The collective dynamics of a stochastic Port-Hamiltonian self-driven agent model in one dimension
Authors
Matthias Ehrhardt, Thomas Kruse, Antoine Tordeux
Abstract
This paper studies the collective motion of self-driven agents in a one-dimensional space with periodic boundaries, using a stochastic Port-Hamiltonian system (PHS) with symmetric nearest-neighbor interactions and additive Brownian noise as an external input. In the case of a quadratic potential the PHS is an Ornstein-Uhlenbeck process for which we explicitly determine the distribution for any time t ≥ 0 and in the limit t → ∞. In particular, we characterize the collective motion by showing that the agents’ positions tend to build exactly one cluster. This is confirmed in simulations that show rapid and coordinated motion among agents, driven by noise, despite the absence of a preferred direction of motion in the model. Remarkably, the theoretical properties observed in the Ornstein-Uhlenbeck process also emerge in simulations of the nonlinear model incorporating a general interaction potential.
Citation
- Journal: ESAIM: Mathematical Modelling and Numerical Analysis
- Year: 2024
- Volume: 58
- Issue: 2
- Pages: 515–544
- Publisher: EDP Sciences
- DOI: 10.1051/m2an/2024004
BibTeX
@article{Ehrhardt_2024,
title={{The collective dynamics of a stochastic Port-Hamiltonian self-driven agent model in one dimension}},
volume={58},
ISSN={2804-7214},
DOI={10.1051/m2an/2024004},
number={2},
journal={ESAIM: Mathematical Modelling and Numerical Analysis},
publisher={EDP Sciences},
author={Ehrhardt, Matthias and Kruse, Thomas and Tordeux, Antoine},
year={2024},
pages={515--544}
}
References
- Acebrón, J. A., Bonilla, L. L., Pérez Vicente, C. J., Ritort, F. & Spigler, R. The Kuramoto model: A simple paradigm for synchronization phenomena. Reviews of Modern Physics vol. 77 137–185 (2005) – 10.1103/revmodphys.77.137
- Ballerini, M. et al. Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study. Proceedings of the National Academy of Sciences vol. 105 1232–1237 (2008) – 10.1073/pnas.0711437105
- Barberis, L. & Peruani, F. Phase separation and emergence of collective motion in a one-dimensional system of active particles. The Journal of Chemical Physics vol. 150 (2019) – 10.1063/1.5085840
- Carlitz, L. Some theorems on Bernoulli numbers of higher order. Pacific Journal of Mathematics vol. 2 127–139 (1952) – 10.2140/pjm.1952.2.127
- Chandler, R. E., Herman, R. & Montroll, E. W. Traffic Dynamics: Studies in Car Following. Operations Research vol. 6 165–184 (1958) – 10.1287/opre.6.2.165
- Chaté, H., Ginelli, F., Grégoire, G., Peruani, F. & Raynaud, F. Modeling collective motion: variations on the Vicsek model. The European Physical Journal B vol. 64 451–456 (2008) – 10.1140/epjb/e2008-00275-9
- Ciuffo, B. et al. Requiem on the positive effects of commercial adaptive cruise control on motorway traffic and recommendations for future automated driving systems. Transportation Research Part C: Emerging Technologies vol. 130 103305 (2021) – 10.1016/j.trc.2021.103305
- Cordoni, F., Di Persio, L. & Muradore, R. Stabilization of bilateral teleoperators with asymmetric stochastic delay. Systems & Control Letters vol. 147 104828 (2021) – 10.1016/j.sysconle.2020.104828
- Cordoni, F., Di Persio, L. & Muradore, R. Stochastic Port-Hamiltonian Systems. Journal of Nonlinear Science vol. 32 (2022) – 10.1007/s00332-022-09853-2
- Cvijović, D. & Srivastava, H. M. Closed-form summation of the Dowker and related sums. Journal of Mathematical Physics vol. 48 (2007) – 10.1063/1.2712895
- Cvijović, D. & Srivastava, H. M. Closed-form summations of Dowker’s and related trigonometric sums. Journal of Physics A: Mathematical and Theoretical vol. 45 374015 (2012) – 10.1088/1751-8113/45/37/374015
- Czirók, A., Barabási, A.-L. & Vicsek, T. Collective Motion of Self-Propelled Particles: Kinetic Phase Transition in One Dimension. Physical Review Letters vol. 82 209–212 (1999) – 10.1103/physrevlett.82.209
- da, F., Carlos, Glasser, L., M. & Kowalenko, V. Generalized cosecant numbers and trigonometric inverse power sums. Applicable Analysis and Discrete Mathematics vol. 12 70–109 (2018) – 10.2298/aadm1801070f
- De, R. & Chakraborty, D. Collective motion: Influence of local behavioural interactions among individuals. Journal of Biosciences vol. 47 (2022) – 10.1007/s12038-022-00277-4
- DEGOND, P., DIMARCO, G. & MAC, T. B. N. HYDRODYNAMICS OF THE KURAMOTO–VICSEK MODEL OF ROTATING SELF-PROPELLED PARTICLES. Mathematical Models and Methods in Applied Sciences vol. 24 277–325 (2013) – 10.1142/s0218202513400095
- Dowker, J. S. Casimir effect around a cone. Physical Review D vol. 36 3095–3101 (1987) – 10.1103/physrevd.36.3095
- Dowker, J. S. Heat kernel expansion on a generalized cone. Journal of Mathematical Physics vol. 30 770–773 (1989) – 10.1063/1.528395
- Dowker, J. S. On Verlinde’s formula for the dimensions of vector bundles on moduli spaces. Journal of Physics A: Mathematical and General vol. 25 2641–2648 (1992) – 10.1088/0305-4470/25/9/033
- Fang, Z. & Gao, C. Stabilization of Input-Disturbed Stochastic Port-Hamiltonian Systems Via Passivity. IEEE Transactions on Automatic Control vol. 62 4159–4166 (2017) – 10.1109/tac.2017.2676619
- Fong C.K., Course Notes in Linear Algebra, MATH 2107, February (2008).
- Gardiner C.W., Handbook of Stochastic Methods, Vol. 3. Springer Berlin (1985).
- Gautrais, J. et al. Deciphering Interactions in Moving Animal Groups. PLoS Computational Biology vol. 8 e1002678 (2012) – 10.1371/journal.pcbi.1002678
- Gazis, D. C., Herman, R. & Rothery, R. W. Nonlinear Follow-the-Leader Models of Traffic Flow. Operations Research vol. 9 545–567 (1961) – 10.1287/opre.9.4.545
- Großmann, R., Aranson, I. S. & Peruani, F. A particle-field approach bridges phase separation and collective motion in active matter. Nature Communications vol. 11 (2020) – 10.1038/s41467-020-18978-5
- Gunter, G. et al. Are Commercially Implemented Adaptive Cruise Control Systems String Stable? IEEE Transactions on Intelligent Transportation Systems vol. 22 6992–7003 (2021) – 10.1109/tits.2020.3000682
- Herman, R., Montroll, E. W., Potts, R. B. & Rothery, R. W. Traffic Dynamics: Analysis of Stability in Car Following. Operations Research vol. 7 86–106 (1959) – 10.1287/opre.7.1.86
- Keta, Y.-E., Jack, R. L. & Berthier, L. Disordered Collective Motion in Dense Assemblies of Persistent Particles. Physical Review Letters vol. 129 (2022) – 10.1103/physrevlett.129.048002
- Khound, P., Will, P., Tordeux, A. & Gronwald, F. Extending the adaptive time gap car-following model to enhance local and string stability for adaptive cruise control systems. Journal of Intelligent Transportation Systems vol. 27 36–56 (2021) – 10.1080/15472450.2021.1983810
- Lamoline, F. & Winkin, J. J. On stochastic port-hamiltonian systems with boundary control and observation. 2017 IEEE 56th Annual Conference on Decision and Control (CDC) 2492–2497 (2017) doi:10.1109/cdc.2017.8264015 – 10.1109/cdc.2017.8264015
- Lamoline F. and Hastir A., On Dirac structure of infinite-dimensional stochastic port-Hamiltonian systems. Preprint: arXiv:2210.06358 (2022).
- Makridis, M., Mattas, K., Anesiadou, A. & Ciuffo, B. OpenACC. An open database of car-following experiments to study the properties of commercial ACC systems. Transportation Research Part C: Emerging Technologies vol. 125 103047 (2021) – 10.1016/j.trc.2021.103047
- Marchetti, M. C. et al. Hydrodynamics of soft active matter. Reviews of Modern Physics vol. 85 1143–1189 (2013) – 10.1103/revmodphys.85.1143
- Marrocco, A. Numerical simulation of chemotactic bacteria aggregation via mixed finite elements. ESAIM: Mathematical Modelling and Numerical Analysis vol. 37 617–630 (2003) – 10.1051/m2an:2003048
- Martin, D. et al. Fluctuation-Induced Phase Separation in Metric and Topological Models of Collective Motion. Physical Review Letters vol. 126 (2021) – 10.1103/physrevlett.126.148001
- Maury, B. & Venel, J. A discrete contact model for crowd motion. ESAIM: Mathematical Modelling and Numerical Analysis vol. 45 145–168 (2010) – 10.1051/m2an/2010035
- Moreno, J. C., Rubio Puzzo, M. L. & Paul, W. Collective dynamics of pedestrians in a corridor: An approach combining social force and Vicsek models. Physical Review E vol. 102 (2020) – 10.1103/physreve.102.022307
- Nemoto, T., Fodor, É., Cates, M. E., Jack, R. L. & Tailleur, J. Optimizing active work: Dynamical phase transitions, collective motion, and jamming. Physical Review E vol. 99 (2019) – 10.1103/physreve.99.022605
- Pavliotis, G. A. Stochastic Processes and Applications. Texts in Applied Mathematics (Springer New York, 2014). doi:10.1007/978-1-4939-1323-7 – 10.1007/978-1-4939-1323-7
- Pipes, L. A. An Operational Analysis of Traffic Dynamics. Journal of Applied Physics vol. 24 274–281 (1953) – 10.1063/1.1721265
- Ramaswamy, S. Active matter. Journal of Statistical Mechanics: Theory and Experiment vol. 2017 054002 (2017) – 10.1088/1742-5468/aa6bc5
- Rashad, R., Califano, F., van der Schaft, A. J. & Stramigioli, S. Twenty years of distributed port-Hamiltonian systems: a literature review. IMA Journal of Mathematical Control and Information vol. 37 1400–1422 (2020) – 10.1093/imamci/dnaa018
- Rüdiger B., Tordeux A. and Ugurcan B., Stability analysis of a stochastic port-Hamiltonian car-following model. Preprint: arXiv:2212.05139 (2022).
- Satoh, S. Input‐to‐state stability of stochastic port‐Hamiltonian systems using stochastic generalized canonical transformations. International Journal of Robust and Nonlinear Control vol. 27 3862–3885 (2017) – 10.1002/rnc.3769
- Satoh, S. & Fujimoto, K. Passivity Based Control of Stochastic Port-Hamiltonian Systems. IEEE Transactions on Automatic Control vol. 58 1139–1153 (2013) – 10.1109/tac.2012.2229791
- Shaebani, M. R., Wysocki, A., Winkler, R. G., Gompper, G. & Rieger, H. Computational models for active matter. Nature Reviews Physics vol. 2 181–199 (2020) – 10.1038/s42254-020-0152-1
- Stern, R. E. et al. Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments. Transportation Research Part C: Emerging Technologies vol. 89 205–221 (2018) – 10.1016/j.trc.2018.02.005
- Tordeux, A. & Totzeck, C. Multi-scale description of pedestrian collective dynamics with port-Hamiltonian systems. Networks and Heterogeneous Media vol. 18 906–929 (2023) – 10.3934/nhm.2023039
- Treiber, M., Kesting, A. & Helbing, D. Delays, inaccuracies and anticipation in microscopic traffic models. Physica A: Statistical Mechanics and its Applications vol. 360 71–88 (2006) – 10.1016/j.physa.2005.05.001
- van der Schaft, A. Port-Hamiltonian systems: an introductory survey. Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006 1339–1365 (2007) doi:10.4171/022-3/65 – 10.4171/022-3/65
- van der Schaft, A. & Jeltsema, D. Port-Hamiltonian Systems Theory: An Introductory Overview. Foundations and Trends® in Systems and Control vol. 1 173–378 (2014) – 10.1561/2600000002
- Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I. & Shochet, O. Novel Type of Phase Transition in a System of Self-Driven Particles. Physical Review Letters vol. 75 1226–1229 (1995) – 10.1103/physrevlett.75.1226
- Vicsek, T. & Zafeiris, A. Collective motion. Physics Reports vol. 517 71–140 (2012) – 10.1016/j.physrep.2012.03.004
- Wang, T., Li, G., Zhang, J., Li, S. & Sun, T. The effect of Headway Variation Tendency on traffic flow: Modeling and stabilization. Physica A: Statistical Mechanics and its Applications vol. 525 566–575 (2019) – 10.1016/j.physa.2019.03.116