Network structure formation in pulsating particle populations

Pattern formations in populations of “pulsating particles” which can autonomously vary their volumes were investigated. It was found that clusters of pulsating particles whose pulsation frequencies depend on the number of other particles in the vicinity form the following dynamical network structures: (i) void type networks exhibiting large voids, and (ii) branching type networks consisting of individual branches. Such global structures are formed through the annihilation and creation of local structures in the particle populations induced by tug-of-war-like interactions among them.

Figure 1

(a) Illustration of pulsating particle, and (b) two-body interaction potential $V$ as a function of $\mid {\mathbf{x}}_{\mathbf{i}}-{\mathbf{x}}_{\mathbf{j}}\mid$.

Figure 2

Phase diagram with snapshots of the spatial structures of the system, for several values of ${\omega }_o$ and $K$.

Figure 3

Typical temporal evolutions of the void type network and the branching type network.

Figure 4

(Color online) Typical temporal evolutions of $L$ for the void type network (blue solid curve) and the branching type network (red dashed curve).

Figure 5

(a)–(c) Illustration of typical local structures. (d) The rotational motions of a three-particle system (one of the particles is filled to show it clearly) in cases where the pulsation phases of two (white) particles are the same (left) and where those of all particles are different (right). (e) Rotational velocity (RV) as a function of the pulsation phase difference between two of the particles (positive RV indicates the clockwise rotation). (f), (g) Examples of the destruction of local structures (junctions) and void shrinking.