Models of traffic flow
(Patrick Dorey)

Description

If the density of cars on a stretch of road is too great, traffic jams can appear almost out of nowhere, and disappear again with a similar lack of apparent cause. This might seem to be so dependent on the behaviours of individual drivers that a mathematical analysis would be impossible, but, working with simplified models, many such features of traffic flow have now been understood.

The aim of this project is to survey some of this work and maybe do some experiments of your own as well. Traffic models tend to fall into two classes - 'fluid-dynamical' models which treat the traffic as a continuous density, and 'microscopic' models which focus on individual cars using ideas taken from statistical mechanics (a nice example which you might examine is known as the 'asymmetric simple exclusion process', or ASEP). Either approach could be emphasised, with the second offering particular opportunities for independent work.

Prerequisites

For the continuous models, the partial differential equations course will be a help; for the ASEP, ideas from 1H probabilty will be relevant. A willingness to use a computer would help in the more experimental side of the project.

Resources

• The video above is from this article in the New Journal of Physics, and was described in a New Scientist article here.
• A good introductory discussion of the fluid-dynamical models is chapter 22 of the book Towing Icebergs, Falling Dominoes, and Other Adventures in Applied Mathematics, by Robert B Banks (Princeton 1998).
  At about the same level is this article in Scientific American 261 (August 1989) 98-101.
• Microscopic models are discussed in great detail in Physics Reports 329 (May 2000) 199-329. While this article is at research level, the introductory sections are worth reading.
• For some material more directly focussed on the ASEP, the first two chapters of this review article by Blythe and Evans are worth a look.
• Further areas to explore include the so-called bus route model...
• Some other on-line links (complete with animations) are here, here and here.