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Velocity statistics of the Nagel-Schreckenberg model
Nicolas Bain1, Thorsten Emig1, Franz-Josef Ulm1
1Multi-Scale Materials Science for Energy and Environment, 〈MSE〉2, UMI 3466, The Joint CNRS-MIT Laboratory, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA.
Abstract:
The statistics of velocities in the cellular automaton model of Nagel and Schreckenberg for traffic are studied. From numerical simulations, we obtain the probability distribution function (PDF) for vehicle velocities and the velocity-velocity (vv) covariance function. We identify the probability to find a standing vehicle as a potential order parameter that signals nicely the transition between free congested flow for a sufficiently large number of velocity states. Our results for the vv covariance function resemble features of a second-order phase transition. We develop a 3-body approximation that allows us to relate the PDFs for velocities and headways. Using this relation, an approximation to the velocity PDF is obtained from the headway PDF observed in simulations. We find a remarkable agreement between this approximation and the velocity PDF obtained from simulations.
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