Journal of Statistical Physics

, Volume 166, Issue 2, pp 211–229 | Cite as

Mean Field Limit and Propagation of Chaos for a Pedestrian Flow Model

  • Li Chen
  • Simone Göttlich
  • Qitao Yin


In this paper a rigorous proof of the mean field limit for a pedestrian flow model in two dimensions is given by using a probabilistic method. The model under investigation is an interacting particle system coupled to the eikonal equation on the microscopic scale. For stochastic initial data, it is proved that the solution of the N-particle pedestrian flow system with properly chosen cut-off converges in the probability sense to the solution of the characteristics of the non-cut-off Vlasov equation. Furthermore, the result on propagation of chaos is also deduced in terms of bounded Lipschitz distance.


Probabilistic method Pedestrian flow Mean field limit Vlasov equation Propagation of chaos 

Mathematics Subject Classification

35Q83 82C22 



This work was financially supported by the DAAD project “DAAD-PPP VR China” (Project-ID: 57215936).


  1. 1.
    Bellomo, N., Dogbé, C.: On the modeling of traffic and crowds: a survey of models, speculations, and perspectives. SIAM Rev. 53(3), 409–463 (2011)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bellomo, N., Piccoli, B., Tosin, A.: Modeling crowd dynamics from a complex system viewpoint. Math. Models Methods Appl. Sci. 22, 1230004 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Boers, N., Pickl, P.: On mean field limits for dynamical systems. J. Stat. Phys. 164(1), 1–16 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Cristiani, E., Piccoli, B., Tosin, A.: Multiscale modeling of granular flows with application to crowd dynamics. Multiscale Model. Simul. 9(1), 155–182 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Colombo, R., Garavello, M., Lécureux-Mercier, M.: A class of nonlocal models for pedestrian traffic. Math. Models Methods Appl. Sci. 22(4), 1150023 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Degond, P., Appert-Rolland, C., Moussaid, M., Pettré, J., Theraulaz, G.: A hierarchy of heuristic-based models of crowd dynamics. J. Stat. Phys. 152(6), 1033–1068 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Etikyala, R., Göttlich, S., Klar, A., Tiwari, S.: Particle methods for pedestrian flow models: from microscopic to nonlocal continuum models. Math. Models Methods Appl. Sci. 24(12), 2503–2523 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hauray, M., Jabin, P.E.: N-particles approximation of the Vlasov equations with singular potential. Arch. Ration. Mech. Anal. 183(3), 489–524 (2007)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Helbing, D., Molnar, P.: Social force model for pedestrian dynamics. Phys. Rev. E 51(5), 4282 (1995)ADSCrossRefGoogle Scholar
  10. 10.
    Hughes, R.L.: A continuum theory for the flow of pedestrians. Transp. Res. Part B 36(6), 507–535 (2002)CrossRefGoogle Scholar
  11. 11.
    Jabin, P.E., Hauray, M.: Particles approximations of Vlasov equations with singular forces: propagation of chaos. arXiv preprint (2014). arXiv:1107.3821
  12. 12.
    Lewin, K.: In: D. Cartwright (eds.) Field Theory in Social Science: Selected Theoretical Papers. Harper and Brothers, New York (1951)Google Scholar
  13. 13.
    Naldi, G., Pareschi, L., Toscani, G.: Mathematical Modeling of Collective Behavior in Socio-economic and Life Sciences. Birkhäuser, Boston (2010)CrossRefMATHGoogle Scholar
  14. 14.
    Philipowski, R.: Interacting diffusions approximating the porous medium equation and propagation of chaos. Stoch. Process. Appl. 117(4), 526–538 (2007)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Piccoli, B., Tosin, A.: Pedestrian flows in bounded domains with obstacles. Contin. Mech. Thermodyn. 21(2), 85–107 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Spohn, H.: Large Scale Dynamics of Interacting Particles. Springer, Berlin (2012)MATHGoogle Scholar
  17. 17.
    Sznitman, A.S.: Topics in propagation of chaos. In: Ecole d’été de probabilités de Saint-Flour XIX-1989. Springer, Berlin, pp. 165–251 (1991)Google Scholar
  18. 18.
    Toscani, G.: Kinetic models of opinion formation. Commun. Math. Sci. 4(3), 481–496 (2006)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MannheimMannheimGermany

Personalised recommendations