Self-propelled Hard Disks Research Articles

A particle-field approach bridges phase separation and collective motion in active matter

Whereas self-propelled hard discs undergo motility-induced phase separation, self-propelled rods exhibit a variety of nonequilibrium phenomena, including clustering, collective motion, and spatio-temporal chaos. In this work, we present a theoretical framework representing active particles by continuum fields. This concept combines the simplicity of alignment-based models, enabling analytical studies, and realistic models that incorporate the shape of self-propelled objects explicitly. By varying particle shape from circular to ellipsoidal, we show how nonequilibrium stresses acting among self-propelled rods destabilize motility-induced phase separation and facilitate orientational ordering, thereby connecting the realms of scalar and vectorial active matter. Though the interaction potential is strictly apolar, both, polar and nematic order may emerge and even coexist. Accordingly, the symmetry of ordered states is a dynamical property in active matter. The presented framework may represent various systems including bacterial colonies, cytoskeletal extracts, or shaken granular media.

Full Phase Diagram of Active Brownian Disks: From Melting to Motility-Induced Phase Separation.

We establish the complete phase diagram of self-propelled hard disks in two spatial dimensions from the analysis of the equation of state and the statistics of local order parameters. The equilibrium melting scenario is maintained at small activities, with coexistence between active liquid and hexatic order, followed by a proper hexatic phase, and a further transition to an active solid. As activity increases, the emergence of hexatic and solid order is shifted towards higher densities. Above a critical activity and for a certain range of packing fractions, the system undergoes motility-induced phase separation and demixes into low and high density phases; the latter can be either disordered (liquid) or ordered (hexatic or solid) depending on the activity.

Spontaneously Flowing Crystal of Self-Propelled Particles.

We experimentally and numerically study the structure and dynamics of a monodisperse packing of spontaneously aligning self-propelled hard disks. The packings are such that their equilibrium counterparts form perfectly ordered hexagonal structures. Experimentally, we first form a perfect crystal in a hexagonal arena which respects the same crystalline symmetry. Frustration of the hexagonal order, obtained by removing a few particles, leads to the formation of a rapidly diffusing "droplet." Removing more particles, the whole system spontaneously forms a macroscopic sheared flow, while conserving an overall crystalline structure. This flowing crystalline structure, which we call a "rheocrystal," is made possible by the condensation of shear along localized stacking faults. Numerical simulations very well reproduce the experimental observations and allow us to explore the parameter space. They demonstrate that the rheocrystal is induced neither by frustration nor by noise. They further show that larger systems flow faster while still remaining ordered.

Synchronization in Dynamical Networks of Locally Coupled Self-Propelled Oscillators

Systems of mobile physical entities exchanging information with their neighborhood can be found in many different situations. The understanding of their emergent cooperative behaviour has become an important issue across disciplines, requiring a general conceptual framework in order to harvest the potential of these systems. We study the synchronization of coupled oscillators in time-evolving networks defined by the positions of self-propelled agents interacting in real space. In order to understand the impact of mobility in the synchronization process on general grounds, we introduce a simple model of self-propelled hard disks performing persistent random walks in 2$d$ space and carrying an internal Kuramoto phase oscillator. For non-interacting particles, self-propulsion accelerates synchronization. The competition between agent mobility and excluded volume interactions gives rise to a richer scenario, leading to an optimal self-propulsion speed. We identify two extreme dynamic regimes where synchronization can be understood from theoretical considerations. A systematic analysis of our model quantifies the departure from the latter ideal situations and characterizes the different mechanisms leading the evolution of the system. We show that the synchronization of locally coupled mobile oscillators generically proceeds through coarsening verifying dynamic scaling and sharing strong similarities with the phase ordering dynamics of the 2$d$ XY model following a quench. Our results shed light into the generic mechanisms leading the synchronization of mobile agents, providing a efficient way to understand more complex or specific situations involving time-dependent networks where synchronization, mobility and excluded volume are at play.

Coupling spin to velocity: collective motion of Hamiltonian polar particles

We propose a conservative two-dimensional particle model in which particles carry a continuous and classical spin. The model includes standard ferromagnetic interactions between spins of two different particles, and a nonstandard coupling between spin and velocity of the same particle inspired by the coupling observed in self-propelled hard discs. Because of this coupling Galilean invariance is broken and the conserved linear momentum associated to translation invariance is not proportional to the velocity of the center of mass. Also, the dynamics is not invariant under a global rotation of the spins alone. This, in principle, leaves room for collective motion and thus raises the question whether collective motion can arise in Hamiltonian systems. We study the statistical mechanics of such a system, and show that, in the fully connected (or mean-field) case, a transition to collective motion does exist in spite of momentum conservation. Interestingly, the velocity of the center of mass, which in the absence of Galilean invariance, is a relevant variable, also feeds back on the magnetization properties, as it acts as an external magnetic field that smoothens the transition. Molecular dynamics simulations of finite size systems indeed reveal a rich phase diagram, with a transition from a disordered to a homogeneous polar phase, but also more complex inhomogeneous phases with local order interrupted by topological defects.

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.

From single-particle to collective effective temperatures in an active fluid of self-propelled particles

We present a comprehensive analysis of effective temperatures based on fluctuation-dissipation relations in a model of an active fluid composed of self-propelled hard disks. We first investigate the relevance of effective temperatures in the dilute and moderately dense fluids. We find that a unique effective temperature does not in general characterize the non-equilibrium dynamics of the active fluid over this broad range of densities, because fluctuation-dissipation relations yield a lengthscale-dependent effective temperature. By contrast, we find that the approach to a non-equilibrium glass transition at very large densities is accompanied by the emergence of a unique effective temperature shared by fluctuations at all lengthscales. This suggests that an effective thermal dynamics generically emerges at long times in very dense suspensions of active particles due to the collective freezing occurring at non-equilibrium glass transitions.

Clustering and heterogeneous dynamics in a kinetic Monte Carlo model of self-propelled hard disks.

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 is increased, the nonequilibrium clusters percolate to form a ramified structure reminiscent of a physical gel. We show that the addition of a finite amount of noise is needed to trigger a nonequilibrium phase separation, showing that demixing in active Brownian particles results from a delicate balance between noise, interparticle interactions, and self-propulsion. We show that self-propulsion has a profound influence on the dynamics of the active fluid. We find that the diffusion constant has a nonmonotonic behavior as self-propulsion is increased at finite density and that activity produces strong deviations from Fickian diffusion that persist over large time scales and length scales, suggesting that systems of active particles generically behave as dynamically heterogeneous systems.

Nonequilibrium Glassy Dynamics of Self-Propelled Hard Disks

We analyze 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 MonteCarlo 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 continuously shifts to larger densities with increasing activity, and the relaxation time is surprisingly well described by an algebraic divergence resulting from the emergence of highly collective dynamics. These results show that dense assemblies of active particles undergo a nonequilibrium glass transition that is profoundly affected by self-propulsion mechanisms.