# Introduction

Methods from physics have been successfully used for the investigation of vehicular traffic for a long time.
On the other hand, pedestrian dynamics has not been studied as extensively. Due to its generically two-dimensional nature, pedestrian motion is more difficult to describe in terms of simple models. However, many interesting collective effects and self-organization phenomena have been observed:

1.) Jamming:

At large densities various kinds of jamming phenomena occur, e.g. when many people try to leave a large room at the same time. This clogging effect is typical for a bottleneck situation.

Other types of jamming occur in the case of counterflow where two groups of pedestrians mutually block each other. This happens typically at high densities and when it is not possible to turn around and move back, e.g. when the flow of people is large.

2.) Lane formation:

In counterflow, i.e. two groups of people moving in opposite directions, a kind of spontaneous symmetry breaking occurs. The motion of the pedestrians can self-organize in such a way that (dynamically varying) lanes are formed where people move in just one direction.

In this way, strong interactions with oncoming pedestrians are reduced and a higher walking speed is possible.

3.) Oscillations:

In counterflow at bottlenecks, e.g. doors, one can observe oscillatory changes of the direction of motion. Once a pedestrian is able to pass the bottleneck it becomes easier for others to follow him in the same direction until somebody is able to pass (e.g. through a fluctuation) the bottleneck in the opposite direction.

4.) Patterns at intersections:

At intersections various collective patterns of motion can be formed. A typical example are short-lived roundabouts which make the motion more efficient. Even if these are connected with small detours the formation of these patterns can be favourable since they allow for a ''smoother'' motion.

5.) Trail formation:

In counterflow, i.e. two groups of people moving in opposite directions, a kind of spontaneous symmetry breaking occurs. The motion of the pedestrians can self-organize in such a way that (dynamically varying) lanes are formed where people move in just one direction. In this way, strong interactions with oncoming pedestrians are reduced and a higher walking speed is possible.

6.) Panics:

In panic situations many counter-intuitive phenomena (e.g. ''faster-is-slower'' and ''freezing-by-heating'' effects) can occur.

We have introduced a new kind of CA model which -- despite its simplicity -- is able to reproduce the observed collective effects. This is essential if one intends to use the model for real applications, e.g. the optimization of evacuation procedures.
The model takes its inspiration from the process of chemotaxis. Some insects create a chemical trace to guide other individuals to food places. This is also the central idea of the active-walker models used for the simulation of trail formation. In our approach the pedestrians also create a trace. In contrast to trail formation and chemotaxis, however, this trace is only virtual although one could assume that it corresponds to some abstract representation of the path in the mind of the pedestrians.
Its main purpose is to transform effects of long-ranged interactions (e.g. following people walking some distance ahead) into a local interaction (with the ''trace''). This allows for a much more efficient simulation on a computer.

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