Some things are just fated. Any article that opens with "In terms of its large-scale behaviors, a crowd of pedestrians can look strikingly similar to many other collections of repulsively interacting particles," is going to be a classic. It is my solemn duty to bring you the low-down on repulsive pedestrians.

Yes, apparently, determining the repulsiveness of your average pedestrian is not just a question of science but, indeed, one of physics. You have to wonder what the unit of repulsiveness is. Fortunately, the paper in question is, unlike me, serious.

Understanding the behavior of pedestrians is a reasonably important problem. It determines where, and how many, emergency exits a building requires. It tells us how the lack of queuing culture slows everything down. It tells us when flow is congested, where people will stand and wait, and how that waiting will make the jam worse.

In physics, we often model a collection of particles by programming in a set of attractive and repulsive interactions. For instance, water molecules are, on average, neutral. But, because oxygen attracts electrons so strongly, the oxygen part of water is slightly negatively charged, while the hydrogen atoms are slightly positively charged. Hence, when two water molecules interact, they may gently attract or repel each other, depending on their relative orientation. However, unless the alignment between the molecules is very precise and not disturbed, the molecules will rotate such that the force between them ends up being an attractive one.

Once the water molecules get close enough to each other, the electrons start to notice each other, and a strong repulsive force takes over. Hence, in water, clusters continually form and break up, with each cluster lasting just a few picoseconds.

It is rather... attractive... to apply these ideas to human pedestrians. Just think of them as particles, moving in a flow of particles. They are driven by a pressure to go from their point of origin to their destination. But, if they get too close to another particle, they are repelled.

This sounds like a good model, but it is entirely artificial. There is no physical or psychological argument from which these repulsive potentials can be derived. As a result, researchers have spent a lot of time guessing rules. But, since pedestrian behavior is rather complicated, these rules can get increasingly complicated, making it difficult to apply them outside of the datasets from which they were derived.

Now a new suggestion about how to manage this complexity has entered the ring. Unlike previous attempts, this one has the benefit of simplicity. The behavior of pedestrians can be predicted by calculating, for each pair of pedestrians, the time to collision. The researchers used video tracking software to calculate the separation between pedestrians. From that, they calculated the probability of finding two pedestrians separated by a given distance. For any given pair of pedestrians, they can also calculate their approach velocity and time to collision.

Examining the data shows several interesting phenomena. First, groups and couples walking together maintain a fairly constant separation (well, duh). Folks that are approaching each other take action to avoid collision—slowing down, changing direction, or both. But the timing of when they do so seems to depend on their speed. From the raw data, it is pretty difficult to predict what a given pedestrian will do.

Instead the researchers analyzed the behavior in terms of how long each pedestrian was from a collision were they to maintain their current trajectory and speed. Looked at this way, all the data appears the same. People always seem to take action such that the time to collision is always greater than about 200-500 milliseconds.

The point is that the distribution function that describes the probability of finding two pedestrians separated by a certain distance is different for every pedestrian when considered in terms of his or her own motion. But, when it is considered as a function of the time to collision, it is the same for everyone. Hence, this one simple probability distribution function may be highly predictive of pedestrian motion.

To demonstrate this, the researchers modeled crowds moving through certain spaces. Each pedestrian was given a starting location, a starting time, a destination, and a desired speed. They showed that this simple rule predicts the classic arc of confusion that forms at bottlenecks (because no one but the British know how to queue). They showed that pedestrians naturally form lanes moving in a common direction. Their model even predicts the garden variety asshat that moves against the flow of a lane. And, finally, they show that if you put a bunch of people in a room with no destination, they will naturally circulate in a particular direction.

So, what makes this different from other models of crowd behavior? The main difference is the simplicity of the model. The strength of the reaction of a pedestrian to another is only given by a single factor: the time to collision. This is then factored into a fake interaction energy, which is used to define a force field in which pedestrians move. This is still as artificial as any other physics-based approach (because these forces are not real, they are generated through our response to the environment), but it has the advantage of being based on measured behavior and minimal fine-tuning. The fact that it successfully predicts some pedestrian behaviors without any fine-tuning means that we might actually be able to use it outside of the datasets used to generate the rules.

Physical Review Letters, 2014, DOI: 10.1103/PhysRevLett.113.238701