Abstract
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)].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
T. Riethmüller, L. Schimanski-Geier, D. Rosenkranz, T. Pöschel, J. Stat. Phys. 86, 421 (1997)
U. Erdmann, W. Ebeling, L. Schimansky-Geier, F. Schweitzer, Eur. Phys. J. B-Cond. Matter 15, 105 (2000)
P. Reimann, R. Kawai, C. Van den Broeck, P. Haenggi, Europhys. Lett. 45, 545 (1999)
R. Mahnke, J. Kaupužs, Phys. Rev. E 59, 117 (1999)
R. Kühne, R. Mahnke, I. Lubashevsky, J. Kaupužs, Phys. Rev. E 65, 66125 (2002)
D. Helbing, Rev. Mod. Phys. 73, 1067 (2001)
D. Helbing, M. Treiber, eprint arxiv:cond-mat/0307219 (2003)
M. Krbalek, D. Helbing, Physica A 333, 370 (2004)
D. Helbing, M. Treiber, A. Kesting, Physica A 363, 62 (2006)
M. Mehta, Random Matrices (Academic Press, 2004)
M. Krbalek, P. Šeba, P. Wagner, Phys. Rev. E 64, 066119 (2001)
D.N. Zubarev, V.A. Morozov, G. Röpke, Statistical Mechanics of Nonequilibrium Processes (Akademie Verlag, Berlin, 1996+1997), Vols. 1, 2
K.L. Klimontovich, Statistical Theory of Open Systems (Kluwer Academic Publishers, Dordrecht, 1995)
W. Ebeling, I. Sokolov, Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems (World Scientific, Singapore, 2005)
M. Krbálek, J. Phys. A: Mathematical and Theoretical 40, 5813 (2007)
T. Antal, G.M. Schütz, Phys. Rev. E 62, 83 (2000)
H. Risken, The Fokker-Planck Equation, 2nd edn. (Springer, Berlin, 1989)
D. Helbing, Eur. Phys. J. B, submitted (2008), e-print http://arxiv.org/abs/0805.3402
M. Bando, K. Hasebe, K. Nakanishi, A. Nakayama, A. Shibata, Y. Sugiyama, J. Phys. I France 5, 1389 (1995)
L. Landau, E. Lifshitz, Fluid Mechanics (Addison Wesley, Reading, MA, 1959)
M. Cross, P. Hohenberg, Rev. Mod. Phys. 65, 872 (1993)
A.A. Zaikin, L. Schimansky-Geier, Phys. Rev. E 58, 4355 (1998)
I. Rehberg, S. Rasenat, M. de la Torre Juárez, W. Schöpf, F. Hörner, G. Ahlers, H.R. Brand, Phys. Rev. Lett. 67, 596 (1991)
M. Wu, G. Ahlers, D. Cannell, Phys. Rev. Lett. 75, 1743 (1995)
M. Treiber, Phys. Rev. E 53, 577 (1996)
W. Schöpf, I. Rehberg, Journal of Fluid Mechanics Digital Archive 271, 235 (2006)
M. Treiber, L. Kramer, Phys. Rev. E 49, 3184 (1994)
M.A. Scherer, G. Ahlers, F. Hörner, I. Rehberg, Phys. Rev. Lett. 85, 3754 (2000)
D. Helbing (2008), eprint arxiv:physics/0805.3402
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Treiber, M., Helbing, D. Hamilton-like statistics in onedimensional driven dissipative many-particle systems. Eur. Phys. J. B 68, 607–618 (2009). https://doi.org/10.1140/epjb/e2009-00121-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1140/epjb/e2009-00121-8