Traffic flow

From Freepedia

The mathematical study of traffic flow, and in particular vehicular traffic flow, is done with the aim to get a better understanding of these phenomena and to hopefully avoid some problems of traffic congestion. Even if the first attempts to give a mathematical theory of traffic flow dates back to the 50s, we still not have a satisfactory and general theory to be applied in real flow conditions.

This is because traffic phenomena are complex and nonlinear, dependending on the interactions of a large number of vehicles. Moreover, vehicles do not interact simply following the laws of mechanics, but thanks to the reactions of a human driver. In particular, they show phenomena of cluster formation and backward propagating shocks of vehicle density. Moreover, fluctuations in measured quantities (e.g. mean velocity of vehicles) are often huge, leading to a difficult understanding of experiments.

So now the modelling of traffic flow is one of the most challenging themes of mathematical physics.

Scientists approach the problem in mainly three ways, correspnding to the three main scales of observation in physics.

• microscopic scale: you write for every vehicle a model, i.e. an equation, that is usually an ODE.
• macroscopic scale: in analogy with fluid dynamics models, you write a system of (PDE) balance laws for some gross quantities of interest, e.g density, mean velocity, flux.
• mesoscopic (kinetic) scale: you define a function <math>f(t,x,V)</math> which expresses the probability of having a vehicle at time <math>t</math> in position <math>x</math> which runs with velocity <math>V</math>. This function, following methods of statistical mechanics, can be computed solving an integro-differential equation, like the Boltzmann Equation.

References

A survey about the state of art in traffic flow modelling

• N. Bellomo, V. Coscia, M. Delitala, On the Mathematical Theory of Vehicular Traffic Flow I. Fluid Dinamic and Kinetic Modelling, Math. Mod. Meth. App. Sc., Vol. 12, No. 12 (2002) 1801-1843

A good book from the physical point of view

• B. Kerner, The Physics of traffic, Springer Verlag (2004)