Dynamics and Thermodynamics of Traffic Flow

Summary

Application of thermodynamics to traffic flow is discussed. On a microscopic level, traffic flow is described by Bando’s optimal velocity model in terms of accelerating and decelerating forces. It allows us to introduce kinetic, potential, as well as a total energy, which is the internal energy of the car system in view of thermodynamics. The total energy is however not conserved, although it has a certain value in any of the two possible stationary states corresponding either to a fixed point or to a limit cycle solution in the space of headways and velocities.

On a mesoscopic level of description, the size n of a car cluster is considered as a stochastic variable in the master equation for the system. Here n=0 corresponds to the fixed point solution with no cluster of the microscopic model, whereas the limit cycle is represented by the coexistence of a car cluster with n>0 and a free flow phase.

The stationary solution obeys a detailed balance condition, which allows us to describe some properties of the model by equilibrium thermodynamics in analogy to the liquid–vapour system. We define the free energy and the chemical potential of the car system. In this sense the behaviour of traffic flow can be described by equilibrium thermodynamics in spite of the fact that it is a driven system.