Journal of Statistical Physics

, Volume 87, Issue 1–2, pp 91–114 | Cite as

Enskog-like kinetic models for vehicular traffic

  • A. Klar
  • R. Wegener


In the present paper a general criticism of kinetic equations for vehicular traffic is given. The necessity of introducing an Enskog-type correction into these equations is shown. An Enskog-like kinetic traffic flow equation is presented and fluid dynamic equations are derived. This derivation yields new coefficients for the standard fluid dynamic equations of vehicular traffic. Numerical simulations for inhomogeneous traffic flow situations are shown together with a comparison between kinetic and fluid dynamic models.

Key Words

Traffic flow Enskog equation Boltzmann equation fluid dynamic models Payne equation simulation of inhomogeneous traffic flow 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D. C. Gazis, R. Herman, and R. Rothery, Nonlinear follow-the-leader models for traffic flow,Operations Res. 9:545 (1961).Google Scholar
  2. 2.
    R. Wiedemann, Simulation des Strassenverkehrsflusses, Schriftenreihe des Instituts für Verkehrswesen der Universität Karlsruhe (1974).Google Scholar
  3. 3.
    D. L. Gerlough and M. J. Huber, Traffic Flow Theory, Transportation Research Board Special Report 1965, Washington, D.C. (1975).Google Scholar
  4. 4.
    G. B. Witham,Linear and Nonlinear Waves, (Wiley, New York, 1974).Google Scholar
  5. 5.
    H. J. Payne,Transport. Res. Rec. 722:68 (1979).Google Scholar
  6. 6.
    R. D. Kühne, Macroscopic freeway model for dense traffic, in9th International Symposium on Transportation and Traffic Theory, N. Vollmuller, ed. (1984), p. 21.Google Scholar
  7. 7.
    B. S. Kerner and P. Konhäuser,Phys. Rev. E 50:54 (1994).Google Scholar
  8. 8.
    D. Helbing, Improved fluid dynamic model for vehicular traffic,Phys. Rev. E 51:3164 (1995).Google Scholar
  9. 9.
    I. Prigogine and R. Herman,Kinetic Theory of Vehicular Traffic (Elsevier, New York, 1971).Google Scholar
  10. 10.
    W. F. Phillips, Kinetic Model for Traffic Flow, National Technical Information Service, Springfield, Virginia (1977).Google Scholar
  11. 11.
    S. L. Paveri-Fontana, On Boltzmann like treatments for traffic flow,Transport. Res. 9:225 (1979).Google Scholar
  12. 12.
    D. Helbing, Gas-kinetic derivation of Navier-Stokes-like traffic equation. Preprint, University of Stuttgart (1995).Google Scholar
  13. 13.
    I. Prigogine and F. C. Andrews, A Boltzmann like approach for traffic flow,Operations Res. 8:789 (1960).Google Scholar
  14. 14.
    M. Lampis,Transport. Sci. 12:16 (1978).Google Scholar
  15. 15.
    P. Nelson, A kinetic model of vehicular traffic and its associated bimodal equilibrium solutions,Transport. Theory Stat. Phys. 24:383 (1995).Google Scholar
  16. 16.
    R. Wegener and A. Klar, A kinetic model for vehicular traffic derived from a stochastic microscopic model.Transport. Theory Stat. Phys. 25:785 (1996).Google Scholar
  17. 17.
    A. Klar, R. D. Kühne, and R. Wegener, Mathematical models for vehicular traffic,Surv. Math. Ind. 6:215 (1996).Google Scholar
  18. 18.
    K. Nagel and A. Schleicher,Parallel Computing 20:125 (1994).Google Scholar
  19. 19.
    M. Schreckenberg, A. Schadschneider, K. Nagel, and N. Ito,Phys. Rev. E 51:2939 (1995).Google Scholar
  20. 20.
    N. Anstett, Entwicklung eines ereignisorientierten Fahrzeug-Folge-Modells zur mikroskopischen Verkehrssimulation, Diplomarbeit Universität Stuttgart/Daimler Benz AG (1992).Google Scholar
  21. 21.
    C. Cercignani,The Boltzmann Equation and its Applications (Springer, New York, 1988).Google Scholar
  22. 22.
    J. H. Ferziger and H. G. Kaper,Mathematical Theory of Transport Processes in Gases (North-Holland, Amsterdam, 1972).Google Scholar
  23. 23.
    C. Cercignani and M. Lampis. On the kinetic theory of a dense gas of rough spheres,J. Stat. Phys. 53:655–672 (1988).Google Scholar
  24. 24.
    J. Lebowitz, J. Percus, and J. Sykes,Phys. Rev. 188:487 (1967).Google Scholar
  25. 25.
    P. Resibois, H-theorem for the (modified) nonlinear enskog equation,J. Stat. Phys. 19:593 (1978).Google Scholar
  26. 26.
    P. Markowich, C. Ringhofer, and C. Schmeiser,Semiconductor Equations (Springer, New York, 1990).Google Scholar
  27. 27.
    R. D. Kühne and S. Rödiger, InProceedings of the 1991 Winter Simulation Conference, Phoenix, Arizona, B. L. Nelson, W. D. Kelton, and G. M. Clark, eds. (IEEE, Piscataway, New Jersey, 1991).Google Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • A. Klar
    • 1
  • R. Wegener
    • 2
  1. 1.Fachbereich MathematikUniversität KaiserslauternKaiserslauternGermany
  2. 2.Institut für Techno- und WirtschaftsmathematikKaiserslauternGermany

Personalised recommendations