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Collective Behavior of Vicsek Particles Without and with Obstacles

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Collective Behavior of Vicsek Particles Without and with Obstacles EPJ manuscript No. (will be inserted by the editor) Collective behavior of Vicsek particles without and with obstacles Raul Martinez1;2;3, Francisco Alarcon1;3, Diego Rogel Rodriguez1;3, Juan Luis Aragones2, and Chantal Valeriani1;3 1 Departamento de estructura de la materia, f´ısicat´ermicay electr´onica,Facultad de Ciencias F´ısicas,Universidad Complutense de Madrid, 28040 Madrid, Spain 2 IFIMAC, Facultad de Ciencias, Universidad Aut´onomade Madrid, Ciudad Universitaria de Cantoblanco, 28049, Madrid, Spain 3 GISC - Grupo Interdisciplinar de Sistemas Complejos, Madrid, Spain the date of receipt and acceptance should be inserted later Abstract. In our work we have studied a two-dimensional suspension of finite-size Vicsek hard-disks, whose time evolution follows an event-driven dynamics between subsequent time steps. Having compared its collective behaviour with the one expected for a system of scalar Vicsek point-like particles, we have analysed the effect of considering two possible bouncing rules between the disks: a Vicsek-like rule and a pseudo-elastic one, focusing on the order-disorder transition. Next, we have added to the two-dimensional suspension of hard-disk Vicsek particles disk-like passive obstacles of two types: either fixed in space or moving according to the same event-driven dynamics. We have performed a detailed analysis of the particles' collective behaviour observed for both fixed and moving obstacles. In the fixed obstacles case, we have observed formation of clusters at low noise, in agreement with previous studies. When using moving passive obstacles, we found that that order of active particles is better destroyed as the drag of obstacles increases. In the no drag limit an interesting result was found: introduction of low drag passive particles can lead in some cases to a more ordered state of active flocking particles than what they show in bulk. PACS. 05.65.+b Self-organized systems, 64.70.qj Dynamics and criticality { 87.18.Gh Cell-cell communi- cation; collective behavior of motile cells 1 Introduction Several studies have focused on the collective behaviour of self-propelled particles in crowded environments [2], Active matter deals with out-of-equilibrium systems of where chemotactic particles show anomalous dynamics on self-driven units (active particles) each able to convert en- a percolating cluster [22], while clustering has been ob- ergy (either stored or from its surroundings) into directed served when chemotactic particles move in a porous medium motion[1,2,3,4,5,6,7]. Due to the particle-particle interac- [23] or in bulk [24]. At the same time, several studies tion and to the particles' interplay with the surrounding have been reported with self-propelled particles interact- medium, active particles are capable of a characteristic ing with obstacles either fixed in space [25,26] or moving collective behaviour, which has raised great interest over due to the collisions with the active particles [21,27]. the last few decades, both theoretically and experimen- Inspired by the above mentioned works, we propose to tally [8,9,10,11,12,13]. study a two dimensional dilute suspension of self-propelled One of the simplest numerical models used to under- Vicsek hard disks and compare them to their point ana- stand order-disorder out-of-equilibrium transitions and fea- log, in the presence of either fixed or mobile hard-disks tures of the collective motion in active suspensions [14,15, obstacles. To start with, we describe the event-driven al- 16] is the Vicsek model [17]. Despite the simple interaction gorithm to simulate hard-disks Vicsek-like particles, con- arXiv:1911.10374v1 [cond-mat.soft] 23 Nov 2019 rules acting between self-propelled point-like particles, the sidering two possible bouncing rules between particles: a model has proven capable of capturing the behavior of sys- Vicsek-like and a pseudo-elastic one. To establish the ef- tems with emergent collective motion, ranging from flocks fect of adding a finite area to the self-propelled disks, we of birds to microorganisms cluster formation. will compare the results with the the Vicsek-points model In order to describe more realistic systems, such as analog, focusing on the order-disorder transition. the formation of schools of fish [18] or to model bacterial motion [19,20,21], modified versions of the Vicsek model Next ,we will simulate Vicsek particles (either point- have been proposed, where self-propelled hard-disks take like or hard disks) interacting with fixed disk-like obsta- the place of self-propelled point-like particles. cles, considering the case where particles' size is much smaller of that of the obstacles. In either case our results Send offprint requests to: [email protected] agree with the ones presented by Chepizhko and Peruani 2 Martinez et al.: Collective behavior of Vicsek particles without and with obstacles [26]. To conclude, we will let the obstacles move, and study 3. The new direction of the ith-particle is θi(t + ∆t) = m the resulting phenomena. θi (t) + ηζi(t), where ζi(t) is a white noise (hζ(t)i = 0, 0 0 The manuscript is organised as follows. In section 2 hζi(t)ζj(t )i = δijδ(t − t ), and ζi 2 [−π; π]) and η the we present the numerical details including bulk systems noise strength (η 2 [0; 1], set at the beginning of each and systems with obstacles. In section 3, we present the run). obtained results and draw our conclusions in section 4. 4. Each particle’s position is then updated according to r(t+ ∆t) = r(t) + v(t + ∆t)∆t. All simulations are performed with periodic boundary 2 Model and simulation details conditions (unless stated otherwise). We make use of re- duced units, with ∆t as the time unit and R the length To start with, in 2.1 we introduce the original version unit (∆t = 1 and R = 1). In all simulations, v∆t < R. of the Vicsek model [17,14]. Next, in section 2.2 we pro- We initialise the system with random positions and ve- pose the Vicsek model for hard-disks, thus presenting two locities, r and θ for each particle, and let it evolve until bouncing rules responsible for the interactions between steady state is reached. All results presented (including them: a Vicsek-like rule, which aligns the velocities of the snapshots) are taken from the steady state. We fix the colliding particles, inspired by the Vicsek model update particles' density ρ = N=L2 to be ρ = 1:953 throughout rule; and a pseudo-elastic rule, inspired by the possibility our simulations. of the existence of an inertial dynamic at short times. To measure the total alignment within the system, we Having discussed the bulk system, we present the nu- compute the order parameter ν [14]: merical details for the system with fixed obstacles. In sec- tion 2.3, the interaction between particles and fixed obsta- v u N !2 N !2 cles is discussed. Two bouncing rules are considered: one 1 u X X ν = t sin(θ ) + cos(θ ) (2) that aligns particles with the wall and another that per- N k k forms an elastic (or mirror) bouncing. The elastic bounc- k=1 k=1 ing rule is inspired by the possibility of the existence of an Whenever ν = 0 particles are disordered and move ran- inertial dynamic at short times, while the aligning bounc- domly in every direction; whereas when ν = 1 particles ing rule is inspired by hydrodynamic interactions with are totally aligned and move as a flock. walls. To study particles aggregation, we detect all clusters in In section 2.4, we present the numerical details for the the system via the following criterion: at a given time step system with moving obstacles: in our model, passive par- in the steady state, two particles belong to the same clus- ticles are desplaced by the collisions with other particles ter if their distance is smaller than the interaction range and slowed down by a drag force (to model the interaction R. The largest cluster nc will just be the cluster with the with the fluid). Two bouncing rules are considered: one largest number of particles. describing collisions between two obstacles and another one for the obstacle - particle collision. Given that for ob- stacles the constant speed condition does not have to be 2.2 Self-propelled Vicsek hard-disks imposed, we consider 1) an elastic bouncing rule for the obstacle-obstacle bouncing rule, and 2) a pseudo-elastic When considering finite-size Vicsek disks (diameter σ) at bouncing rule for the active particle-obstacle.s packing fraction φ = (Nπσ2)=(4L2). To conclude the method section, we will present the Disks' velocities are updated at intervals of ∆t as pre- chosen parameter range 2.5. scribed by the Vicsek model. However, between these in- tervals they follow an event-driven algorithm [28] 1 and can interact with each others via two different bouncing 2.1 Self-propelled Vicsek point-like particles rules: a Vicsek-like and a pseudo-elastic-like bouncing rule. The Vicsek-like bouncing rule tries to mimic a In the scalar Vicsek model [15,14], the system consists of Vicsek interaction between two disks. As schematically N identical point particles in a two dimensional square box represented in figure 1, it is applied as follows: of edge L. All particles propel at a constant speed v, thus the velocity vi of the ith-particle at time t is completely 1. Given the velocities just before the collision of two parti- − − m determined by its direction of motion θi cles, say v1 and v2 for particle 1 and 2 respectively, θ m − − is computed as θ = (θ1 + θ2 )=2 (left panels in figure vi = v(^ı cos θi(t) +^ sin θi(t)): (1) 1, top and bottom).
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