Skip to content

Pedestrians as interacting particles

February 8, 2015

I generally don’t like to use this blog to discuss my own scientific projects.  (Physics as a whole is much more interesting than my own meager contributions to it.)  But there is a recent project I was involved in that is getting a decent amount of attention from the popular press.  So I thought there might be some value in giving a description of it from the horse’s mouth (or, at least, the mouth of one of the horses).


The basic question behind this project is this:
Is it possible to model a crowd of people as a collection of interacting particles?
If so, what kind of “particles” are they?

Now, this question may seem silly to you.  Obviously, a human being isn’t a “particle” like an electron or a billiard ball.  Human motion (in most cases) arises from the workings of the human brain, and not from random physical forces.  So what would motivate someone to talk about humans as interacting particles?

The answer, at least for me personally, is that the motivation comes from watching videos like this one:


Basically, when you watch the movement of crowds at a large enough scale, the motion starts to look beautiful and familiar.  Perhaps something like the flow of a liquid:


Or maybe a granular fluid:


These apparent similarities are pretty exciting to a person like me (and to many others who are perhaps not a lot like me), in part because they imply the possibility of bringing old knowledge to a new frontier.  We have centuries of knowledge that was built to describe the physics of fluids and many-body systems, and now there is the hope that we can adapt it to say something useful about human crowds.

But all that those videos really demonstrate is that crowds and particle systems are visually similar, and so the question remains: is there really a good analogy between particle systems and human crowds?  If so, what kind of “particles” are we?


When you come down to it, the defining feature of a particle system is the “interaction law”, which is the equation that relates the energy E of interaction between two particles to their relative positions.  For example, for two electrons the interaction law is the Coulomb law E \propto e^2/r, while for two neutral atoms it is something like the Lennard-Jones potential, E \propto C_1/r^{12} - C_2/r^6.

So, in some sense, the question of “how do we describe human crowds as particle systems?” comes down to the question “what is the interaction law between pedestrians?”.


In fact, scientists have been interested in that question for a few decades now.  And, generally, the way they have approached it is to make some hypothesis about how the interaction law E(r) should look, and then make a big computer simulation of pedestrians following that law and see if the simulation behaves correctly.

This approach has gotten us pretty far — it has helped to save lives in Mecca and given us fantastic CGI crowd animations in movies and video games.  But it has also given rise to a sort of messy situation, scientifically, in which there are many competing models for crowds and their interaction law, with no generally accepted way of adjudicating between them.

What my colleagues and I eventually figured out is that we could take a different approach to this problem.  Instead of guessing what we thought the interaction law should be and then checking how well our guess worked, we decided to look first at real data and see what the data was telling us.  If there is, indeed, a universally correct equation for the interaction law between people, then it should be encoded in the data.

I’ll spare you the technical details of our data-digging, but the basic idea is like this.  In many-body physics, there is a general rule (the Boltzmann law) that relates energy to probability.  This law says, in short, that in a large system, any configuration of particles having a large energy is exponentially rare: its probability is proportional to \exp(-E\times \text{const.}).  So, for a crowd of people, looking at the relative abundance of different configurations of people tells you something about how much “energy” is associated with that configuration.  This allows you to infer the correct quantitative form of the interaction energy, by correlating the properties of different configurations with their relative abundance in the dataset.

So what we did is to first amass a large amount of crowd data.  This data was generally in the form of digitized video footage of people walking around crowded areas.  For example, we had data from students milling around on college campuses, shoppers walking around shopping streets, and even a few controlled experiments where people were recorded walking out of a crowded room.


An example of some of the data we looked at, showing different pedestrian “tracks”. The color corresponds to the time-averaged pedestrian density.


When we finished the analysis, there were two results that jumped out from the data very clearly: one obvious, and one surprising.

The first result is that the interaction law between pedestrians is not a function only on their relative distance r.

In hindsight, this result is  pretty obvious.  For example, two people walking headfirst into each other will feel a large “force” that compels them to move out of each other’s way.  On the other hand, two people walking side-by-side may feel no such force, even if they are relatively close to each other.

The implication of this result, though, is important.  It means that human pedestrians are very different from other, non-sentient “particles.”  An electron, for example, feels a force that is based on the physical proximity of other electric charges to itself at that particular instant.  Humans, on the other hand, respond to the world not as it is currently, but as they anticipate it to be in the near future.


Given the first, obvious result — that humans are not particles — the second result is much more surprising.  What we found is that there is, in fact, a very consistent interaction law between pedestrians in a crowd.  And it looks like this:

(\text{interaction energy}) \propto 1/(\text{projected time to collision})^2.

In other words, as pedestrians navigate around each other, they base their movements not on the physical distance between each other, but on the extrapolated time \tau to an upcoming collision.  What’s more, the form of their interaction energy has the very simple form E \propto 1/\tau^2.  That this interaction law could have such a remarkably simple form was quite surprising, and that it holds across a whole range of different environments, densities, and cultures, was even more surprising.

EtauIt’s remarkable that something so mathematically simple could describe what is essentially a psychological phenomenon.

That, in a nutshell, was the finding of our paper.  My colleagues went on to show how this simple rule could immediately be used to make fast and accurate simulations of pedestrian crowds.  You can check out some of their simulation videos here, but I’ll also embed one of my favorite ones:

Here, two groups of people are asked to walk perpendicularly past each other.  They manage to resolve their imminent collisions by spontaneously forming diagonal “stripes” that cut through each other.


What’s nice about this result is that it gives us, with some confidence, the first major building block needed for making a real theory of human crowds.  Now that we know the nature of the interaction between two individuals, we can start putting together a kinematic theory of how crowds move in the aggregate.  This has all sorts of practical importance, in terms of understanding and predicting crowd disasters before they happen, but it also opens up a variety of fun problems to the language of condensed matter physics.  Maybe, in addition to describing the bulk “flow” of crowds, we can talk about the emergent features (“quasiparticles“) of crowds, like lane formation and mosh pit vortices.

It will be fun to see how this field develops in the near future.



Of course, most of the credit for this work belongs to my co-authors, Ioannis Karamouzas and Stephen Guy, at the Applied Motion Lab at the University of Minnesota.  They did the hard work of suggesting the problem, doing (most of) the data analysis, and writing the computer simulations.  My role was mostly to insist on making the problem “sound more like physics.”


6 Comments leave one →
  1. February 20, 2015 3:37 pm

    Always fascinating to me when physics and mathematics intersect and even align with “the human experience”. Finding such a graceful equation to describe the dynamic is even more fascinating.

    Great Job!
    Arcadia Star

  2. asthearmyofhelaman permalink
    April 6, 2015 4:06 pm

    Very cool! I would think about this (at least tangentially) when I would walk between classes in college and would get stuck in traffic flows of people walking through bottle-necks, and up stairs, or when north/south traffic would cross east/west traffic, and when I would see wear patterns on the stairs and landings of the aging library. I’m interested to see where your research takes you and what applications you come up with!

  3. April 12, 2015 2:13 am

    Once again the eerie effectiveness of math! XD

    I spent a lot of time on the Los Angeles freeway system, and I used to pass that time wondering about the dynamics of freeway traffic — how waves of “excitation” seem to pass down the channel. It would have been fun to capture some large-scale freeway footage for analysis.

    • Brian permalink*
      April 12, 2015 10:23 am

      Yes, I am very much interested in doing this same kind of analysis for freeway traffic. Maybe soon!

      • April 12, 2015 1:37 pm

        By the time I left L.A. the freeway traffic pattern was mostly: Stopped 🙂

        One of your other posts had a link from a commenter to a “Jamatron” study (IIRC). Excellent link! I’d always wondered what caused those waves of braking. Looks like one answer is: Being in traffic. Very interesting!


  1. How thick is the atmosphere? A derivation of the Boltmzann distribution | Gravity and Levity

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: