Order to Chaos: The Fascinating Math That All Pedestrians Need to Know
taylorvich writes:
https://newatlas.com/science/pedestrian-walking-math/
Have you ever wondered why walking from point A to B can be easy in some places, and incredibly frustrating in others? Well, scientists were curious about it too, and have now worked out the mathematical equation that underpins how we navigate sidewalks, and why, collectively, it descends into chaos so quickly.
Massachusetts Institute of Technology (MIT) researchers have uncovered the scientific logic at the root of pedestrian traffic flow, and it's more fascinating than you might think. And it's not the number of people, but their individual behavior, that drives it.
In 2021, MIT instructor of applied mathematics Karol Bacik, who is also trained in fluid dynamics and granular flow, published research on crowd flow and social distancing. This gave him and his team the idea to study pedestrian behavior more acutely. Two years later, Bacik and researchers looked at "lane formation" - which is observed in particles, grains and ultimately people. Essentially, they found that if it looked like a lane was forming in a crowd, people would either join it or walk parallel to it.
Back then, they used the model of a crosswalk to develop a theory as to how much "angular spread" - pedestrians veering off these invisible lanes - it would take for order to descend into chaos. And they found that all it took was someone to veer off this lane by around 13 degrees to sow the seeds of collective disorder.
"Now we're asking, how robust is this mechanism?" Bacik said. "Does it only work in this very idealized situation, or can lane formation tolerate some imperfections, such as some people not going perfectly straight, as they might do in a crowd?"
In the new study, the team switched focus from what degree it takes to cause disorder, to instead looking at the moment this disorder is triggered.
"If you think about the whole crowd flowing, rather than individuals, you can use fluid-like descriptions," Bacik explained. "It's this art of averaging, where, even if some people may cross more assertively than others, these effects are likely to average out in a sufficiently large crowd. If you only care about the global characteristics like, are there lanes or not, then you can make predictions without detailed knowledge of everyone in the crowd."
Using fluid flow, the researchers altered the parameters to apply to the crosswalk and pedestrians, and simulated the foot traffic with a group of people crossing the floor of a gymnasium as they would a road.
Out of this, they found that pedestrians in a crosswalk are more likely to form lanes, when they walk relatively straight across, from opposite directions. This order largely holds until people start veering across at more extreme angles. Then, the equation predicts that the pedestrian flow is likely to be disordered, with few to no lanes forming.
The study was published in the Proceedings of the National Academy of Sciences: https://dx.doi.org/10.1073/pnas.2420697122
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