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A pedestrian flow model with stochastic velocities: Microscopic and macroscopic approaches
Department of Mathematics, University of Mannheim, 68131 Mannheim, Germany |
We investigate a stochastic model hierarchy for pedestrian flow. Starting from a microscopic social force model, where the pedestrians switch randomly between the two states stop-or-go, we derive an associated macroscopic model of conservation law type. Therefore we use a kinetic mean-field equation and introduce a new problem-oriented closure function. Numerical experiments are presented to compare the above models and to show their similarities.
References:
[1] |
D. Armbruster, S. Martin and A. Thatcher,
Elastic and inelastic collisions of swarms, Physica D: Nonlinear Phenomena, 344 (2017), 45-57.
doi: 10.1016/j.physd.2016.11.008. |
[2] |
D. Armbruster, S. Motsch and A. Thatcher,
Swarming in bounded domains, Physica D: Nonlinear Phenomena, 344 (2017), 58-67.
doi: 10.1016/j.physd.2016.11.009. |
[3] |
H. Bauer,
Probability Theory, vol. 23 of De Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 1996, Translated from the fourth (1991) German edition by Robert B. Burckel and revised by the author.
doi: 10.1515/9783110814668. |
[4] |
N. Bellomo, C. Bianca and V. Coscia,
On the modeling of crowd dynamics: An overview and research perspectives, S$\vec{\rm e}$MA J., 54 (2011), 25-46.
|
[5] |
C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, vol. 106 of Applied Mathematical Sciences, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4419-8524-8. |
[6] |
L. Chen, S. Göttlich and Q. Yin,
Mean field limit and propagation of chaos for a pedestrian flow model, Journal of Statistical Physics, 166 (2017), 211-229.
doi: 10.1007/s10955-016-1679-5. |
[7] |
A. Chertock, A. Kurganov, A. Polizzi and I. Timofeyev,
Pedestrian flow models with slowdown interactions, Math. Models Methods Appl. Sci., 24 (2014), 249-275.
doi: 10.1142/S0218202513400083. |
[8] |
E. Cristiani, B. Piccoli and A. Tosin,
Multiscale Modeling of Pedestrian Dynamics, vol. 12 of MS&A. Modeling, Simulation and Applications, Springer, Cham, 2014.
doi: 10.1007/978-3-319-06620-2. |
[9] |
P. Degond, C. Appert-Rolland, M. Moussaïd, J. Pettré and G. Theraulaz,
A hierarchy of heuristic-based models of crowd dynamics, J. Stat. Phys., 152 (2013), 1033-1068.
doi: 10.1007/s10955-013-0805-x. |
[10] |
P. Degond and C. Ringhofer,
Stochastic dynamics of long supply chains with random breakdowns, SIAM J. Appl. Math., 68 (2007), 59-79.
doi: 10.1137/060674302. |
[11] |
P. Degond, C. Appert-Rolland, J. Pettré and G. Theraulaz,
Vision-based macroscopic pedestrian models, Kinet. Relat. Models, 6 (2013), 809-839.
doi: 10.3934/krm.2013.6.809. |
[12] |
G. Dimarco and S. Motsch,
Self-alignment driven by jump processes: Macroscopic limit and numerical investigation, Math. Models Methods Appl. Sci., 26 (2016), 1385-1410.
doi: 10.1142/S0218202516500330. |
[13] |
R. Etikyala, S. Göttlich, A. Klar and S. Tiwari,
Particle methods for pedestrian flow models: From microscopic to nonlocal continuum models, Math. Models Methods Appl. Sci., 24 (2014), 2503-2523.
doi: 10.1142/S0218202514500274. |
[14] |
I. I. Gikhman and A. V. Skorokhod, The Theory of Stochastic Processes. Ⅱ, Classics in Mathematics, Springer-Verlag, Berlin, 2004, Translated from the Russian by S. Kotz, Reprint of the 1975 edition.
doi: 10.1007/978-3-642-61921-2. |
[15] |
D. Helbing, A fluid dynamic model for the movement of pedestrians, Complex Systems, 6 (1992), 391-415, arXiv: cond-mat/9805213. |
[16] |
D. Helbing and P. Molnár, Social force model for pedestrian dynamics, Physical Review E, 51 (1998), 4282-4286, arXiv: cond-mat/9805244.
doi: 10.1103/PhysRevE.51.4282. |
[17] |
R. L. Hughes,
A continuum theory for the flow of pedestrians, Transportation Research Part B: Methodological, 36 (2002), 507-535.
doi: 10.1016/S0191-2615(01)00015-7. |
[18] |
P.-E. Jabin,
Macroscopic limit of Vlasov type equations with friction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 651-672.
doi: 10.1016/S0294-1449(00)00118-9. |
[19] |
P.-E. Jabin,
Various levels of models for aerosols, Math. Models Methods Appl. Sci., 12 (2002), 903-919.
doi: 10.1142/S0218202502001957. |
[20] |
A. Jüngel, Transport Equations for Semiconductors, vol. 773 of Lecture Notes in Physics, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-89526-8. |
[21] |
A. Klar, F. Schneider and O. Tse,
Approximate models for stochastic dynamic systems with velocities on the sphere and associated fokker-planck equations, Kinetic and Related Models, 7 (2014), 509-529.
doi: 10.3934/krm.2014.7.509. |
[22] |
R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.
doi: 10.1017/CBO9780511791253. |
[23] |
B. Piccoli and A. Tosin,
Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738.
doi: 10.1007/s00205-010-0366-y. |
[24] |
B. Piccoli and A. Tosin,
Pedestrian flows in bounded domains with obstacles, Contin. Mech. Thermodyn., 21 (2009), 85-107.
doi: 10.1007/s00161-009-0100-x. |
[25] |
M. Schultz, Stochastic transition model for pedestrian dynamics, in Pedestrian and Evacuation Dynamics 2012, Springer International Publishing, (2013), 971-985, arXiv: 1210.5554.
doi: 10.1007/978-3-319-02447-9_81. |
[26] |
A. Tordeux and A. Schadschneider, A stochastic optimal velocity model for pedestrian flow, in Parallel Processing and Applied Mathematics, Springer International Publishing, 9574 (2016), 528-538.
doi: 10.1007/978-3-319-32152-3_49. |
[27] |
A. Tordeux and A. Schadschneider, White and relaxed noises in optimal velocity models for pedestrian flow with stop-and-go waves, Journal of Physics A: Mathematical and Theoretical, 49 (2016), 185101, 16pp.
doi: 10.1088/1751-8113/49/18/185101. |
show all references
References:
[1] |
D. Armbruster, S. Martin and A. Thatcher,
Elastic and inelastic collisions of swarms, Physica D: Nonlinear Phenomena, 344 (2017), 45-57.
doi: 10.1016/j.physd.2016.11.008. |
[2] |
D. Armbruster, S. Motsch and A. Thatcher,
Swarming in bounded domains, Physica D: Nonlinear Phenomena, 344 (2017), 58-67.
doi: 10.1016/j.physd.2016.11.009. |
[3] |
H. Bauer,
Probability Theory, vol. 23 of De Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 1996, Translated from the fourth (1991) German edition by Robert B. Burckel and revised by the author.
doi: 10.1515/9783110814668. |
[4] |
N. Bellomo, C. Bianca and V. Coscia,
On the modeling of crowd dynamics: An overview and research perspectives, S$\vec{\rm e}$MA J., 54 (2011), 25-46.
|
[5] |
C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, vol. 106 of Applied Mathematical Sciences, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4419-8524-8. |
[6] |
L. Chen, S. Göttlich and Q. Yin,
Mean field limit and propagation of chaos for a pedestrian flow model, Journal of Statistical Physics, 166 (2017), 211-229.
doi: 10.1007/s10955-016-1679-5. |
[7] |
A. Chertock, A. Kurganov, A. Polizzi and I. Timofeyev,
Pedestrian flow models with slowdown interactions, Math. Models Methods Appl. Sci., 24 (2014), 249-275.
doi: 10.1142/S0218202513400083. |
[8] |
E. Cristiani, B. Piccoli and A. Tosin,
Multiscale Modeling of Pedestrian Dynamics, vol. 12 of MS&A. Modeling, Simulation and Applications, Springer, Cham, 2014.
doi: 10.1007/978-3-319-06620-2. |
[9] |
P. Degond, C. Appert-Rolland, M. Moussaïd, J. Pettré and G. Theraulaz,
A hierarchy of heuristic-based models of crowd dynamics, J. Stat. Phys., 152 (2013), 1033-1068.
doi: 10.1007/s10955-013-0805-x. |
[10] |
P. Degond and C. Ringhofer,
Stochastic dynamics of long supply chains with random breakdowns, SIAM J. Appl. Math., 68 (2007), 59-79.
doi: 10.1137/060674302. |
[11] |
P. Degond, C. Appert-Rolland, J. Pettré and G. Theraulaz,
Vision-based macroscopic pedestrian models, Kinet. Relat. Models, 6 (2013), 809-839.
doi: 10.3934/krm.2013.6.809. |
[12] |
G. Dimarco and S. Motsch,
Self-alignment driven by jump processes: Macroscopic limit and numerical investigation, Math. Models Methods Appl. Sci., 26 (2016), 1385-1410.
doi: 10.1142/S0218202516500330. |
[13] |
R. Etikyala, S. Göttlich, A. Klar and S. Tiwari,
Particle methods for pedestrian flow models: From microscopic to nonlocal continuum models, Math. Models Methods Appl. Sci., 24 (2014), 2503-2523.
doi: 10.1142/S0218202514500274. |
[14] |
I. I. Gikhman and A. V. Skorokhod, The Theory of Stochastic Processes. Ⅱ, Classics in Mathematics, Springer-Verlag, Berlin, 2004, Translated from the Russian by S. Kotz, Reprint of the 1975 edition.
doi: 10.1007/978-3-642-61921-2. |
[15] |
D. Helbing, A fluid dynamic model for the movement of pedestrians, Complex Systems, 6 (1992), 391-415, arXiv: cond-mat/9805213. |
[16] |
D. Helbing and P. Molnár, Social force model for pedestrian dynamics, Physical Review E, 51 (1998), 4282-4286, arXiv: cond-mat/9805244.
doi: 10.1103/PhysRevE.51.4282. |
[17] |
R. L. Hughes,
A continuum theory for the flow of pedestrians, Transportation Research Part B: Methodological, 36 (2002), 507-535.
doi: 10.1016/S0191-2615(01)00015-7. |
[18] |
P.-E. Jabin,
Macroscopic limit of Vlasov type equations with friction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 651-672.
doi: 10.1016/S0294-1449(00)00118-9. |
[19] |
P.-E. Jabin,
Various levels of models for aerosols, Math. Models Methods Appl. Sci., 12 (2002), 903-919.
doi: 10.1142/S0218202502001957. |
[20] |
A. Jüngel, Transport Equations for Semiconductors, vol. 773 of Lecture Notes in Physics, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-89526-8. |
[21] |
A. Klar, F. Schneider and O. Tse,
Approximate models for stochastic dynamic systems with velocities on the sphere and associated fokker-planck equations, Kinetic and Related Models, 7 (2014), 509-529.
doi: 10.3934/krm.2014.7.509. |
[22] |
R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.
doi: 10.1017/CBO9780511791253. |
[23] |
B. Piccoli and A. Tosin,
Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738.
doi: 10.1007/s00205-010-0366-y. |
[24] |
B. Piccoli and A. Tosin,
Pedestrian flows in bounded domains with obstacles, Contin. Mech. Thermodyn., 21 (2009), 85-107.
doi: 10.1007/s00161-009-0100-x. |
[25] |
M. Schultz, Stochastic transition model for pedestrian dynamics, in Pedestrian and Evacuation Dynamics 2012, Springer International Publishing, (2013), 971-985, arXiv: 1210.5554.
doi: 10.1007/978-3-319-02447-9_81. |
[26] |
A. Tordeux and A. Schadschneider, A stochastic optimal velocity model for pedestrian flow, in Parallel Processing and Applied Mathematics, Springer International Publishing, 9574 (2016), 528-538.
doi: 10.1007/978-3-319-32152-3_49. |
[27] |
A. Tordeux and A. Schadschneider, White and relaxed noises in optimal velocity models for pedestrian flow with stop-and-go waves, Journal of Physics A: Mathematical and Theoretical, 49 (2016), 185101, 16pp.
doi: 10.1088/1751-8113/49/18/185101. |






EOOC | ||
0.3251 | - | |
|
0.1755 | 0.8897 |
|
0.0717 | 1.2919 |
EOOC | ||
0.3251 | - | |
|
0.1755 | 0.8897 |
|
0.0717 | 1.2919 |
EOOC | ||
|
0.4457 | - |
|
0.2215 | 1.0085 |
|
0.0889 | 1.3170 |
EOOC | ||
|
0.4457 | - |
|
0.2215 | 1.0085 |
|
0.0889 | 1.3170 |
EOOC | ||
|
0.5203 | - |
|
0.2873 | 0.8567 |
|
0.1153 | 1.3176 |
EOOC | ||
|
0.5203 | - |
|
0.2873 | 0.8567 |
|
0.1153 | 1.3176 |
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