Rule-based Automaton with Local Interactions

While previous approaches on propelled particle systems focussed on the long-time dynam- ics [105, 28, 29, 22, 40, 23, 11], e.g. the emergence of the polar state, or concentrated on the analysis of the ensuing stationary patterns, the transient assembly processes leading to collective motion remain largely elusive. In the introduction (section 1.2) we explained that growing clusters act as “vacuum cleaners”, which successively take up surrounding particles and align them to the horde. Since clusters already exist on the isotopic side of the phase boundary it is not clear which criteria clusters must meet to induce a transition (in time) to a polar state.

To investigate this question, we designed a rule-based automaton for (self-)propelled particles moving on a hexagonal lattice which is efficient enough to deal with large system sizes and particle numbers, thereby offering good statistics to study the time-dependent dynamics on short and long time scales. In our model, particles of lengthLand width equal

to one inter-lattice spacing, interact via an effective excluded volume interaction and a local polar alignment field. Both enter into the model by supplementing the probabilities for the unperturbed persistent random walk by appropriate factors [for details on the model please refer to caption of Fig. 2.18(a)]. Since particles in the model move with constant speed and interactions solely change the outcome with respect to the particles’ orientations, the automaton model is restricted to dilute conditions where caging effects or the emergence of bond orientational and translational order can be neglected.

In the absence of excluded volume interactions, we find—in agreement with other agent- based models with ferromagnetic interactions, e.g. [105, 28, 22, 29, 40, 23]—a phase tran- sition to collective motion above a critical packing fraction. Interestingly, strong excluded volume interactions increase this critical packing fraction or even inhibit the development of collective motion. This breakdown of collective motion at large particle densities has also been found in other models [67], publication B.9 and publication B.8. A heuris- tic explanation for this finding is the following: While ferromagnetic interactions tend to build-up parallel orientations, repulsive interactions lead to anti-correlations in the veloc- ity. Therefore, it is only a matter of the strength of the ferromagnetic to the repulsive interaction whether collective motion emerges or not21_.

Analyzing the dynamics of the pattern forming processes of N particles, we find that collective motion is accompanied by a gain in free volume,ρ₋γ, whereρ=N L/A[withA: area of simulation domain, and N _∼ 106_{] is the (reduced) particle density and γ denotes}

the fraction of occupied lattice sites. In the absence of polar alignment processes between the filaments or for too strong excluded volume interactions, we find that γ fluctuates around a constant value depending on the strength of the excluded volume interaction [see Fig. 2.18(b), inset]. Adding polar alignment interactions leads to qualitatively different behavior: There is a threshold density ρc, where the gain in free volume ρ−γ jumps to

a value much larger than the corresponding value for a non-interacting system. Moreover, measuring the polar order parameters, it turns out that this jump actually coincides with the onset of polar order. The jump in ρ₋γ may, therefore, be taken as a signature to map out the model’s phase diagram (not shown, see publication B.4). The gain in free volume also clearly indicates that filaments must have formed some clusters much denser than expected for a purely statistical (random) overlap of filaments. Fig. 2.18(b) shows that these clusters are formed by a nucleation process: There is a lag-phase during which γ remains largely close to a value obtained in the absence of polar alignment processes. Subsequently, there is a time window [Tnuc, Tstat], where the available free volume fraction

ρ₋γ increases towards a higher stationary value. The time to reach the final polar steady state diverges as a power law Tstat ∝(ρ−ρc)−ζ with ζ ≈1.

Due to the fact that small clusters are continuously assembled and disassembled dur- ing the lag-phase, the sudden decrease of γ at Tnuc indicates that a notably larger cluster

develops. Until such an event occurs, the overall macroscopic state, characterized here by 21_{This argument only applies to models with equations of motion solely for particles’ velocities. Particles} with e.g.a direction and a velocity can sustain strong repulsive interactions, while keeping orientational correlations. An example is the vibrated polar disk assay, which is discussed in section 2.2.

2.3 Modeling Propelled Particle Systems under Dilute Conditions 35

the gain in free volume, is unchanged. A systematic analysis of the cluster mass distribu- tion during the lag-phase corroborates our finding that the onset of collective motion is triggered by a single nucleation events (details on evaluation of cluster mass distribution

seepublication B.4), which manifests itself in a coherently moving cluster of sufficiently

large size/mass. We conclude that collective motion close to the critical density is induced by a rare event, i.e. the nucleation of a cluster of sufficiently large mass, and not by a wide-spread coarsening process of polarized domains.

Figure 2.18: (a) Illustration of the rule-based lattice gas automaton: The particles’ dynamics are assumed to be fully determined by its head, while the tail strictly follows the head’s trail. The persistent random walk is implemented by a stochastic process, where each filament’s headx0moves to one of its five neighboring sitesxα according to a fixed set

of conditional probabilitiesPprw(xα|x0). Excluded volume: For a lattice site that is already

k-times occupied, the probability for further occupation is reduced by a Boltzmann factor, e−k _{, where characterizes the penalty for multiple occupations. Formally, for → ∞,}

the limit of strict excluded volume is obtained. Alignment interaction: Each particle is assigned an alignment field u(x, t) at its occupied and neighboring lattice sites (see black arrows); it is directed along the particles contour. Overlapping alignment fields of different particles are averaged. In a collision event, the alignment field modifies the transition probability to move from x0 to x by the Boltzmann factor, αcosϕ, where ϕ is the relative

angle between the alignment field and the direction of motion of the respective collision partner, ϕ = ∠(u(x),x−x0). Update rule: Given a configuration of N filaments we

use random sequential updating and move a chosen head from position x0 to a target

positionxwith probabilityP(x_|x0;u, k)∝Pprw(x|x0)·e−k·αcosϕ(u). (b) Lag-phase and

nucleation: Time traces of the ratio of occupied lattice sites γ for ρ = 0.6, and a set of

values: = 0,0.5,1 with α = 5, and = 1 with α = 1, from bottom to top. Close to the phase boundary from the isotropic (ISO) to the polar ordered state (PO) [ = 1; (red)], a lag-phase exists, where the system waits for the nucleation of a cluster that triggers the emergence of polar order; the time when nucleation occurs Tnuc and the time whenγ

becomes stationaryTstat are indicated by vertical dotted or dashed lines. Inset: for α= 1,

2.3 Modeling Propelled Particle Systems under Dilute Conditions 37