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What physics can teach us about Pamplona’s annual running of the bulls

Physicists from Argentina and Spain have studied the pedestrian dynamics at the annual running of the bulls in Pamplona.

In his first and arguably most famous novel, The Sun Also Rises, Ernest Hemingway wrote tersely but lovingly about the annual running of the bulls in Pamplona, Spain, during the seven-day Festival of San Fermín. “One man fell, rolled to the gutter and lay quiet,” the protagonist, Jake, observes while watching from a balcony. “But the bulls went right on and did not notice him. They were all running together.” A new analysis of the physics behind the dynamics of the crowds running from the bulls takes the probability of falling into account, according to a recent paper published in the Proceedings of the National Academy of Sciences.

Local legend holds that the running of the bulls dates back to northeastern Spain it the early 14th century. Cattle herders at the time found that hurrying the beasts through the streets was an efficient way to move livestock from fields or barges to the market—or the bullfighting ring. Young men started racing in front of the charging bulls, competing to see who could make it safely to the bull pens without being overtaken (or worse, trampled and gored). The race eventually became part of the San Fermín festival and is now the festival’s most popular event.

Revelers run with Nunez del Cuvillo's fighting bulls during the eighth day of the San Fermin Running of the Bulls festival on July 13, 2017, in Pamplona, Spain.
Enlarge / Revelers run with Nunez del Cuvillo’s fighting bulls during the eighth day of the San Fermin Running of the Bulls festival on July 13, 2017, in Pamplona, Spain.

Gari Garaialde/Getty Images

Per official records, 15 people have died during the running of the bulls in Pamplona since 1910, usually from being gored. Sometimes bystanders can be injured or killed, too, especially if they’re trying to capture live footage of the event with their smartphones. That happened during a different bull run, this time in Villaseca de la Sagra, Spain. In 2015, a 32-year-old man was gored from behind while attempting to share his experience by way of a smartphone recording. The victim died of neck and thigh wounds. Hot tip: maybe running away from charging bulls isn’t the best time to try to snap a selfie.

As I’ve written previously, pedestrian traffic is a fascinating case study in dynamic collective behavior, and hence it holds much interest for physicists. (Back in September, a pair of then-recent papers on pedestrian traffic dynamics were honored with Ig Nobel Prizes in physics and kinetics, respectively.) Physicists typically model such systems as interacting matter particles, with social forces acting on people in similar ways to physical forces. But modeling such a complex system is difficult, in part because of a dearth of high-quality experimental data. This is especially true in the case of pedestrians fleeing from real danger—say, six bulls charging through the streets.

Although the running of the bulls is often offered as an example of so-called “competitive pedestrian dynamics,” Daniel Parisi—a co-author of the recent PNAS paper and a physicist at the Instituto Tecnologico de Buenos Aires in Argentina—noticed that the scenario had not yet been studied in detail. So he and his colleagues set about to rectify that deficiency. “Runners, first waiting for and then escaping from bulls, constitute a fascinating annual scenario of real fleeing pedestrians, becoming an invaluable opportunity for studying and understanding extreme pedestrian dynamics,” the authors wrote.

The researchers recorded two runs on two consecutive days on July 8 and 9, 2019, in two different spots on Estafeta Street. (The Pamplona running of the bulls was canceled in 2020 and 2021 because of the pandemic.) They were able to extract the trajectories of individual bulls and runners from those recordings for analysis. Runners (and spectators) began the morning milling about the route to the bull ring. Once the ring doors were opened, most of the spectators moved inside. That reduced street congestion since mostly just the runners remained, waiting for the bulls to be released.

The team observed a shockwave of running pedestrians moving at high speed a few seconds before the first bulls arrived, triggering the start of the race. Once the runners in front and the bulls behind them passed, a wake of runners with decreasing speed formed behind. Within 40 or 50 seconds, the system returned to a more normal situation of pedestrians walking at a leisurely place along the street.

Parisi et al. knew that many prior studies on pedestrian systems relied on a fundamental speed-density diagram to model such systems. The speed of a given group of pedestrians usually decreases as the density of the crowd increased. “Under ordinary circumstances, this behavior can be explained because people try to avoid physical contact and slow down when the available space reduces,” they wrote in their paper.

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