The European Physical Journal B

, Volume 68, Issue 4, pp 607–618 | Cite as

Hamilton-like statistics in onedimensional driven dissipative many-particle systems

  • M. Treiber
  • D. HelbingEmail author
Interdisciplinary Physics


This contribution presents a derivation of the steady-state distribution of velocities and distances of driven particles on a onedimensional periodic ring, using a Fokker-Planck formalism. We will compare two different situations: (i) symmetrical interaction forces fulfilling Newton’s law of “actio = reactio” and (ii) asymmetric, forwardly directed interactions as, for example in vehicular traffic. Surprisingly, the steady-state velocity and distance distributions for asymmetric interactions and driving terms agree with the equilibrium distributions of classical many-particle systems with symmetrical interactions, if the system is large enough. This analytical result is confirmed by computer simulations and establishes the possibility of approximating the steady state statistics in driven many-particle systems by Hamiltonian systems. Our finding is also useful to understand the various departure time distributions of queueing systems as a possible effect of interactions among the elements in the respective queue [Physica A 363, 62 (2006)].


05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion 05.10.Gg Stochastic analysis methods 47.70.-n Reactive and radiative flows 89.40.-a Transportation 


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  1. T. Riethmüller, L. Schimanski-Geier, D. Rosenkranz, T. Pöschel, J. Stat. Phys. 86, 421 (1997) Google Scholar
  2. U. Erdmann, W. Ebeling, L. Schimansky-Geier, F. Schweitzer, Eur. Phys. J. B-Cond. Matter 15, 105 (2000) Google Scholar
  3. P. Reimann, R. Kawai, C. Van den Broeck, P. Haenggi, Europhys. Lett. 45, 545 (1999) Google Scholar
  4. R. Mahnke, J. Kaupužs, Phys. Rev. E 59, 117 (1999) Google Scholar
  5. R. Kühne, R. Mahnke, I. Lubashevsky, J. Kaupužs, Phys. Rev. E 65, 66125 (2002) Google Scholar
  6. D. Helbing, Rev. Mod. Phys. 73, 1067 (2001) Google Scholar
  7. D. Helbing, M. Treiber, eprint arxiv:cond-mat/0307219 (2003) Google Scholar
  8. M. Krbalek, D. Helbing, Physica A 333, 370 (2004) Google Scholar
  9. D. Helbing, M. Treiber, A. Kesting, Physica A 363, 62 (2006) Google Scholar
  10. M. Mehta, Random Matrices (Academic Press, 2004) Google Scholar
  11. M. Krbalek, P. Šeba, P. Wagner, Phys. Rev. E 64, 066119 (2001) Google Scholar
  12. D.N. Zubarev, V.A. Morozov, G. Röpke, Statistical Mechanics of Nonequilibrium Processes (Akademie Verlag, Berlin, 1996+1997), Vols. 1, 2 Google Scholar
  13. K.L. Klimontovich, Statistical Theory of Open Systems (Kluwer Academic Publishers, Dordrecht, 1995) Google Scholar
  14. W. Ebeling, I. Sokolov, Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems (World Scientific, Singapore, 2005) Google Scholar
  15. M. Krbálek, J. Phys. A: Mathematical and Theoretical 40, 5813 (2007) Google Scholar
  16. T. Antal, G.M. Schütz, Phys. Rev. E 62, 83 (2000) Google Scholar
  17. H. Risken, The Fokker-Planck Equation, 2nd edn. (Springer, Berlin, 1989) Google Scholar
  18. D. Helbing, Eur. Phys. J. B, submitted (2008), e-print Google Scholar
  19. M. Bando, K. Hasebe, K. Nakanishi, A. Nakayama, A. Shibata, Y. Sugiyama, J. Phys. I France 5, 1389 (1995) Google Scholar
  20. L. Landau, E. Lifshitz, Fluid Mechanics (Addison Wesley, Reading, MA, 1959) Google Scholar
  21. M. Cross, P. Hohenberg, Rev. Mod. Phys. 65, 872 (1993) Google Scholar
  22. A.A. Zaikin, L. Schimansky-Geier, Phys. Rev. E 58, 4355 (1998) Google Scholar
  23. I. Rehberg, S. Rasenat, M. de la Torre Juárez, W. Schöpf, F. Hörner, G. Ahlers, H.R. Brand, Phys. Rev. Lett. 67, 596 (1991) Google Scholar
  24. M. Wu, G. Ahlers, D. Cannell, Phys. Rev. Lett. 75, 1743 (1995) Google Scholar
  25. M. Treiber, Phys. Rev. E 53, 577 (1996) Google Scholar
  26. W. Schöpf, I. Rehberg, Journal of Fluid Mechanics Digital Archive 271, 235 (2006) Google Scholar
  27. M. Treiber, L. Kramer, Phys. Rev. E 49, 3184 (1994) Google Scholar
  28. M.A. Scherer, G. Ahlers, F. Hörner, I. Rehberg, Phys. Rev. Lett. 85, 3754 (2000) Google Scholar
  29. D. Helbing (2008), eprint arxiv:physics/0805.3402 Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Institute for Economics and Traffic, Dresden University of TechnologyDresdenGermany
  2. 2.ETH Zurich, UNO D11, Universitätstr. 41ZurichSwitzerland

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