Transition to synchrony in chiral active particles

I study deterministic dynamics of chiral active particles in two dimensions. Particles are considered as discs interacting with elastic repulsive forces. An ensemble of particles, started from random initial conditions, demonstrates chaotic collisions resulting in their normal diffusion. This chaos is transient, as rather abruptly a synchronous collisionless state establishes. The life time of chaos grows exponentially with the number of particles. External forcing (periodic or chaotic) is shown to facilitate the synchronization transition.


Introduction
Active or self-propelled particles are a subject of active current research (see introductory review [1]). For microscopic particles, random interactions with an environment are essential, and one speaks about active Brownian particles, subject to noisy forces. For macroscopic particles, existence of random forces is not so obvious, but here one also often introduces them to model observed fluctuations in the motions of animals and birds (cf. famous Vicsek model [2]). A large class of active particles constitute chiral active particles, natural trajectories of which are not straight lines, but circles. The origin of chirality can be asymmetry in the particle shape or in the propulsion mechanism; also external magnetic field may lead to circular motion (see discussion and examples in [1]).
There are different models for interaction of active particles. In many cases, inspired by the Vicsek model [2], one assumes an aligning interaction: neighboring particles "prefer" to align their velocities. In the context of chiral Brownian (i.e. noise-driven) particles, such an aligning interaction can lead to an appearance of synchronized rotating clusters (see, e.g., Refs. [3][4][5][6]). In this paper, I consider deterministic particles without alignment forces. The main finding is that this system demonstrates a transition from supertransient chaos to synchronous clusters.
I introduce the basic model in Section 2. In Section 3 I demonstrate that while for small times, circling colliding active particles demonstrate chaos leading to their diffusion, at large times a transition to a synchronous state without collisions occur. The life time of chaos grows exponentially with the number of particles, what allows to speak about supertransients, typical for spatio-temporal chaos. In section 4 I demonstrate that external periodic or random forcing induces a transition to a collisionless state already at small times. I conclude with discussion in Section 5.

Model formulation
I consider active particles in two dimensions, with circular natural trajectories. The two components of velocity (v =ẋ, u =ẏ) for one particle obey following equations: Here W is a steady speed, which a particle attends if the other forces F x,y vanish; parameter µ describes the rate of the relaxation to this steady speed; m is the particle's mass. The velocity field rotates with frequency ω. In a steady state, an isolated particle rotates on a circle of radius W/ω with frequency ω.
It is worth noting that Eq. (1) is widely used in synchronization and coupled oscillators studies as the Stuart-Landau model (see, e.g., [7]). In this interpretation two variables u, v describe a state of an autonomous oscillating system close to the Hopf bifurcation point. It is also well-known that under a periodic or random force this oscillator synchronizes [7], this effect will be explored in Section 4.
Below I consider two types of forces acting on particles. The first force is the interaction between the particles. I assume, following Ref. [8], a conservative repulsing interaction governed by a truncated Lennard-Jones potential, which depends on the distance R between the particles: This potential takes from the full Lennard-Jones potential only its repulsing part, and the attracting part is absent. Thus, this potential mimics not-so-hard discs with radius σ/2, which repulse each other when collide, and do not interact aside of collisions. I stress here that the interactions have no any alignment action (the latter is often assumed in models of Vicseck type). If there where no activity and chirality (i.e. ω = µ = 0 in (1)), then the model reduces to a Hamiltonian one of elastically colliding disks. In the case of hard disks it is believed that the dynamics is fully chaotic (the Boltzmann-Sinai hypothesis), although the proof [9] has some restrictions. For smooth potentials, stable periodic orbits in the Hamiltonian dynamics may appear [10]. Much less is known for colliding active particles, but our simulations in Section 3 indicate for chaos. In addition to interaction forces described by (2), I will consider external forces specified in Section 4.
Below I study numerically an ensemble of active particles with circular orbits (1),(2) in a periodic geometry, i.e., on a torus L × L. I fix µ = 2, W = ω = 1, σ = 0.2, m = 1, and = 0.1 throughout the paper. The main parameters to explore is the number of the particles N and the density ρ = N L −2 .

Quasistationary chaotic state
Because collisions of hard disk have a scattering property with an essential degree of instability (like dispersive billiards), one can expect chaos in colliding active particle described by Eqs. (1), (2). Due to multiple collisions, velocity of a particle is random, so its motion is a diffusion in two dimensions. I illustrate this with figures 1,2. They show trajectories of N = 20 particles in a particular run. Up to time T st ≈ 27000 (I measure time in periods of rotations) motion of all particles is irregular.
Quantitatively, a good characteristics of irregularity is the mean diffusion constant of the particles. It can be calculated from the mean squared displacement after a large time interval Diffusion in the system is normal, as the graph of (x(T ) − x(0)) 2 + (y(T ) − y(0)) 2 versus time interval T (Fig. 3(a)) shows. Because D is an intensive quantity, one can expect that it depends on the intensive parameter -density of particles ρ, but not on the x(t) Figure 2. Trajectories of particles for the same data as Fig. 1. number of particles N . However, calculations of the diffusion constant Fig. 3(b) show, that for a small number of particles a significant depletion of the diffusion constant is observed. I attribute this to correlations which appear in small communities where the same particles collide many times. In large ensembles, a particle has again and again new neighbors, so that correlations effectively disappear (see nearly coinciding values for N = 30 and N = 40 in Fig. 3(b)).

Spontaneous transition to synchrony
The main observation of this work is that the chaos described above is in fact a transient state, it evolves eventually into a configuration without collisions; such a transition can be seen at T st ≈ 27000 in Fig. 1. Indeed, a set of particles (1) has  an absorbing synchronous state where all the velocities are equal: v 1 = v 2 = . . . = v N , u 1 = u 2 = . . . = u N . In this state the particles rotate synchronously, the distances between them remain constant, and they do not collide. Thus this state continues forever. Such a state can exist also in a Hamiltonian setup, but there it can occur only for specially constructed initial conditions. If collisions in a set of Hamiltonian discs occur, they cannot disappear, because of the reversibility of the dynamics. For active particles with a non-Hamiltonian dynamics, there is no such a restriction. I stress that the state in Fig. 1 at T > T st is not completely synchronous, as the phases of rotation of different particles do not coincide. It is sufficient to achieve a state where collisions disappear, such a regime, which I call absorbing collisionless state, continues forever. I illustrate the collissionless synchronous state corresponding to T > T st of Fig. 1 in Fig. 4. Fig. 1 shows that the transition to synchrony is quite abrupt, so this is an example of type-II supertransients according to classification of [11].
The time at which a synchronous absorbing state appears depends on the initial configuration of particles, it fluctuates highly. In Fig. 5 I show dependence of the mean time to achieve a collisionless state on the number of particles and on density ρ. The main feature is that the dependence on the density is rather weak, but the life time of a chaotic state grows exponentially with the number of particles N . Thus, this system belongs to a class of extended systems with chaotic supertransients [12,13]. Figure 5 also shows that the law of exponential growth with the number of particles only weakly depends on the density parameter.

Driven transition to synchrony
Noninteracting particles are described by a set of effective Stuart-Landau oscillators (1). Thus, such an ensemble can be synchronized by an external periodic or random force, as described in the theory of synchronization [7]. One can expect that the same happens also in the presence of the collisions, although the synchronization onset can be retarded due to them. I Illustrate the effect of a periodic force F x = γ cos(ωt) on the ensemble in Fig. 6. A small force has almost no effect, the mean life time of the chaotic state is almost the same as for autonomous particles. At large values of γ, the transition occurs within a few periods, the mean life time is almost the same for different N and ρ. Noteworthy, at intermediate force amplitudes 0.01 < γ < 0.03, the reduction of the mean life time of chaos is mostly pronounced for systems with low density (see curves with ρ = 1 for N = 20 and N = 30).
Common noise is another source of synchrony in ensembles of uncoupled oscillators [7]. Here I report on numerical experiments where noise was in the form of a Poissonian sequence of delta-pulses:  where t n are Poissonian time events appearing with rate τ −1 , and a n are independent amplitudes of the pulses taken from a Gaussian distribution with standard deviation σ. Mean life times of chaos are shown in Fig. 7. Remarkably, under noisy force the mean life time of chaos is nearly a constant, it does depend on the number of particles and on the density. Furthermore, for small number of particles common noise retards transition to synchrony compared to the spontaneous one. For large number of particles the effect is opposite, here for N = 30 noise reduces the life time of chaos by a factor larger than 20.

Conclusion
In this paper I studied an ensemble of active particles with circular natural trajectories, subject to elastic collisions. This setup differs from previous studies in two aspects: (i) the basic dynamics is purely deterministic (except for the special considered case of a common random force), and (ii) there are no aligning interactions. I show that the autonomous system demonstrates supertransient chaos. Starting with random initial conditions, for a long interval of time, chaotic dynamics is observed, leading to a normal diffusion of the particles. However, the final state is a regime without collisions, where neighboring particles effectively synchronize their circular rotations. I demonstrate that the life time of chaos depends on the density of particles, and grows exponentially with the number of them. Furthermore, I show that the transition to synchronous rotations can be induced by an external periodic force, common for all particles (this effect has been previously reported in other context in Refs. [14,15]). Also I demonstrate that a random common force also leads to a synchronous dynamics without collisions. Forced transitions have a certain transient time which for large forces does not depend on the number of particles.