Traffic of Ants on a Trail: A Stochastic Modelling and Zero Range Process

  • Katsuhiro Nishinari
  • Andreas Schadschneider
  • Debashish Chowdhury
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3305)

Abstract

Recently we have proposed a stochastic cellular automaton model of ants on a trail and investigated its unusual flow-density relation by using a mean field theory and computer simulations. In this paper, we study the model in detail by utilizing the analogy with the zero range process, which is known as one of the exactly solvable stochastic models. We show that our theory can quantitatively account for the unusual non-monotonic dependence of the average speed of the ants on their density for finite lattices with periodic boundary conditions. Moreover, we argue that the flow-density diagram exhibits a second order phase transition at the critial density only in a limiting case.

Keywords

Cellular Automaton Average Speed Thermodynamic Limit Order Phase Transition Open Boundary Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Wolfram, S.: Theory and Applications of Cellular Automata. World Scientific, Singapore (1986)MATHGoogle Scholar
  2. 2.
    Chopard, B., Droz, M.: Cellular Automata Modelling of Physical Systems. Cambridge University Press, Cambridge (1998)CrossRefGoogle Scholar
  3. 3.
    Chowdhury, D., Santen, L., Schadschneider, A.: Statistical physics of vehicular traffic and some related systems. Phys. Rep. 329, 199–329 (2000)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Chowdhury, D., Nishinari, K., Schadschneider, A.: Self-organized patterns and traffic flow in colonies of organisms:from bacteria and social insects to vertebrates. Phase Transitions 77, 601–624 (2004)CrossRefGoogle Scholar
  5. 5.
    Chowdhury, D., Guttal, V., Nishinari, K., Schadschneider, A.: A cellular-automata model of flow in ant trails: non-monotonic variation of speed with density. J. Phys. A: Math. Gen. 35, L573–L577 (2002)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Burd, M., Archer, D., Aranwela, N., Stradling, D.J.: Traffic dynamics of the leaf cutting ant. American Natur. 159, 283–293 (2002)CrossRefGoogle Scholar
  7. 7.
    Evans, M.R., Blythe, R.A.: Nonequilibrium dynamics in low-dimensional systems. Physica A 313, 110–152 (2002)MATHCrossRefGoogle Scholar
  8. 8.
    Nishinari, K., Chowdhury, D., Schadschneider, A.: Cluster formation and anomalous fundamental diagaram in an ant trail model. Phys. Rev. E 67, 036120 (2003)Google Scholar
  9. 9.
    Spitzer, F.: Interaction of markov processes. Advances in Math. 5, 246–290 (1970)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Evans, M.R.: Phase transitions in one-dimensional nonequilibrium systems. Braz. J. Phys. 30, 42–57 (2000)CrossRefGoogle Scholar
  11. 11.
    Camazine, S., Deneubourg, J.L., Franks, N.R., Sneyd, J., Theraulaz, G., Bonabeau, E.: Self-organization in Biological Systems. Princeton University Press, Prinston (2001)Google Scholar
  12. 12.
    Mikhailov, A.S., Calenbuhr, V.: From Cells to Societies. Springer, Berlin (2002)MATHGoogle Scholar
  13. 13.
    Kunwar, A., John, A., Nishinari, K., Schadschneider, A., Chowdhury, D.: Collective traffic-like movement of ants on a trail – dynamical phases and phase transitions (submitted for publication)Google Scholar
  14. 14.
    Nishinari, K., Takahashi, D.: Analytical properties of ultradiscrete Burgers equation and rule-184 cellular automaton. J. Phys. A: Math. Gen. 31, 5439–5450 (1998)MATHCrossRefGoogle Scholar
  15. 15.
    Nagel, K., Schreckenberg, M.: A cellular automaton model for freeway traffic. J. Phys. I 2, 2221–2229 (1992)CrossRefGoogle Scholar
  16. 16.
    Evans, M.R.: Exact Steady States of Disordered Hopping Particle Models with Parallel and Ordered Sequential Dynamics. J. Phys. A: Math. Gen. 30, 5669–5685 (1997)MATHCrossRefGoogle Scholar
  17. 17.
    O’Loan, O.J., Evans, M.R., Cates, M.E.: Jamming Transition in a Homogeneous One-Dimensional System: the Bus Route Model. Phys. Rev. E 58, 1404–1418 (1998); see also Europhys. Lett. 42, 137–142 (1998)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Katsuhiro Nishinari
    • 1
  • Andreas Schadschneider
    • 2
  • Debashish Chowdhury
    • 3
  1. 1.Department of Applied Mathematics and InformaticsRyukoku UniversityShigaJapan
  2. 2.Institut für Theoretische PhysikUniversität zu KölnKölnGermany
  3. 3.Department of PhysicsIndian Institute of TechnologyKanpurIndia

Personalised recommendations