A Fluid-Dynamic Model for the Movement of Pedestrians

A fluid-dynamic description for the collective movement of pedestrians is developed on the basis of a Boltzmann-like gas-kinetic model. The differences between these pedestr ian-specific equat ions and those for ordinary fluids are worked out ; they concern, for example, the mechanism of relaxat ion to equilibrium, the role of "pressure," the special influence of internal friction, and the origin of "temperatur e." Some interesting results are derived that can be compared to real situat ions-for example, the development of walking lanes and of pedestri an jams, the propagation of waves, and behavior on a dance floor. Possible applications ofthe model to townand traffic-planning are outl ined. 1. Introduct ion Previous publications on the behavior of pedestrians have been predominantly empirical (often in the sense of regression analyses) , and were int end ed to facilitat e planning of efficient t raffic [16, 24, 34J . While there also exist theoretical approaches to pedestrian movement [2, 3, 7, 9, 12, 29, 32, 33], most t heoret ical work has been done in the related to pic of aut omobile traffic (see, for example, [1,6,8,26, 27, 28]). In particular , some Boltzmann-like (gas-kineti c) approaches have been developed [1, 26, 27]. The author has observed that footprints of pedestrian crowds in the snow and quick-motion pict ures of pedest rians resemble fluid streamlines. It is the object of this paper to give a suitable explanat ion of the fluid-dynamic properties of pedest rian crowds. Henderson was the first to apply gas-kinetic and fluid-dynamic models to empirical data of pedestrian crowds [12, 13, 14, 15J. His work , however , began with t he conventional theory for ordinary fluids , and assumed a conservation of mome ntum and energy. In cont rast to Hend erson's approach, this ar ticle develops a special theory for pedestrianswithout makin g use of unrealistic conservati on assumptions. Our pr ocedure is describ ed as follows. Pedestrians will be distinguished into groups of different typ es f1 of motion , nor mally represent ing different intended directions of walking . At a time t the pedest rians of each typ e f1 of