How can public spaces remain safe when large crowds move through them? Engineers and researchers who study these environments often rely on physical models borrowed from fluid dynamics — a branch of physics that describes the collective motion of fluids, whose behaviour emerges from the interactions of many particles.
But a new study published in the Journal of Statistical Physics: Theory and Experiment (JSTAT) highlights a crucial issue: the way data are collected and measured within these models lacks standardisation and may overlook important features of human collective behaviour. Unlike particles, people are living agents with individual decisions and complex interactions, making their movement harder to capture with traditional approaches.
In their study, the authors propose and experimentally test new methods to address these limitations, comparing them with more established techniques. Their results point the way toward clearer methodological guidelines and the development of more reliable tools for those involved in the design and management of public spaces.
Humans as physical particles (or not?)
“Our field of research is pedestrian dynamics,” explains Juliane Adrian, researcher at the Institute for Advanced Simulation 7: Civil Safety Research, Forschungszentrum Jülich (Germany), and first author of the study. “We want to understand when a situation turns from normal to dangerous,” Adrian says. “But humans are not physical particles. They have free will.”
Traditionally, the analysis starts from the so-called flow equation, which combines the density of people and their speed. Based on this relationship, researchers build a “fundamental diagram” of a given space, used to assess under which conditions congestion or dangerous crowding may occur.
“There is a certain point where you have an optimum flow at an optimum density,” Adrian explains. “But at some point, this tips over: if the density increases further, the flow decreases because people can no longer move freely. They need to stop, take detours, and adjust their speed because there are so many other people around.”
For this reason, obtaining a realistic fundamental diagram is crucial for correctly interpreting crowd dynamics and designing safer spaces. But this is also where the problem lies.
How data are measured matters
“The flow equation and the fundamental diagram are not problematic in themselves,” Adrian explains. “The real issue is how we measure quantities like speed, density and flow.”
By reviewing the existing literature, Adrian and colleagues found what she describes as “a general lack of consensus” in measurement approaches — something that can lead to significant differences in how even very similar situations are interpreted.
Traditional methods tend to work well in simple scenarios, for instance when everyone moves in the same direction. But real crowds are rarely that simple. “There might be inhomogeneities in the crowd. There might be counter movement,” Adrian says. “For example, in a bidirectional stream, some people walk in one direction while others move in the opposite one. And people may also change their mind, turn around, or move back and forth. On top of that, pedestrian crowds can reach very high densities.”
In these more complex situations, standard approaches can become unreliable — and may even detect movement where there is effectively none. “If it’s really dense, even if people stand still, there might still be movement in the crowd,” Adrian explains. “People might lean or move slightly, so if you measure their speed, it can appear as motion in different directions — even in the opposite one.”
The experiments
In their experiments, Adrian and colleagues recorded groups of people walking in controlled environments — such as corridors or open spaces — using overhead video cameras. Dedicated software was then used to reconstruct the individual trajectories of each participant, treated in a simplified way as moving points.
Starting from these trajectories, the researchers calculated key quantities such as speed, density and flow. The novelty, however, lies in how these quantities are defined and measured.
To describe crowd movement consistently, Adrian and colleagues adopted an approach based on Voronoi cells — dividing space so that each person is assigned the area closest to them. This makes it possible to define quantities like density, speed and flow consistently at the same point in space. “We divide the space into regions to define these quantities consistently,” Adrian explains.
A crucial aspect of the method is that all quantities are measured at the same place and at the same moment. This avoids a key limitation of traditional approaches, where density, speed and flow are calculated in different ways — for instance, density over an area and flow over time — making them difficult to compare directly. “We can have measurements at the same location, same time point,” Adrian explains.
On this basis, the researchers build continuous fields similar to those used in fluid physics and apply the continuity equation — a “conservation law that ensures that no pedestrian simply appears or disappears,” as Adrian puts it — to describe how people are distributed and move through space.
Finally, Adrian and colleagues compared their approach with traditional methods using experimental data, showing that the differences become particularly significant at high densities, when crowds are more prone to congestion and complex collective behaviours.
A more accurate description of collective motion
Based on their work, Adrian and colleagues conclude that how density, speed and flow are measured in crowds matters far more than previously assumed — especially in critical situations.
In particular, they show that traditional methods — which combine averages taken over space (for density) and over time (for flow) — can produce inconsistent or even misleading results, especially when crowds become dense or start to congest. Under these conditions, the different quantities are no longer fully compatible, and the so-called fundamental diagram — the relationship between density, speed and flow — can become distorted.
Their approach instead provides a more reliable description of collective motion, capturing effects that are often hidden by standard methods, such as local slowdowns, oscillations, or even complete crowd standstills.
The paper “Pedestrian Flow Analysis in High-Density Crowds: Continuity Equation with Voronoi-Based Fields” by Juliane Adrian, Ann Katrin Boomers, Sarah Paetzke and Armin Seyfried is now available in JSTAT.
Journal of Statistical Mechanics Theory and Experiment
Experimental study
Pedestrian Flow Analysis in High-Density Crowds: Continuity Equation with Voronoi-Based Fields