Abstract
We study statistical properties of a family of maps acting in the space of integer valued sequences, which model dynamics of simple deterministic traffic flows. We obtain asymptotic (as time goes to infinity) properties of trajectories of those maps corresponding to arbitrary initial configurations in terms of statistics of densities of various patterns and describe weak attractors of these systems and the rate of convergence to them. Previously only the so called regular initial configurations (having a density with only finite fluctuations of partial sums around it) in the case of a slow particles model (with the maximal velocity 1) have been studied rigorously. Applying ideas borrowed from substitution dynamics we are able to reduce the analysis of the traffic flow models corresponding to the multi-lane traffic and to the flow with fast particles (with velocities greater than 1) to the simplest case of the flow with the one-lane traffic and slow particles, where the crucial technical step is the derivation of the exact life-time for a given cluster of particles. Applications to the optimal redirection of the multi-lane traffic flow and a model of a pedestrian going in a slowly moving crowd are discussed as well.
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REFERENCES
V. Belitsky and P. A. Ferrari, Ballistic annihilation and deterministic surface growth, J. Stat. Phys. 80:517-543 (1995).
V. Belitsky, J. Krug, E. Jordao Neves, and G. M. Schutz, A cellular automaton model for two-lane traffic, J. Stat. Phys. 103:945-971 (2001).
M. Blank, Variational principles in the analysis of traffic flows. (Why it is worth to go against the flow), Markov Process. Related Fields 6:287-304 (2000).
M. Blank, Dynamics of traffic jams: Order and chaos, Moscow Math. J. 1:1-26 (2001).
D. Chowdhury, L. Santen, and A. Schadschneider, Statistical physics of vehicular traffic and some related systems, Phys. Rep. 329:199-329 (2000).
A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, 2nd edn. (Springer, New York, 1998).
B. Derrida, J. L. Lebowitz, and E. R. Speer, Shock profiles for the asymmetric simple exclusion process in one dimension, J. Stat. Phys. 89:135-166 (1997).
H. Fuks, Exact results for deterministic cellular automata traffic models, Phys. Rev. E 60:197-202 (1999).
L. Gray and D. Griffeath, The ergodic theory of traffic jams, J. Stat. Phys. 105:413-452 (2001).
J. Krug and H. Spohn, Universality classes for deterministic surface growth, Phys. Rev. A 38:4271-4283 (1988).
T. M. Liggett, Stochastic Interacting Systems: Contact, Voter, and Exclusion Processes (Springer-Verlag, New York, 1999).
J. Milnor, On the concept of attractor, Comm. Math. Phys. 99:177-195 (1985).
K. Nagel and H. J. Herrmann, Deterministic models for traffic jams, Physica A 199:254-269 (1993).
K. Nagel and M. Schreckenberg, A cellular automaton model for freeway traffic, J. Physique I 2:2221-2229 (1992).
K. Nishinari and D. Takahashi, Analytical properties of ultradiscrete Burgers equation and rule-184 cellular automaton, J. Phys. A 31:5439-5450 (1998).
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Blank, M. Ergodic Properties of a Simple Deterministic Traffic Flow Model. Journal of Statistical Physics 111, 903–930 (2003). https://doi.org/10.1023/A:1022806500731
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DOI: https://doi.org/10.1023/A:1022806500731