with which we compute the phase diagrams of the APM for q=4 and q=6 and analyze the flocking dynamics in the coexistence region, where the high-density (polar liquid) phase forms a fluctuating stripe of coherently moving particles on the background of the low-density (gas) phase. A reorientation transition of the phase-separated profiles from transversal band motion to longitudinal lane formation is found, which is absent in the Vicsek model and the active Ising model. The origin of this reorientation transition is revealed by a stability analysis: for large velocities the transverse diffusivity approaches zero and stabilizes lanes. Computer simulations corroborate the analytical predictions of the flocking and reorientation transitions and validate the phase diagrams of the APM.

Competing active and passive interactions drive amoebalike crystallites and ordered bands in active colloids

Swimmers and self-propelled particles are physical models for the collective behavior and motility of a wide variety of living systems, such as bacteria colonies, bird flocks, and fish schools. Such artificial active materials are amenable to physical models which reveal the microscopic mechanisms underlying the collective behavior. Here we study colloids in a dc electric field. Our quasi-two-dimensional system of electrically driven particles exhibits a rich and exotic phase behavior exhibiting passive crystallites, motile crystallites, an active gas, and banding. Amongst these are two mesophases, reminiscent of systems with competing interactions. At low field strengths activity suppresses demixing, leading to motile crystallites. Meanwhile, at high field strengths, activity drives partial demixing to traveling bands. We parametrize a particulate simulation model which reproduces the experimentally observed phases.

Phases and excitations of active rod-bead mixtures: simulations and experiments

We present a large-scale numerical study, supplemented by experimental observations, on a quasi-two-dimensional active system of polar rods and spherical beads confined between two horizontal plates and energised by vertical vibration. For a low rod concentration Φr, our observations are consistent with a direct phase transition, as bead concentration Φb is increased, from the isotropic phase to a homogeneous flock. For Φr above a threshold value, an ordered band dense in both rods and beads occurs between the disordered phase and the homogeneous flock, in both experiments and simulations. Within the size ranges accessible, we observe only a single band, whose width increases with Φr. Deep in the ordered state, we observe broken-symmetry "sound" modes and giant number fluctuations. The direction-dependent sound speeds and the scaling of fluctuations are consistent with the predictions of field theories of flocking; sound damping rates show departures from such theories, but the range of wavenumbers explored is modest. At very high densities, we see phase separation into rod-rich and bead-rich regions, both of which move coherently.

Traveling bands, clouds, and vortices of chiral active matter

We consider stochastic dynamics of self-propelled particles with nonlocal normalized alignment interactions subject to phase lag. The role of the lag is to indirectly generate chirality into particle motion. To understand large-scale behavior, we derive a continuum description of an active Brownian particle flow with macroscopic scaling in the form of a partial differential equation for a one-particle probability density function. Due to indirect chirality, we find a spatially homogeneous nonstationary analytic solution for this class of equations. Our development of kinetic and hydrodynamic theories towards such a solution reveals the existence of a wide variety of spatially nonhomogeneous patterns reminiscent of traveling bands, clouds, and vortical structures of linear active matter. Our model may thereby serve as the basis for understanding the nature of chiral active media and designing multiagent swarms with designated behavior.

Flocking transition within the framework of Kuramoto paradigm for synchronization: Clustering and the role of the range of interaction

A Kuramoto-type approach to address flocking phenomena is presented. First, we analyze a simple generalization of the Kuramoto model for interacting active particles, which is able to show the flocking transition (the emergence of coordinated movements in a group of interacting self-propelled agents). In the case of all-to-all interaction, the proposed model reduces to the Kuramoto model for phase synchronization of identical motionless noisy oscillators. In general, the nature of this non-equilibrium phase transition depends on the range of interaction between the particles. Namely, for a small range of interaction, the transition is first order, while for a larger range of interaction, it is a second order transition. Moreover, for larger interaction ranges, the system exhibits the same features as in the case of all-to-all interaction, showing a spatially homogeneous flux when flocking phenomenon has emerged, while for lower interaction ranges, the flocking transition is characterized by cluster formation. We compute the phase diagram of the model, where we distinguish three phases as a function of the range of interaction and the effective coupling strength: a disordered phase, a spatially homogeneous flocking phase, and a cluster-flocking phase. Then, we present a general discussion about the applicability of this way of modeling to more realistic and general situations, ending with a brief presentation of a second example (a second model with a conservative interaction) where the flocking transition may be studied within the framework that we are proposing.

Noise-induced effects in collective dynamics and inferring local interactions from data

In animal groups, individual decisions are best characterized by probabilistic rules. Furthermore, animals of many species live in small groups. Probabilistic interactions among small numbers of individuals lead to a so-called intrinsic noise at the group level. Theory predicts that the strength of intrinsic noise is not a constant but often depends on the collective state of the group; hence, it is also called a state-dependent noise or a multiplicative noise. Surprisingly, such noise may produce collective order. However, only a few empirical studies on collective behaviour have paid attention to such effects owing to the lack of methods that enable us to connect data with theory. Here, we demonstrate a method to characterize the role of stochasticity directly from high-resolution time-series data of collective dynamics. We do this by employing two well-studied individual-based toy models of collective behaviour. We argue that the group-level noise may encode important information about the underlying processes at the individual scale. In summary, we describe a method that enables us to establish connections between empirical data of animal (or cellular) collectives and the phenomenon of noise-induced states, a field that is otherwise largely limited to the theoretical literature. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.

Otherwise identical particles with differing, fixed speeds demix under time-reversible dynamics

In recent years situations where elsewise identical particles demix when different degrees of freedom do not thermalize have become a research focus in nonequilibrium statistical mechanics. The majority of these models are formulated in the context of active particles, but the phenomenon also occurs for particles without driving. All the models studied so far share the property that they do not obey microscopic reversibility, and it may be thought that this is a necessary condition for such demixing to occur. We show here that such a demixing transition also occurs in a mixture of otherwise identical particles moving at two fixed but different speeds according to a time-reversible quasi-Newtonian dynamics. The mechanical instability underlying this behavior is generated by the lack of thermalization between the two subsystems, which is shared by all systems showing this behavior.

Stealth Entropy Production in Active Field Theories near Ising Critical Points

We address the steady-state entropy production rate (EPR) of active scalar ϕ^{4} theories, which lack time-reversal symmetry, close to a phase-separation critical point. We consider both nonconserved (model A) and conserved (model B) dynamics at Gaussian level, and also address the former at leading order in ε=4-d. In each case, activity is irrelevant in the RG sense: the active model lies in the same (dynamic Ising) universality class as its time-reversible counterpart. Hence one might expect that activity brings no new critical behavior. Here we show instead that, on approach to criticality in these models, the singular part of the EPR per (diverging) spacetime correlation volume either remains finite or itself diverges. A nontrivial critical scaling for entropy production thus ranks among universal dynamic Ising-class properties.

Polar pattern formation induced by contact following locomotion in a multicellular system

Biophysical mechanisms underlying collective cell migration of eukaryotic cells have been studied extensively in recent years. One mechanism that induces cells to correlate their motions is contact inhibition of locomotion, by which cells migrating away from the contact site. Here, we report that tail-following behavior at the contact site, termed contact following locomotion (CFL), can induce a non-trivial collective behavior in migrating cells. We show the emergence of a traveling band showing polar order in a mutant Dictyostelium cell that lacks chemotactic activity. We find that CFL is the cell-cell interaction underlying this phenomenon, enabling a theoretical description of how this traveling band forms. We further show that the polar order phase consists of subpopulations that exhibit characteristic transversal motions with respect to the direction of band propagation. These findings describe a novel mechanism of collective cell migration involving cell-cell interactions capable of inducing traveling band with polar order.

From scalar to polar active matter: Connecting simulations with mean-field theory

We study numerically the phase behavior of self-propelled elliptical particles interacting through the "hard" repulsive Gay-Berne potential at infinite Péclet number. Changing a single parameter, the aspect ratio, allows us to continuously go from discoid active Brownian particles to elongated polar rods. Discoids show phase separation, which changes to a cluster state of polar domains, which then form polar bands as the aspect ratio is increased. From the simulations, we identify and extract the two effective parameters entering the mean-field description: the force imbalance coefficient and the effective coupling to the local polarization. These two coefficients are sufficient to obtain a complete and consistent picture, unifying the paradigms of scalar and polar active matter.

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.

Multiple Particle Correlation Analysis of Many-Particle Systems: Formalism and Application to Active Matter

We introduce a fast spatial point pattern analysis technique that is suitable for systems of many identical particles giving rise to multiparticle correlations up to arbitrary order. The obtained correlation parameters allow us to quantify the quality of mean field assumptions or theories that incorporate correlations of limited order. We study the Vicsek model of self-propelled particles and create a correlation map marking the required correlation order for each point in phase space incorporating up to ten-particle correlations. We find that multiparticle correlations are important even in a large part of the disordered phase. Furthermore, the two-particle correlation parameter serves as an excellent order parameter to locate both phase transitions of the system, whereas two different order parameters were required before.

Active binary mixtures of fast and slow hard spheres

We computationally studied the phase behavior and dynamics of binary mixtures of active particles, where each species had distinct activities leading to distinct velocities, fast and slow. We obtained phase diagrams demonstrating motility-induced phase separation (MIPS) upon varying the activity and concentration of each species, and extended current kinetic theory of active/passive mixtures to active/active mixtures. We discovered two regimes of behavior quantified through the participation of each species in the dense phase compared to their monodisperse counterparts. In regime I (active/passive and active/weakly-active), we found that the dense phase was segregated by particle type into domains of fast and slow particles. Moreover, fast particles were suppressed from entering the dense phase while slow particles were enhanced entering the dense phase, compared to monodisperse systems of all-fast or all-slow particles. These effects decayed asymptotically as the activity of the slow species increased, approaching the activity of the fast species until they were negligible (regime II). In regime II, the dense phase was homogeneously mixed and each species participated in the dense phase as if it were it a monodisperse system (i.e. not mixed at all). Finally, we showed that a weighted average of constituent particle activities, which we term the net activity, defines a binodal for the MIPS transition in active/active binary mixtures. We examined the critical point of the transition and found a critical exponent (β = 0.45) in agreement with similar studies on monodisperse systems, and distinct from equilibrium systems.

Spontaneous Velocity Alignment in Motility-Induced Phase Separation

We study a system of purely repulsive spherical self-propelled particles in the minimal setup inducing motility-induced phase separation (MIPS). We show that, even if explicit alignment interactions are absent, a growing order in the velocities of the clustered particles accompanies MIPS. Particles arrange into aligned or vortexlike domains whose size increases as the persistence of the self-propulsion grows, an effect that is quantified studying the spatial correlation function of the velocities. We explain the velocity alignment by unveiling a hidden alignment interaction of the Vicsek-like form, induced by the interplay between steric interactions and self-propulsion.

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.

Enhanced Orientational Ordering Induced by an Active yet Isotropic Bath

Can a bath of isotropic but active particles promote ordering of anisotropic but passive particles? In this Letter, we uncover a fluctuation-driven mechanism by which this is possible. Somewhat counterintuitively, we show that the passive particles tend to be more ordered upon increasing the noise strength of the active isotropic bath. We first demonstrate this in a general dynamical model for a nonconserved order parameter (model A) coupled to an active isotropic field and then concentrate on two examples: (i) a collection of polar rods on a substrate in an active isotropic bath and (ii) a passive apolar suspension in a momentum conserved, actively forced but isotropic fluid, which are relevant for current research in active systems. Our theory, which is relevant for understanding ordering transitions in out-of-equilibrium systems can be tested in experiments, for instance, by introducing a low concentration of passive rodlike objects in active isotropic fluids and, since it is applicable to any nonconserved dynamical field, may have applications far beyond active matter.

Swimmer Suspensions on Substrates: Anomalous Stability and Long-Range Order

We present a comprehensive theory of the dynamics and fluctuations of a two-dimensional suspension of polar active particles in an incompressible fluid confined to a substrate. We show that, depending on the sign of a single parameter, a state with polar orientational order is anomalously stable (or anomalously unstable), with a nonzero relaxation (or growth) rate for angular fluctuations, not parallel to the ordering direction, at zero wave number. This screening of the broken-symmetry mode in the stable state does lead to conventional rather than giant number fluctuations as argued by Bricard et al., Nature 503, 95 (2013), but their bend instability in a splay-stable flock does not exist and the polar phase has long-range order in two dimensions. Our theory also describes confined three-dimensional thin-film suspensions of active polar particles as well as dense compressible active polar rods, and predicts a flocking transition without a banding instability.

Nonquenched rotators ease flocking and memorize it

We introduce a minimal model for a two-dimensional polar flock with nonquenched rotators and show that the rotators make the usual macroscopic long-range order of the flock more robust than the clean system. The rotators memorize the flock-information which helps in establishing the robustness. Moreover, the memory of the rotators assists in probing the moving flock. We also formulate a hydrodynamic framework for the microscopic model that makes our study comprehensive. Using linearized hydrodynamics, it is shown that the presence of such nonquenched heterogeneities increases the sound speeds of the flock. The enhanced sound speeds lead to faster convection of information and consequently the robust ordering in the system. We argue that similar nonquenched heterogeneities may be useful in monitoring and controlling large crowds.

Thermodynamic uncertainty relation for underdamped Langevin systems driven by a velocity-dependent force

Recently, it has been shown that there is a trade-off relation between thermodynamic cost and current fluctuations, referred to as the thermodynamic uncertainty relation (TUR). The TUR has been derived for various processes, such as discrete-time Markov jump processes and overdamped Langevin dynamics. For underdamped dynamics, it has recently been reported that some modification is necessary for application of the TUR. However, the previous TUR for underdamped dynamics is not applicable to a system driven by a velocity-dependent force. In this study, we present a TUR, applicable to a system driven by a velocity-dependent force in the context of underdamped Langevin dynamics, by extending the theory of Vu and Hasegawa [Phys. Rev. E 100, 032130 (2019)2470-004510.1103/PhysRevE.100.032130]. We show that our TUR accurately describes the trade-off properties of a molecular refrigerator (cold damping), Brownian dynamics in a magnetic field, and an active particle system.

Three-body interactions drive the transition to polar order in a simple flocking model

A large class of mesoscopic or macroscopic flocking theories are coarse-grained from microscopic models that feature binary interactions as the chief aligning mechanism. However, while such theories seemingly predict the existence of polar order with just binary interactions, actomyosin motility assay experiments show that binary interactions are insufficient to obtain polar order, especially at high densities. To resolve this paradox, here we introduce a solvable one-dimensional flocking model and derive its stochastic hydrodynamics. We show that two-body interactions are insufficient to generate polar order unless the noise is non-Gaussian. We show that noisy three-body interactions in the microscopic theory allow us to capture all essential dynamical features of the flocking transition, in systems that achieve orientational order above a critical density.

Transport coefficients of self-propelled particles: Reverse perturbations and transverse current correlations

The reverse perturbation method [Phys. Rev. E 59, 4894 (1999)1063-651X10.1103/PhysRevE.59.4894] for shearing simple liquids and measuring their viscosity is extended to the Vicsek model (VM) of active particles [Phys. Rev. Lett. 75, 1226 (1995)PRLTAO0031-900710.1103/PhysRevLett.75.1226] and its metric-free version. The sheared systems exhibit a phenomenon that is similar to the skin effect of an alternating electric current: Momentum that is fed into the boundaries of a layer decays mostly exponentially toward the center of the layer. It is shown how two transport coefficients, i.e., the shear viscosity ν and the momentum amplification coefficient λ, can be obtained by fitting this decay with an analytical solution of the hydrodynamic equations for the VM. The viscosity of the VM consists of two parts, a kinetic and a collisional contribution. While analytical predictions already exist for the former, a novel expression for the collisional part is derived by an Enskog-like kinetic theory. To verify the predictions for the transport coefficients, Green-Kubo relations were evaluated and transverse current correlations were measured in independent simulations. Not too far to the transition to collective motion, we find excellent agreement between the different measurements of the transport coefficients. However, the measured values of ν and 1-λ are always slightly higher than the mean-field predictions, even at large mean free paths and at state points quite far from the threshold to collective motion, that is, far in the disordered phase. These findings seem to indicate that the mean-field assumption of molecular chaos is much less reliable in systems with velocity-alignment rules such as the VM, compared to models obeying detailed balance such as multiparticle collision dynamics.

Tuning the self-organization of confined active particles by the steepness of the trap

We consider the collective dynamics of self-propelling particles in two dimensions. They can align themselves according to the direction of propulsion of their neighbours, together with small rotational fluctuations. They also interact with each other via soft, isotropic, repulsive potentials. The particles are confined in a circular trap. The steepness of the trap is tuneable. The average packing fraction of the particles is low. When the trap is steep, particles flock along its boundary. They form a polar cluster that spreads over the boundary. The cluster is not spatially ordered. We show that when the steepness is decreased beyond a threshold value, the cluster becomes round and compact and eventually spatial order (hexagonal) emerges in addition to the pre-established polar order. We investigate the kinetics of such ordering. We find that while rotating around the centre of the trap along its circular boundary, the cluster needs to roll around its centre of mass to be spatially ordered. We have studied the stability of the order when the trap is suddenly switched off. We find that for the particles with velocity alignment interaction, the decay of the spatial order is much slower than the particles without the alignment interaction.

Flip motion of solitary wave in an Ising-type Vicsek model

An Ising-type Vicsek model is proposed for collective motion and sudden direction change in a population of self-propelled particles. Particles move on a linear lattice with velocity +1 or -1 in the one-dimensional model. The probability of the velocity of a particle at the next step is determined by the number difference of the right- and left-moving particles at the present lattice site and its nearest-neighboring sites. A solitary wave appears also in our model similarly to previous models. In some parameter range, the moving direction of the solitary wave sometimes changes rather suddenly, which is like the sudden change of direction of a flock of birds. We study the average reversal time of traveling direction numerically and compare the results with a mean-field theory. The one-dimensional model is generalized to a two-dimensional model. Flip motion of a bandlike soliton is observed in the two-dimensional model.

Clustering and anisotropic correlated percolation in polar flocks

We study clustering and percolation phenomena in the Vicsek model, taken here in its capacity of prototypical model for dry aligning active matter. Our results show that the order-disorder transition is not related in any way to a percolation transition, contrary to some earlier claims. We study geometric percolation in each of the phases at play, but we mostly focus on the ordered Toner-Tu phase, where we find that the long-range correlations of density fluctuations give rise to an anisotropic percolation transition.

Interrupted Motility Induced Phase Separation in Aligning Active Colloids

Switching on high activity in a relatively dense system of active Janus colloids, we observe fast clustering, followed by cluster aggregation towards full phase separation. The phase separation process is however interrupted when large enough clusters start breaking apart. Following the cluster size distribution as a function of time, we identify three successive dynamical regimes. Tracking both the particle positions and orientations, we characterize the structural ordering and alignment in the growing clusters and thereby unveil the mechanisms at play in these regimes. In particular, we identify how alignment between the neighboring particles is responsible for the interruption of the full phase separation. Our large scale quantification of the phase separation kinetics in active colloids points towards the new physics observed when both alignment and short-range repulsions are present.

Self-propelled Vicsek particles at low speed and low density

We study through numerical simulation the Vicsek model for very low speeds and densities. We consider scalar noise in two and three dimensions and vector noise in three dimensions. We focus on the behavior of the critical noise with density and speed, trying to clarify seemingly contradictory earlier results. We find that, for scalar noise, the critical noise is a power law in both density and speed, but although we confirm the density exponent in two dimensions, we find a speed exponent different from earlier reports (we consider lower speeds than previous studies). On the other hand, for the vector noise case we find that the dependence of the critical noise cannot be separated as a product of power laws in speed and density. Finally, we study the dependence of the relaxation time with speed. At the critical point we find a power law, with the same exponent in two and three dimensions.

Additivity and density fluctuations in Vicsek-like models of self-propelled particles

We study coarse-grained density fluctuations in the disordered phase of the paradigmatic Vicsek-like models of self-propelled particles with alignment interactions and random self-propulsion velocities. By numerically integrating a fluctuation-response relation-the direct consequence of an additivity property-we compute logarithm of the large-deviation probabilities of the coarse-grained subsystem density, while the system is in the disordered fluid phase with vanishing macroscopic velocity. The large-deviation probabilities, computed within additivity, agree remarkably well with that obtained from direct microscopic simulations of the models. Our results provide evidence of the existence of an equilibriumlike chemical potential, which governs the coarse-grained density fluctuations in the Vicsek-like models. Moreover, comparison of the particle-number fluctuations among several self-propelled particle systems suggests a common mechanism through which the number fluctuations arise in such systems.

Interpretable machine learning for inferring the phase boundaries in a nonequilibrium system

Still under debate is the question of whether machine learning is capable of going beyond black-box modeling for complex physical systems. We investigate the generalizing and interpretability properties of learning algorithms. To this end, we use supervised and unsupervised learning to infer the phase boundaries of the active Ising model, starting from an ensemble of configurations of the system. We illustrate that unsupervised learning techniques are powerful at identifying the phase boundaries in the control parameter space, even in situations of phase coexistence. It is demonstrated that supervised learning with neural networks is capable of learning the characteristics of the phase diagram, such that the knowledge obtained at a limited set of control variables can be used to determine the phase boundaries across the phase diagram. In this way, we show that properly designed supervised learning provides predictive power to regions in the phase diagram that are not included in the training phase of the algorithm. We stress the importance of introducing interpretability methods in order to perform a physically relevant classification of the phases with deep learning.

Activity-induced mixing and phase transitions of self-propelled swimmers

We study the mixing of active swimmers. Two different types of swimmers (modeled as particles) are placed initially in two boxes with an interconnection between them. The mixing of swimmers happens as they move with their own self-propelled forces. The self-propelled force is constant and the direction of the exerted thrust is governed by the neighboring swimmers. Overall mixing of the swimmers depends on the magnitude of the exerted thrust, the initial packing fraction, and the activity level. Different nonequilibrium states are also identified depending on the exerted thrust and the initial packing fraction of the swimmers.

Optimizing active work: Dynamical phase transitions, collective motion, and jamming

Active work measures how far the local self-forcing of active particles translates into real motion. Using population Monte Carlo methods, we investigate large deviations in the active work for repulsive active Brownian disks. Minimizing the active work generically results in dynamical arrest; in contrast, despite the lack of aligning interactions, trajectories of high active work correspond to a collectively moving, aligned state. We use heuristic and analytic arguments to explain the origin of dynamical phase transitions separating the arrested, typical, and aligned regimes.

Contrarian compulsions produce exotic time-dependent flocking of active particles

Animals having a tendency to align their velocities to an average of those of their neighbors may flock as illustrated by the Vicsek model and its variants. If, in addition, they feel a systematic contrarian trend, the result may be a time periodic adjustment of the flock or period doubling in time. These exotic phases are predicted from kinetic theory and numerically found in a modified two-dimensional Vicsek model of self-propelled particles. Numerical simulations demonstrate striking effects of alignment noise on the polarization order parameter measuring particle flocking: maximum polarization length is achieved at an optimal nonzero noise level. When contrarian compulsions are more likely than conformist ones, nonuniform polarized phases appear as the noise surpasses threshold.

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.

Stationary superstatistics distributions of trapped run-and-tumble particles

We present an analysis of the stationary distributions of run-and-tumble particles trapped in external potentials in terms of a thermophoretic potential that emerges when trapped active motion is mapped to trapped passive Brownian motion in a fictitious inhomogeneous thermal bath. We elaborate on the meaning of the non-Boltzmann-Gibbs stationary distributions that emerge as a consequence of the persistent motion of active particles. These stationary distributions are interpreted as a class of distributions in nonequilibrium statistical mechanics known as superstatistics. Our analysis provides an original insight on the link between the intrinsic nonequilibrium nature of active motion and the well-known concept of local equilibrium used in nonequilibrium statistical mechanics and contributes to the understanding of the validity of the concept of effective temperature. Particular cases of interest, regarding specific trapping potentials used in other theoretical or experimental studies, are discussed. We point out as an unprecedented effect, the emergence of new modes of the stationary distribution as a consequence of the coupling of persistent motion in a trapping potential that varies highly enough with position.

Exact Hydrodynamic Description of Active Lattice Gases

We introduce lattice gas models of active matter systems whose coarse-grained "hydrodynamic" description can be derived exactly. We illustrate our approach by considering two systems exhibiting two of the most studied collective behaviors in active matter: the motility-induced phase separation and the transition to collective motion. In both cases, we derive coupled partial differential equations describing the dynamics of the local density and polarization fields and show how they quantitatively predict the emerging properties of the macroscopic lattice gases.

Self-Propelled Particles with Velocity Reversals and Ferromagnetic Alignment: Active Matter Class with Second-Order Transition to Quasi-Long-Range Polar Order

We introduce and study in two dimensions a new class of dry, aligning active matter that exhibits a direct transition to orientational order, without the phase-separation phenomenology usually observed in this context. Characterized by self-propelled particles with velocity reversals and a ferromagnetic alignment of polarities, systems in this class display quasi-long-range polar order with continuously varying scaling exponents, yet a numerical study of the transition leads to conclude that it does not belong to the Berezinskii-Kosterlitz-Thouless universality class but is best described as a standard critical point with an algebraic divergence of correlations. We rationalize these findings by showing that the interplay between order and density changes the role of defects.

Cold Active Motion: How Time-Independent Disorder Affects the Motion of Self-Propelled Agents

Assemblages of self-propelled particles, often termed active matter, exhibit collective behavior due to competition between neighbor alignment and noise-induced decoherence. However, very little is known of how the quenched (i.e., time-independent) disorder impacts active motion. Here we report on the effects of quenched disorder on the dynamics of self-propelled point particles. We identified three major types of quenched disorder relevant in the context of active matter: random torque, force, and stress. We demonstrate that even in the absence of external fluctuations ("cold active matter"), quenched disorder results in nontrivial dynamic phases not present in their "hot" counterpart. In particular, by analyzing when the equations of motion exhibit a Hamiltonian structure and when attractors may be present, we identify in which scenarios particle trapping, i.e., the asymptotic convergence of particle trajectories to bounded areas in space ("traps"), can and cannot occur. Our study provides new fundamental insights into active systems realized by self-propelled particles on natural and synthetic disordered substrates.

Optimized Diffusion of Run-and-Tumble Particles in Crowded Environments

We study the transport of self-propelled particles in dynamic complex environments. To obtain exact results, we introduce a model of run-and-tumble particles (RTPs) moving in discrete time on a d-dimensional cubic lattice in the presence of diffusing hard-core obstacles. We derive an explicit expression for the diffusivity of the RTP, which is exact in the limit of low density of fixed obstacles. To do so, we introduce a generalization of Kac's theorem on the mean return times of Markov processes, which we expect to be relevant for a large class of lattice gas problems. Our results show the diffusivity of RTPs to be nonmonotonic in the tumbling probability for low enough obstacle mobility. These results prove the potential for the optimization of the transport of RTPs in crowded and disordered environments with applications to motile artificial and biological systems.

Flow-induced phase separation of active particles is controlled by boundary conditions

Active particles, including swimming microorganisms, autophoretic colloids, and droplets, are known to self-organize into ordered structures at fluid-solid boundaries. The entrainment of particles in the attractive parts of their spontaneous flows has been postulated as a possible mechanism underlying this phenomenon. Here, combining experiments, theory, and numerical simulations, we demonstrate the validity of this flow-induced ordering mechanism in a suspension of active emulsion droplets. We show that the mechanism can be controlled, with a variety of resultant ordered structures, by simply altering hydrodynamic boundary conditions. Thus, for flow in Hele-Shaw cells, metastable lines or stable traveling bands can be obtained by varying the cell height. Similarly, for flow bounded by a plane, dynamic crystallites are formed. At a no-slip wall, the crystallites are characterized by a continuous out-of-plane flux of particles that circulate and re-enter at the crystallite edges, thereby stabilizing them. At an interface where the tangential stress vanishes, the crystallites are strictly 2D, with no out-of-plane flux. We rationalize these experimental results by calculating, in each case, the slow viscous flow produced by the droplets and the long-ranged, many-body active forces and torques between them. The results of numerical simulations of motion under the action of the active forces and torques are in excellent agreement with experiments. Our work elucidates the mechanism of flow-induced phase separation in active fluids, particularly active colloidal suspensions, and demonstrates its control by boundaries, suggesting routes to geometric and topological phenomena in an active matter.

Collective motion of active Brownian particles with polar alignment

We present a comprehensive computational study of the collective behavior emerging from the competition between self-propulsion, excluded volume interactions and velocity-alignment in a two-dimensional model of active particles. We consider an extension of the active brownian particles model where the self-propulsion direction of the particles aligns with the one of their neighbors. We analyze the onset of collective motion (flocking) in a low-density regime (10% surface area) and show that it is mainly controlled by the strength of velocity-alignment interactions: the competition between self-propulsion and crowding effects plays a minor role in the emergence of flocking. However, above the flocking threshold, the system presents a richer pattern formation scenario than analogous models without alignment interactions (active brownian particles) or excluded volume effects (Vicsek-like models). Depending on the parameter regime, the structure of the system is characterized by either a broad distribution of finite-sized polar clusters or the presence of an amorphous, highly fluctuating, large-scale traveling structure which can take a lane-like or band-like form (and usually a hybrid structure which is halfway in between both). We establish a phase diagram that summarizes collective behavior of polar active brownian particles and propose a generic mechanism to describe the complexity of the large-scale structures observed in systems of repulsive self-propelled particles.

Active colloids with collective mobility status and research opportunities

The collective mobility of active matter (self-propelled objects that transduce energy into mechanical work to drive their motion, most commonly through fluids) constitutes a new frontier in science and achievable technology. This review surveys the current status of the research field, what kinds of new scientific problems can be tackled in the short term, and what long-term directions are envisioned. We focus on: (1) attempts to formulate design principles to tailor active particles; (2) attempts to design principles according to which active particles interact under circumstances where particle-particle interactions of traditional colloid science are augmented by a family of nonequilibrium effects discussed here; (3) attempts to design intended patterns of collective behavior and dynamic assembly; (4) speculative links to equilibrium thermodynamics. In each aspect, we assess achievements, limitations, and research opportunities.

Low-noise phase of a two-dimensional active nematic system

We consider a collection of self-driven apolar particles on a substrate that organize into an active nematic phase at sufficiently high density or low noise. Using the dynamical renormalization group, we systematically study the two-dimensional fluctuating ordered phase in a coarse-grained hydrodynamic description involving both the nematic director and the conserved density field. In the presence of noise, we show that the system always displays only quasi-long-ranged orientational order beyond a crossover scale. A careful analysis of the nonlinearities permitted by symmetry reveals that activity is dangerously irrelevant over the linearized description, allowing giant number fluctuations to persist although now with strong finite-size effects and a nonuniversal scaling exponent. Nonlinear effects from the active currents lead to power-law correlations in the density field, thereby preventing macroscopic phase separation in the thermodynamic limit.

Lattice Model to Derive the Fluctuating Hydrodynamics of Active Particles with Inertia

We derive the hydrodynamic equations with fluctuating currents for the density, momentum, and energy fields for an active system in the dilute limit. In our model, nonoverdamped self-propelled particles (such as grains or birds) move on a lattice, interacting by means of aligning dissipative forces and excluded volume repulsion. Our macroscopic equations, in a specific case, reproduce a transition line from a disordered phase to a swarming phase and a linear dispersion law accounting for underdamped wave propagation. Numerical simulations up to a packing fraction ∼10% are in fair agreement with the theory, including the macroscopic noise amplitudes. At a higher packing fraction, a dense-diluted coexistence emerges. We underline the analogies with the granular kinetic theories, elucidating the relation between the active swarming phase and granular shear instability.

Long-lived force patterns and deformation waves at repulsive epithelial boundaries

For an organism to develop and maintain homeostasis, cell types with distinct functions must often be separated by physical boundaries. The formation and maintenance of such boundaries are commonly attributed to mechanisms restricted to the cells lining the boundary. Here we show that, besides these local subcellular mechanisms, the formation and maintenance of tissue boundaries involves long-lived, long-ranged mechanical events. Following contact between two epithelial monolayers expressing, respectively, EphB2 and its ligand ephrinB1, both monolayers exhibit oscillatory patterns of traction forces and intercellular stresses that tend to pull cell-matrix adhesions away from the boundary. With time, monolayers jam, accompanied by the emergence of deformation waves that propagate away from the boundary. This phenomenon is not specific to EphB2/ephrinB1 repulsion but is also present during the formation of boundaries with an inert interface and during fusion of homotypic epithelial layers. Our findings thus unveil a global physical mechanism that sustains tissue separation independently of the biochemical and mechanical features of the local tissue boundary.

Collective Behavior of Chiral Active Matter: Pattern Formation and Enhanced Flocking

We generalize the Vicsek model to describe the collective behavior of polar circle swimmers with local alignment interactions. While the phase transition leading to collective motion in 2D (flocking) occurs at the same interaction to noise ratio as for linear swimmers, as we show, circular motion enhances the polarization in the ordered phase (enhanced flocking) and induces secondary instabilities leading to structure formation. Slow rotations promote macroscopic droplets with late time sizes proportional to the system size (indicating phase separation) whereas fast rotations generate patterns consisting of phase synchronized microflocks with a controllable characteristic size proportional to the average single-particle swimming radius. Our results defy the viewpoint that monofrequent rotations form a vapid extension of the Vicsek model and establish a generic route to pattern formation in chiral active matter with possible applications for understanding and designing rotating microflocks.

Fore-aft asymmetric flocking

We show that fore-aft asymmetry, a generic feature of living organisms and some active matter systems, can have a strong influence on the collective properties of even the simplest flocking models. Specifically, an arbitrarily weak asymmetry favoring front neighbors changes qualitatively the phase diagram of the Vicsek model. A region where many sharp traveling band solutions coexist is present at low noise strength, below the Toner-Tu liquid, at odds with the phase-separation scenario well describing the usual isotropic model. Inside this region, a "banded-liquid" phase with algebraic density distribution coexists with band solutions. Linear stability analysis at the hydrodynamic level suggests that these results are generic and not specific to the Vicsek model.