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

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Abstract

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.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bellomo, N., Piccoli, B., Tosin, A.: Modeling crowd dynamics from a complex system viewpoint. Math. Models Methods Appl. Sci. 22, 1230004 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Boers, N., Pickl, P.: On mean field limits for dynamical systems. J. Stat. Phys. 164(1), 1–16 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Helbing, D., Molnar, P.: Social force model for pedestrian dynamics. Phys. Rev. E 51(5), 4282 (1995)

    Article  ADS  Google Scholar 

  10. Hughes, R.L.: A continuum theory for the flow of pedestrians. Transp. Res. Part B 36(6), 507–535 (2002)

    Article  Google Scholar 

  11. Jabin, P.E., Hauray, M.: Particles approximations of Vlasov equations with singular forces: propagation of chaos. arXiv preprint (2014). arXiv:1107.3821

  12. Lewin, K.: In: D. Cartwright (eds.) Field Theory in Social Science: Selected Theoretical Papers. Harper and Brothers, New York (1951)

  13. Naldi, G., Pareschi, L., Toscani, G.: Mathematical Modeling of Collective Behavior in Socio-economic and Life Sciences. Birkhäuser, Boston (2010)

    Book  MATH  Google Scholar 

  14. Philipowski, R.: Interacting diffusions approximating the porous medium equation and propagation of chaos. Stoch. Process. Appl. 117(4), 526–538 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Piccoli, B., Tosin, A.: Pedestrian flows in bounded domains with obstacles. Contin. Mech. Thermodyn. 21(2), 85–107 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Spohn, H.: Large Scale Dynamics of Interacting Particles. Springer, Berlin (2012)

    MATH  Google Scholar 

  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)

  18. Toscani, G.: Kinetic models of opinion formation. Commun. Math. Sci. 4(3), 481–496 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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

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Correspondence to Li Chen.

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Chen, L., Göttlich, S. & Yin, Q. Mean Field Limit and Propagation of Chaos for a Pedestrian Flow Model. J Stat Phys 166, 211–229 (2017). https://doi.org/10.1007/s10955-016-1679-5

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  • DOI: https://doi.org/10.1007/s10955-016-1679-5

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