Dynamics and Thermodynamics of Traffic Flow

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 to

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Lule˚ a University of Technology, Department of Physics, SE–97187 Lule˚ a, Sweden [email protected] Rostock University, Institute of Physics, D–18051 Rostock, Germany [email protected]; [email protected] Institute of Mathematics and Computer Science, University of Latvia, LV–1459 Riga, Latvia [email protected]

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.

1 Introduction It has already been manifested by Mahnke et al. [1] that the thermodynamic approach can be applied to such a many–particle driven system as traffic flow. Investigations of the complex system called vehicular traffic can be done on the microscopic level (dynamical equations like Bando’s optimal velocity model), on the mesoscopic level by stochastic equations [2] as well as macroscopically applying equilibrium statistical physics [3]. Recently Krb´ alek [4] uses the one–dimensional thermodynamical traffic gas to predict clearance (also called spacing or headway) comparing with empirical traffic data. It has been stated that the derived exact formula for thermal–equilibrium spacing distribution of a traffic gas with a short–range repulsive two–body potential

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Hans Weber, Reinhard Mahnke, Christof Liebe, and Jevgenijs Kaupuˇzs

V (Δx) = 1/Δx is in successful agreement with highway traffic clearance distributions. Depending on vehicular density c the temperature T (or its inverse β), which acts as a fit parameter, changes its value.

2 Bando’s Optimal Velocity Model: Dynamics The dynamical system called Bando’s optimal velocity model [5, 6] is a set of 2N coupled differential equations acting on a microscopic level, i. e. each car i = 1, . . . , N has its own position x and velocity v equation. The ensemble of cars is subjected to periodic boundary conditions, see Fig. 1.

Fig. 1. One–dimension