A crowd at a metro station during rush hour. Some hurry toward the escalator, others slow down at the departure board, still others weave around a column. Each step is not merely a reaction to what's right in front of your nose. A person estimates in advance where a neighbor will be a second from now and adjusts their route accordingly. Physicists call this forward-looking behavior anticipatory. Until now, crowd models were built differently: people in them acted like gas particles — bouncing off each other, colliding, reacting only to the immediate surroundings. The new work adds anticipation itself to the equation.
The authors turned this anticipation into a mathematical construct. An ordinary pedestrian moves in our three-dimensional world, but their calculation of the future unfolds along an additional, imaginary dimension, like a thread stretching forward in time. Physically, such a thread resembles a polymer chain (a long molecule like DNA) that bends under the impacts of water molecules. Here, the 'impacts' are the uncertainty about what a neighbor will do in the next instant. The further into the future a pedestrian looks, the more their imaginary thread trembles. Beyond a certain limit, the anticipation horizon, the future becomes too blurry, and it's simpler to treat it as an averaged background.
The model was tested on real scenarios. People in the simulation navigate through obstacle-filled spaces, and do so almost like real people. Exiting a packed train car, where everyone squeezes through their own gap, also looks natural. The secret is that the model doesn't impose rigid rules like 'keep to the right.' It simply gives each agent the ability to assess future inconveniences and choose a path with fewer of them. Thus, the operational level (avoiding a puddle) and the tactical level (choosing a lane) merge into one.
This approach opens the door to simple and flexible simulations. It's easy to add new details to the model — for example, fatigue or the influence of a phone in hand. This means architects and transport engineers can more accurately predict how people will behave in a new building or station before it's even built.
In polymer physics, there's a concept called 'persistence length'—the distance over which a chain maintains its direction. For a pedestrian in a crowd, the analogue turned out to be the anticipation horizon: beyond it, the future is so hazy that a person stops factoring it into their steps.
Isaac Asimov, in 'Foundation,' described psychohistory — a science that predicts the behavior of vast masses of people. The new model takes a step toward a similar goal, but for small crowds: it predicts not the fate of an empire, but how a hundred people will exit a train car.