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A Hierarchy of Heuristic-Based Models of Crowd Dynamics

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Abstract

We derive a hierarchy of kinetic and macroscopic models from a noisy variant of the heuristic behavioral Individual-Based Model of Ngai et al. (Disaster Med. Public Health Prep. 3:191–195, 2009) where pedestrians are supposed to have constant speeds. This IBM supposes that pedestrians seek the best compromise between navigation towards their target and collisions avoidance. We first propose a kinetic model for the probability distribution function of pedestrians. Then, we derive fluid models and propose three different closure relations. The first two closures assume that the velocity distribution function is either a Dirac delta or a von Mises-Fisher distribution respectively. The third closure results from a hydrodynamic limit associated to a Local Thermodynamical Equilibrium. We develop an analogy between this equilibrium and Nash equilibria in a game theoretic framework. In each case, we discuss the features of the models and their suitability for practical use.

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Acknowledgements

This work has been supported by the French ‘Agence Nationale pour la Recherche (ANR)’ in the frame of the contracts ‘Pedigree’ (ANR-08-SYSC-015-01) and ‘CBDif-Fr’ (ANR-08-BLAN-0333-01).

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Authors and Affiliations

  1. UPS, INSA, UT1, UTM, Institut de Mathématiques de Toulouse, Université de Toulouse, 31062, Toulouse, France

    P. Degond

  2. CNRS, Institut de Mathématiques de Toulouse UMR 5219, 31062, Toulouse, France

    P. Degond

  3. Laboratoire de Physique Théorique, Université Paris Sud, bâtiment 210, 91405, Orsay cedex, France

    C. Appert-Rolland

  4. CNRS, UMR 8627, Laboratoire de Physique Théorique, 91405, Orsay, France

    C. Appert-Rolland

  5. Center for Adaptive Behavior and Cognition, Max Planck Institute for Human Development, Lentzeallee 94, 14195, Berlin, Germany

    M. Moussaïd

  6. INRIA Rennes – Bretagne Atlantique, Campus de Beaulieu, 35042, Rennes, France

    J. Pettré

  7. Centre de Recherches sur la Cognition Animale, UMR-CNRS 5169, Université Paul Sabatier, Bât 4R3, 118 Route de Narbonne, 31062, Toulouse cedex 9, France

    G. Theraulaz

  8. CNRS, Centre de Recherches sur la Cognition Animale, 31062, Toulouse, France

    G. Theraulaz

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  1. P. Degond
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  2. C. Appert-Rolland
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Correspondence to P. Degond.

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Degond, P., Appert-Rolland, C., Moussaïd, M. et al. A Hierarchy of Heuristic-Based Models of Crowd Dynamics. J Stat Phys 152, 1033–1068 (2013). https://doi.org/10.1007/s10955-013-0805-x

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