Swarmalators

Swarmalators[1] are generalizations of phase oscillators[2] that swarm around in space as they synchronize in time.

They were introduced to model the diverse real-world systems which both sync and swarm, such as vinegar eels,[3] magnetic domain walls,[4] and Japanese tree frogs.

Swarmalation[6] occurs in diverse parts of Nature and technology some of which are discussed below.

Sperm, vinegar eels and potentially other swimmers such as C elegans swarm through space via the rhythmic beating of their tails.

This can lead to vortex arrays,[7] trains [8] metachronal waves [9] and other collective effects.

The hold great promise as memory devices in next generation spintronics.

model,[11] a domain wall can be described by its center of mass

Experiments reveal that the interaction between two such domain walls leads to rich spatiotemporal behaviors some of which is captured by the 1D swarmalator model listed above.

During courtship rituals, male Japanese Tree frogs attract the attention of females by croaking rhythmically.

Evidence[13] suggests this (anti)-synchronization influences the inter-frog spatial dynamics, making them swarmalators.

The resultant "sync-selected self-assembly"[15] gives rise novel superstructure with potential use in biomedicine contexts such as targeted drug delivery, bio imaging, and bio-sensing.

[16] Quincke rollers [citation needed] are a class of active particle that exhibits self-propelled motion in a fluid due to an electrohydrodynamic phenomenon known as the Quincke effect.

[17] This effect occurs when a dielectric (non-conducting) particle is subject to an electric field.

The rotation of the particle, combined with frictional interactions with the surrounding fluid and surface, leads to a rolling motion.

[19] Embryonic cells are the foundational building blocks of an embryo, undergoing division and differentiation to form the complex structures of an organism.

In the context of swarmalators, embryonic cells display a unique blend of synchronization and swarming behaviors.

[20] They coordinate their movements and genetic expression patterns in response to various cues, a process essential for proper tissue formation and organ development.

This linking of sync and self-assembly make embryonic cells a compelling example of a real-world swarmalators.

The linking of sync and swarming defines a new kind of bio-inspired algorithm which several potential applications.

This 2D swarmalator model in generic form is The spatial dynamics combine pairwise interaction

modified by a spatial term so the synchronization becomes position dependent.

In short, the swarmalators model the interaction between self-synchronization and self-assembly in space.

The above can be considered a blending of the aggregation model introduced from biological swarming[23] (the spatial part) and the Kuramoto model of phase oscillators (the phase part).

The model above produces five collective states[24] depicted in Figure 1: To demarcate where each state arises and disappears as a parameters are changed, the rainbow order parameters, where

are the (random) natural frequencies of the i-th swarmalator and are drawn from certain distributions

For instance, the model with natural frequencies can be solved by defining the sum/difference coordinates

and the rainbow order parameters are the equivalent of the 2D model For unimodal distribution of

such as the Cauchy distribution, the model exhibits four collective states depicted in the figure on the right.

Note in each state, the swarmalators split into a locked/drifting sub-populations, just like the Kuramoto model.

For the Kuramoto model (top row), the sync order parameter

Examples of real world swarmalators plotted in a (discipline, number of particles) plane
Figure 1: A depiction of the five collective states of the swarmalator model.
Figure 2: The rainbow order parameters and can discern the various collective states. See main text for definitions of these quantities.
Collective states of 1D swarmalator model, where swarmalators are depicted as dots in the plane.
Top row, sync order parameter versus phase coupling strength for the Kuramoto model with Cauchy natural frequencies with zero center and unit width. Bottom row, rainbow order parameters for the 1D swarmalator model also with unit Cauchy natural frequencies and . The black dotted lines mark the transitions between the async state, phase wave, and mixed state in that order. Data were collecting using an RK4 solver with timestep for time units with swarmalators.