Interacting particle systems on a lattice
Systems of interacting particles on a lattice are established in statistical physics since the 1970s and the pionner work of F. Spitzer and M.T. Liggett (see the book Liggett,1985). The systems are Markovian processes where particles jump according to exponential clocks with jump rates that depend on the current system state. For instance instance, a particle jump to the nearest site if it is free for the exclusion process (Spitzer, 1970), according to the state departure site for the zero-range process (Liggett, 1974) and the random average process (Roussignol, 1980), and according to the states of departure and arrival sites for the misanthrope process (Cocozza-Thivent, 1985).
Many pedestrian and traffic models by exclusion, zero-range, random average, or misanthrope processes exist in the literature. They can be seen as extension in continuous time of stochastic cellular automata models. Some processes (e.g. the zero-range) have product-form invariant distributions, making possible analytic descriptions of the model dynamics. In addition, event-based simulation allow efficient computing of trajectories in continuous time without discretisation scheme. This makes Markovian interacting particle systems a good candidate for large-scale simulation of complex geometries and traffic networks.
- 2023
- I. Ba and A. Tordeux, "Signalized and unsignalized road traffic intersection models: A comprehensive benchmark analysis", Collective Dynamics, vol. 8, pp. A144, 2023.
- 2021
- I. Ba and A. Tordeux, "Comparing Macroscopic First Order Models of Regulated and Unregulated Road Traffic Intersections" in Proceeding of 30th European Safety and Reliability (ESREL) Conference, 2021.
- 2018
- A. Tordeux, G. Lämmel, F. S. Hänseler and B. Steffen, "A mesoscopic model for large-scale simulation of pedestrian dynamics", Transportation Research Part C: Emerging Technologies, vol. 93, pp. 128-147, 2018. Elsevier.
Further references
A Tordeux, M Roussignol, J-P Lebacque and S Lassarre (2014) A stochastic jump process applied to traffic flow modelling. Transportmetrica 10(4):350–375.
R Eymard, M Roussignol and A Tordeux (2012) Convergence of a misanthrope process to the entropy solution of 1D problems. Stoch Proc Appl 122(11):3648–3679.
A Tordeux, S Lassarre, M Roussignol and A Schadschneider (2011) Stationary state properties of a traffic flow model mixing stochastic transport and car-following. Proceedings of Traffic and Granular Flow’11, Springer, pp 35–46. Presentation
T M Liggett (1985). Interacting particle systems (Vol. 2). New York: Springer.
C Cocozza-Thivent (1985). Processus des misanthropes. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 70(4), 509-523.
E D Andjel (1982). Invariant measures for the zero range process. The Annals of Probability, 10(3), 525-547.
M Roussignol (1980). Un processus de saut sur R à une infinité de particules. In Annales de l'IHP Probabilités et statistiques (Vol. 16, No. 2, pp. 101-108).
F Spitzer (1991). Interaction of Markov processes. In Random Walks, Brownian Motion, and Interacting Particle Systems (pp. 66-110). Birkhäuser, Boston, MA.
Here is a presentation about the mesoscopic pedestrian model for large-scale simulation (2018) and here is a presentation about interactive particle systems for traffic flow (2010).
