Kinetic Model for Vehicular Traffic with Continuum Velocity and Mean Field Interactions
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Vehicular trafficShort range interactionsLong range interactionsKinetic theory
Calvo, J.; Nieto, J.; Zagour, M. Kinetic Model for Vehicular Traffic with Continuum Velocity and Mean Field Interactions. Symmetry 2019, 11, 1093. [doi:10.3390/sym11091093]
SponsorshipJ.C. and J.N. are partially supported by Junta de Andalucía Project P12-FQM-954 and MINECO Project RTI2018-098850-B-I00. J.C. is supported by Universidad de Granada (“Plan propio de investigación, programa 9”) through FEDER funds. M.Z. was supported by CNRST (Morocco), project “Modèles Mathématiques appliqués à l’environnement, à l’imagerie médicale et aux biosystèmes”.
This paper is concerned with the modeling and mathematical analysis of vehicular traffic phenomena. We adopt a kinetic theory point of view, under which the microscopic state of each vehicle is described by: (i) position, (ii) velocity and also (iii) activity, an additional varible that we use to describe the quality of the driver-vehicle micro-system. We use methods coming from game theory to describe interactions at the microscopic scale, thus constructing new models within the framework of the Kinetic Theory of Active Particles; the resulting models incorporate some of the symmetries that are commonly found in the mathematical models of the kinetic theory of gases. Short-range interactions and mean field interactions are introduced and modeled to depict velocity changes related to passing phenomena. Our main goal is twofold: (i) to use continuum-velocity variables and (ii) to introduce a non-local acceleration term modeling mean field interactions, related to, for example, the presence of tollgates or traffic highlights.