Researchers analyze flocking behavior on curved surfaces

September 7, 2017 by Julie Cohen, University of California - Santa Barbara
Steady flocks on a sphere and a catenoid. Credit: Suraj Shankar

A murmuration of starlings. The phrase reads like something from literature or the title of an arthouse film. In fact, it is meant to describe the phenomenon that results when hundreds, sometimes thousands, of these birds fly in swooping, intricately coordinated patterns through the sky.

Or in more technical terms, flocking.

But birds are not the only creatures that flock. Such behavior also takes place on a microscopic scale, such as when bacteria roam the folds of the gut. Yet bird or bacteria, all flocking has one prerequisite: The form of the entity must be elongated with a "head" and "tail" to align and move with neighbors in an ordered state.

Physicists study flocking to better understand dynamic organization at various scales, often as a way to expand their knowledge of the rapidly developing field of active matter. Case in point is a new analysis by a group of theoretical physicists, including Mark Bowick, deputy director of UC Santa Barbara's Kavli Institute for Theoretical Physics (KITP).

Generalizing the standard model of flocking motion to the curved surface of a rather than the usual linear plane or flat three-dimensional space, Bowick's team found that instead of spreading out uniformly over the whole sphere, arrowlike agents spontaneously order into circular bands centered on the equator. The team's findings appear in the journal Physical Review X.

"Whether it's bacteria swarming, cells roaming or energy-consuming 'arrows' flying, these systems share universal characteristics independent of the precise size and structure of the agents as well as their detailed interactions," said corresponding author Bowick, who is on leave from Syracuse University while in his role at KITP. "The ordered states of these systems are never perfectly uniform, so fluctuations in density generate sound, much in the same way that wind instruments create music."

On curved surfaces, the team, which includes KITP general member Cristina Marchetti and KITP graduate fellow Suraj Shankar, found "special" sound modes that don't dissipate and flow around obstacles. According to Bowick, these special modes correspond to special harmonics or tones that don't mix with all the other harmonics.

He also noted that these modes are special precisely because the band geometry of the equator is very different from the planar geometry of a flat surface. For example, a particle moving on a ring comes back to its starting point even though it moves along a "straight" path. This doesn't happen on a plane, where entities continue forever in a straight line, never to return, unless they encounter an edge. This feature is a direct consequence of the very different topology of the sphere and the plane.

"Even though a sphere itself has no edge, the swarming patterns have an edge—the edge of the band," Bowick said. "So simply by locally consuming energy, active agents on the sphere spontaneously swarm and create an edge."

The authors also analyzed another curved shape, an hourglass-shaped figure called a catenoid. Unlike a sphere on which parallel lines converge, the catenoid's concave curvature causes parallels to diverge. This opposite curvature pushes the flocking entities and associated sound waves to the top and bottom edges of the hourglass, leaving the middle bare—the opposite of what happens on a sphere.

"Just the fact that these systems flock is pretty remarkable because they dynamically generate motion," said Shankar, a doctoral student in the soft matter program in Syracuse University's physics department. "But they are far richer systems than we expected because they also generate these 'topologically protected' sound modes."

Explore further: Bacteria never swim alone

More information: Topological Sound and Flocking on Curved Surfaces, Phys. Rev. X 7, 031039 – Published 7 September 2017, … 03/PhysRevX.7.031039

Related Stories

Bacteria never swim alone

July 14, 2017

Many animal species display flocking behaviour, but the fact that microorganisms do is not as well known. Researchers at Lund University in Sweden have now shown that algae and bacteria form flocks at very low concentrations ...

Scientists discover novel way to 'heal' defects in materials

October 2, 2012

(—In a paper just published in Nature Materials, a team of researchers that includes William T.M. Irvine, assistant professor in physics at the University of Chicago, has succeeded in creating a defect in the structure ...

Researchers investigate how light behaves in curved space

January 15, 2016

To investigate the influence of gravity on the propagation of light, researchers usually have to examine astronomical length scales and huge masses. However, physicists at Friedrich-Alexander-Universität Erlangen-Nürnberg ...

Physicists grow pleats in two-dimensional curved spaces

December 23, 2010

( -- A design feature well known in skirts and trousers has now been identified in curved, two-dimensional crystals. As University of Chicago physicist William Irvine and his colleagues report in this week’s ...

Quantum cycles power cold-atom pump

May 26, 2016

The idea of a pump is at least as old as the ancient Greek philosopher and scientist Archimedes. More than 2000 years ago, Archimedes allegedly invented a corkscrew pump that could lift water up an incline with the turn of ...

Recommended for you

New study could hold key to hack-proof systems

July 17, 2018

Major data breaches have made worldwide headlines of late but an international consortium of scientists—including a professor from Heriot-Watt—have developed a new technique that could result in hack-proof systems.


Please sign in to add a comment. Registration is free, and takes less than a minute. Read more

Click here to reset your password.
Sign in to get notified via email when new comments are made.