Pedestrian Crowd Dynamics | mysimulator.uk

People as particles with a will of their own

A crowd is not a fluid and it is not a gas, but the physics-based approach to modelling one borrows heavily from both. In 1995, Dirk Helbing and Péter Molnár proposed the social-force model: treat each pedestrian as a point particle subject to Newton's second law, but let one of the "forces" acting on it come from the pedestrian's own intention to reach a goal, not from any physical field. Combine that self-driven force with realistic short-range repulsion from other people and from walls, and remarkably lifelike crowd behaviour falls out of the simulation without anyone scripting it in.

live demo · a crowd self-organising through a corridor● LIVE

Three forces, added together

Every pedestrian's acceleration is the sum of a driving term and repulsion terms, all evaluated every simulation step:

m · dv/dt = F_drive + Σ F_person(other pedestrians) + Σ F_wall(obstacles)

F_drive  = (v_desired · direction_to_goal - v_current) / τ      // relaxation toward the goal
F_person = A · exp((r_ij - d_ij) / B) · n_ij                    // exponential repulsion, short range
F_wall   = same exponential form, measured against the nearest obstacle point

The driving term F_drive is not a physical force at all — it is a mathematical stand-in for intention, pulling the pedestrian's velocity toward their desired speed and direction over a relaxation time τ, typically around half a second. The repulsion term F_person decays exponentially with distance d_ij between two pedestrians, so it is strong at close range — modelling the discomfort of someone standing in your personal space — and negligible once people are a few body-widths apart, which is what keeps the model from turning every crowd into a rigid, evenly spaced lattice.

Patterns nobody programmed in

The striking thing about the social-force model is how many real crowd phenomena appear without being explicitly coded — they emerge from thousands of simple pairwise repulsions:

LANE FORMATION   two-way corridor traffic spontaneously splits into
                 lanes moving in alternating directions

STOP-AND-GO      dense flow through a narrow passage self-organises
WAVES            into oscillating clusters of movement and stalling

ARCHING /        at a bottleneck, pressure builds into an arch-shaped
CLOGGING         cluster around the opening, intermittently releasing

FASTER-IS-SLOWER above a critical desired speed at an exit, higher
EFFECT           urgency increases jamming and REDUCES total outflow

Lane formation happens because it is locally cheaper, in terms of repulsive force encountered, to fall in behind someone already walking your direction than to keep pushing through oncoming pedestrians — a purely local, force-minimising choice that produces a large-scale organised pattern. The faster-is-slower effect is the most consequential of these for building safety: as desired speed rises at a narrow exit, the repulsive forces between people trying to funnel through increase, producing friction-like contact forces that clog the opening — meaning that in a panic, everyone trying to move faster can make the whole crowd's exit rate go down, not up.

Where this actually gets used

Variants of the social-force model are used directly by structural and fire-safety engineers to test whether a stadium, subway platform, or building floor plan can evacuate within a required time, feeding into egress requirements before construction. Urban planners use it to design station concourses and pedestrian crossings that avoid dangerous bottlenecks under peak load, and film and game studios use closely related crowd-simulation techniques to animate large numbers of individually behaving background characters without hand-animating each one. The model's core insight — that a small number of simple, local interaction rules can reproduce complex, large-scale collective behaviour — is also one of the more direct real-world illustrations of emergence in complex systems.

Frequently asked questions

Why do pedestrian lanes form automatically in a busy corridor?

Lanes are a self-organised pattern, not something anyone plans. When people moving in opposite directions repel each other, it is locally cheaper to fall in behind someone already going your way than to keep pushing through oncoming traffic, since that avoids most close, force-generating encounters. That local preference, repeated over many pedestrians, spontaneously segregates the flow into alternating lanes without any central coordination.

What exactly is the faster-is-slower effect?

It is the finding, both in the social-force model and in real evacuation data, that pushing individual desired walking speed above a certain point at a narrow exit makes the overall outflow slower, not faster. Higher desired speed increases the repulsive forces at the bottleneck, which increases friction-like contact and clogging, so more people trying to squeeze through faster ends up jamming the exit and reducing throughput.

Is the social-force model still used for real building safety design?

Yes, in modified and extended forms. Modern evacuation software builds on the same core idea — a goal-directed driving force plus short-range repulsion from other people and obstacles — often combined with route-choice models, and is used to test stadium exits, stations and building floor plans against fire-code egress requirements before they are built.

Try it live

Everything above runs in your browser — open Pedestrian Crowds and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.

▶ Open Pedestrian Crowds simulation

Pedestrian Crowds: The Physics Behind Lane Formation and Deadly Bottlenecks