Advertisement

Journal of Statistical Physics

, Volume 147, Issue 6, pp 1113–1144 | Cite as

One-Dimensional Particle Processes with Acceleration/Braking Asymmetry

  • Cyril FurtlehnerEmail author
  • Jean-Marc Lasgouttes
  • Maxim Samsonov
Article

Abstract

The slow-to-start mechanism is known to play an important role in the particular shape of the Fundamental Diagram of traffic and to be associated to hysteresis effects of traffic flow. We study this question in the context of exclusion and queueing processes, by including an asymmetry between deceleration and acceleration in the formulation of these processes. For exclusions processes, this corresponds to a multi-class process with transition asymmetry between different speed levels, while for queueing processes we consider non-reversible stochastic dependency of the service rate w.r.t. the number of clients. The relationship between these 2 families of models is analyzed on the ring geometry, along with their steady state properties. Spatial condensation phenomena and metastability are observed, depending on the level of the aforementioned asymmetry. In addition, we provide a large deviation formulation of the fundamental diagram which includes the level of fluctuations, in the canonical ensemble when the stationary state is expressed as a product form of such generalized queues.

Keywords

Exclusion processes Zero range processes Fundamental diagram of traffic Spatial condensation 

Notes

Acknowledgements

We thank Guy Fayolle for useful discussions. We also thank referees for fruitful comments and references. This work was supported by the French National Research Agency (ANR) grant No ANR-08-SYSC-017.

References

  1. 1.
    Appert, C., Santen, L.: Boundary induced phase transitions in driven lattice gases with meta-stable states. Phys. Rev. Lett. 86, 2498 (2001) ADSCrossRefGoogle Scholar
  2. 2.
    Barlović, R., Santen, L., Schadschneider, A., Schreckenberg, M.: Metastable states in cellular automata for traffic flow. Eur. Phys. J. B 5, 793 (1998) ADSCrossRefGoogle Scholar
  3. 3.
    Blank, M.: Hysteresis phenomenon in deterministic traffic flows. J. Stat. Phys. 120, 627–658 (2005) MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Cantini, L.: Algebraic Bethe ansatz for the two species ASEP with different hopping rates. J. Phys. A, Math. Theor. 41, 095001 (2008) MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Derrida, B., Evans, M.R., Hakim, V., Pasquier, V.: Exact solution for 1d asymmetric exclusion model using a matrix formulation. J. Phys. A, Math. Gen. 26, 1493–1517 (1993) MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. 6.
    Evans, M., Kafri, Y., Sugden, K., Tailleur, J.: Phase diagrams of two-lane driven diffusive systems. J. Stat. Mech. Theory Exp. 2011(06), 06009 (2011) CrossRefGoogle Scholar
  7. 7.
    Evans, M.R., Majumdar, S.N., Zia, R.K.P.: Canonical analysis of condensation in factorized steady states. J. Stat. Phys. 123(2), 357–390 (2006) MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. 8.
    Fayolle, G., Furtlehner, C.: Dynamical windings of random walks and exclusion models. J. Stat. Phys. 114, 229–260 (2004) MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. 9.
    Fayolle, G., Lasgouttes, J.M.: Asymptotics and scalings for large closed product-form networks via the Central Limit Theorem. Markov Process. Relat. Fields 2(2), 317–348 (1996) MathSciNetzbMATHGoogle Scholar
  10. 10.
    Furtlehner, C., Lasgouttes, J.: A queueing theory approach for a multi-speed exclusion process. In: Traffic and Granular Flow ’07, pp. 129–138 (2007) Google Scholar
  11. 11.
    Golinelli, G., Mallick, K.: The asymmetric simple exclusion process: an integrable model for non-equilibrium statistical mechanics. J. Phys. A, Math. Gen. 39(41), 12679 (2006) MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. 12.
    Harris, C.: Queues with state-dependent stochastic service rate. Oper. Res. 15, 117–130 (1967) ADSzbMATHCrossRefGoogle Scholar
  13. 13.
    Kafri, Y., Levine, E., Mukamel, D., Schütz, G.M., Török, J.: Criterion for phase separation in one-dimensional driven systems. Phys. Rev. Lett. 89, 035702 (2002) ADSCrossRefGoogle Scholar
  14. 14.
    Karimipour, V.: A multi-species ASEP and its relation to traffic flow. Phys. Rev. E 59, 205 (1999) MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    Kaupuz̃s, J., Mahnke, R., Harris, R.J.: Zero-range model of traffic flow. Phys. Rev. E 72, 056125 (2005) ADSCrossRefGoogle Scholar
  16. 16.
    Kelly, F.P.: Reversibility and Stochastic Networks. Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1979) zbMATHGoogle Scholar
  17. 17.
    Kerner, B.: The Physics of Traffic. Springer, Berlin (2005) Google Scholar
  18. 18.
    Kipnis, C., Landim, C.: Scaling Limits of Interacting Particles Systems. Springer, Berlin (1999) Google Scholar
  19. 19.
    Liggett, T.M.: Interacting Particle Systems. Springer, Berlin (2005) zbMATHGoogle Scholar
  20. 20.
    Nagel, K., Paczuski, M.: Emergent traffic jams. Phys. Rev. E 51(4), 2909–2918 (1995) ADSCrossRefGoogle Scholar
  21. 21.
    Nagel, K., Schreckenberg, M.: A cellular automaton model for freeway traffic. J. Phys. I 2, 2221–2229 (1992) CrossRefGoogle Scholar
  22. 22.
    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) ADSCrossRefGoogle Scholar
  23. 23.
    Samsonov, M., Furtlehner, C., Lasgouttes, J.: Exactly solvable stochastic processes for traffic modelling. Tech. Rep. 7278, INRIA (2010) Google Scholar
  24. 24.
    Schönhof, M., Helbing, D.: Criticism of three-phase traffic theory. Transp. Res. 43, 784–797 (2009) CrossRefGoogle Scholar
  25. 25.
    Schreckenberg, M., Schadschneider, A., Nagel, K., Ito, N.: Discrete stochastic models for traffic flow. Phys. Rev. E 51, 2339 (1995) ADSCrossRefGoogle Scholar
  26. 26.
    Schutz, G.M., Harris, R.J.: Hydrodynamics of the zero-range process in the condensation regime. J. Stat. Phys. 127, 419 (2007) MathSciNetADSCrossRefGoogle Scholar
  27. 27.
    Spitzer, F.: Interaction of Markov processes. Adv. Math. 5, 246 (1970) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Sugiyama, Y., et al.: Traffic jams without bottlenecks: experimental evidence for the physical mechanism of the formation of a jam. New J. Phys. 10, 1–7 (2008) MathSciNetCrossRefGoogle Scholar
  29. 29.
    Tóth, B., Valkó, B.: Onsager relations and Eulerian hydrodynamic limit for systems with several conservation laws. J. Stat. Phys. 112, 497–521 (2003) zbMATHCrossRefGoogle Scholar
  30. 30.
    Touchette, H.: The large deviation approach to statistical mechanics. Phys. Rep. 478, 1–69 (2009) MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Cyril Furtlehner
    • 1
    Email author
  • Jean-Marc Lasgouttes
    • 2
  • Maxim Samsonov
    • 1
  1. 1.INRIA Saclay–Île-de-France – LRI, Bat. 490Université Paris-SudOrsay cedexFrance
  2. 2.INRIA Paris–Rocquencourt – Domaine de VoluceauLe Chesnay cedexFrance

Personalised recommendations