Hidden order in chaotic crowds

Have you ever wondered how pedestrians “know” to cross lanes when moving through a crowd, without the subject being discussed or even consciously thought about?

A new theory developed by mathematicians at the University of Bath led by Professor Tim Rogers explains this phenomenon and is able to predict when the stripes will be curved as well as straight. The theory can even describe the slope of a busy lane when people have a habit of passing on one side rather than the other (for example, in a situation where they are often reminded to “pass on the right”).

This mathematical analysis unifies conflicting views on the origin of streak formation and reveals a new class of structures that may go unnoticed in everyday life. The discovery, reported in Scienceis a major advance in the interdisciplinary science of “active matter”—the study of group behaviors in interacting populations ranging in scale from bacteria to herds of animals.

Tested in arenas

To test their theory, the researchers asked a group of volunteers to walk through an experimental arena that mimicked different layouts, with changes to the entrance and exit gates.

An arena was created in the style of King’s Cross station in London. When the researchers reviewed the video from the experiment, they noticed mathematical patterns forming in real life.

Professor Rogers said, “At a glance, a crowd of pedestrians trying to pass through two gates may look disorderly, but when you look more closely, you see the hidden structure. Depending on the layout of the space, you can observe either the classical straight stripes or more complex curved patterns such as ellipses, parabolas and exaggerations.”

Lane formation

Single-envelope processions formed at busy zebra crossings are just one example of lane formation, and this study is likely to have implications for a number of scientific disciplines, particularly the fields of physics and biology. Similar structures can also form from inanimate molecules, such as charged particles or organelles in a cell.

Until now, scientists have offered many different explanations for why human crowds and other active systems naturally self-organize into strips, but none of these theories have been verified. The Bath team used a new analytical approach, inspired by Albert Einstein’s theory of Brownian motion, which makes testable predictions.

Encouraged by the way their theory agreed with numerical simulations of colliding particles, they teamed up with Professor Bogdan Bacik – an experimentalist from the Academy of Physical Education in Katowice, Poland – and carried out a series of experiments (like the one modeled on King’s Cross) using human crowds.

Lead author Dr. Karol Bacik said: “Strip formation does not require conscious thought – the participants in the experiment were not aware that they had arranged themselves into well-defined mathematical curves.”

“Class emerges spontaneously when two groups with different goals intersect in a crowded space and try to avoid colliding with each other. The cumulative effect of many individual decisions inadvertently leads to the formation of lanes.”

The researchers also looked at the effects of externally imposed traffic rules—that is, they instructed participants to pass others on the right. Consistent with the theoretical prediction, the addition of this rule changed the band structure.

“When pedestrians prefer right turns, the lanes end up leaning and that causes frustration that slows people down,” Dr Bacik said.

“What we have developed is a pure mathematical theory that predicts the propensity for banding in any given system,” Professor Rogers said. “We now know there is a lot more structure than we previously thought.”