# Three-body encounter rate in one dimension

Walking long and mildly busy streets in Budapest, I noticed that 3-body encounters were surprisingly common. 2-body encounters are when two people walking in opposite directions meet, or they walk in the same direction and one passes the other. These seemed rare enough (only a few per minute on my daily route), so I got curious how come a relatively large fraction of them involved a third person walking independently. The full mathematical description of the problem can be found here and a simple applet to calculate the encounter rate using a toy model can be found here.

In more detail, a 3-body encounter is defined as the moment in time at which the largest pairwise separation among the three bodies is the smallest, and such an encounter is considered “interesting” if this separation is below some threshold ε. In the paper, I calculated the rate of 3-body encounters undergone by a reference body with velocity v0, given that all bodies move at constant speeds in one dimension, their density is n per unit length, and the probability density function of their velocities is f(v). The result is that the rate linearly depends on ε and quadratically on n.

In figure shows the 2- and 3-body encounter rates as a function of the reference body’s speed in a toy model representing pedestrian movement. The other bodies’ velocity distribution is made up of two rectangular functions with widths of 2 km h-1 around ±5 km h-1 (also n = 120 km-1 representing a mildly busy street in Budapest, and ε = 0.5 m representing a lower limit on comfortable personal space).