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The first attempts to give a mathematical theory of traffic flow dated back to the 50s, but to this day we still do not have a satisfactory and general theory to be applied in real flow conditions.

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

Thus, 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: at a first level, every vehicle is considered as an individual, and therefore for everyone is written an equation, that is usually an ODE .

  • macroscopic scale: in analogy with Fluid Dynamics models, it is something more useful to write a system of ( PDE ) balance laws for some gross quantities of interest, e.g the density of vehicles or their mean velocity.

  • mesoscopic (kinetic) scale: a third, intermediate, possibility, is to define a function f(t,x,V) which expresses the probability of having a vehicle at time t in position x which runs with velocity V. This function, following methods of Statistical Mechanics , can be computed solving an integro-differential equation, like the Boltzmann Equation .



SEE ALSO

Traffic Wave

Microsimulation


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 useful book from the physical point of view:

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