  Human crowds often bear a striking resemblance to interacting particle
systems, and this has prompted many researchers to describe pedestrian dynamics
in terms of interaction forces and potential energies. The correct quantitative
form of this interaction, however, has remained an open question. Here, we
introduce a novel statistical-mechanical approach to directly measure the
interaction energy between pedestrians. This analysis, when applied to a large
collection of human motion data, reveals a simple power law interaction that is
based not on the physical separation between pedestrians but on their projected
time to a potential future collision, and is therefore fundamentally
anticipatory in nature. Remarkably, this simple law is able to describe human
interactions across a wide variety of situations, speeds and densities. We
further show, through simulations, that the interaction law we identify is
sufficient to reproduce many known crowd phenomena.
