Advertisement

Dynamical Universality Class of the Nagel–Schreckenberg and Related Models

  • Andreas SchadschneiderEmail author
  • Johannes Schmidt
  • Jan de Gier
  • Gunter M. Schütz
Conference paper

Abstract

Models for vehicular traffic fall into distinct dynamical universality classes of non-equilibrium systems. Such models share model-independent aspects of their dynamics, such as current fluctuations. Up to now the universality class of the Nagel–Schreckenberg (NaSch) model was not known except for the special case \(v_{\max }=1\). In this case the model corresponds to the ASEP (asymmetric simple exclusion process) which belongs to the Kardar–Parisi–Zhang (KPZ) class characterized by the dynamical exponent z = 3∕2. We have shown that the NaSch model for general \(v_{\max }\) also belongs to the KPZ class. Here we demonstrate that the universality class is not changed by extending the model to a two-lane NaSch model with dynamical lane changing rules. As an application we estimate the relaxation time to the (generally unknown) stationary state.

Notes

Acknowledgements

Support by the German Science Foundation (Grant SCHA 636/8-2) and the Bonn-Cologne Graduate School of Physics and Astronomy (BCGS) is gratefully acknowledged. J.S. thanks the Australian Research Council Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS) for funding that allowed him to visit The University of Melbourne where parts of this work were done.

References

  1. 1.
    Bain, N., Emig, T., Ulm, F.J., Schreckenberg, M.: Velocity statistics of the Nagel-Schreckenberg model. Phys. Rev. E 93, 022305 (2016)CrossRefGoogle Scholar
  2. 2.
    Brockfeld, E., Barlovic, R., Schadschneider, A., Schreckenberg, M.: Optimizing traffic lights in a cellular automaton model for city traffic. Phys. Rev. E 64, 056132 (2001)CrossRefGoogle Scholar
  3. 3.
    Chowdhury, D., Schadschneider, A.: Self-organization of traffic jams in cities: effects of stochastic dynamics and signal periods. Phys. Rev. E 59, R1311 (1999)CrossRefGoogle Scholar
  4. 4.
    Csányi, G., Kertész, J.: Scaling behaviour in discrete traffic models. J. Phys. A 28, L427 (1995). Erratum: 29, 471 (1996)Google Scholar
  5. 5.
    Kardar, M., Parisi, G., Zhang, Y.C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56, 889 (1986)CrossRefGoogle Scholar
  6. 6.
    Krug, J., Neiss, R., Schadschneider, A., Schmidt, J.: Logarithmic superdiffusion in two dimensional driven lattice gases. J. Stat. Phys. 172, 493 (2018)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Nagel, K., Schreckenberg, M.: A cellular automaton model for freeway traffic. J. Phys. I Fr. 2, 2221 (1992)CrossRefGoogle Scholar
  8. 8.
    Popkov, V., Schadschneider, A., Schmidt, J., Schütz, G.M.: Fibonacci family of dynamical universality classes. Proc. Natl. Acad. Sci. U. S. A. 112, 12645 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Popkov, V., Schadschneider, A., Schmidt, J., Schtz, G.M.: Exact scaling solution of the mode coupling equations for non-linear fluctuating hydrodynamics in one dimension. J. Stat. Mech. 093211 (2016)Google Scholar
  10. 10.
    Prähofer, M., Spohn, H. (2004). http://www.m5.ma.tum.de/kpz
  11. 11.
    Prähofer, M., Spohn, H.: Exact scaling functions for one-dimensional stationary KPZ growth. J. Stat. Phys. 115, 255 (2004)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Sasvári, M., Kertész, J.: On cellular automata models of single lane traffic. Phys. Rev. E 56, 4104 (1997)CrossRefGoogle Scholar
  13. 13.
    Schadschneider, A., Chowdhury, D., Nishinari, K.: Stochastic Transport in Complex Systems: From Molecules to Vehicles. Elsevier, Oxford (2010)zbMATHGoogle Scholar
  14. 14.
    Schmidt, J., Schadschneider, A., de Gier, J., Schtz, G.M.: KPZ universality of the Nagel-Schreckenberg model (2019, in preparation)Google Scholar
  15. 15.
    Spohn, H.: Nonlinear fluctuating hydrodynamics for anharmonic chains. J. Stat. Phys. 154, 1191 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Zhang, L., Garoni, T., de Gier, J.: A comparative study of macroscopic fundamental diagrams of urban road networks governed by different traffic signal systems. Transp. Res. B 49, 1 (2013)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Andreas Schadschneider
    • 1
    Email author
  • Johannes Schmidt
    • 1
  • Jan de Gier
    • 2
  • Gunter M. Schütz
    • 3
  1. 1.Institut für Theoretische PhysikUniversität zu KölnKölnGermany
  2. 2.ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS), School of Mathematics and StatisticsThe University of MelbourneMelbourneAustralia
  3. 3.Institute of Complex SystemsForschungszentrum JülichJülichGermany

Personalised recommendations