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From generalized kinetic theory to discrete velocity modeling of vehicular traffic. A stochastic game approach.
No full textThis paper reports about vehicular traffic modeling by methods of the discrete kinetic theory. The purpose is to detail a reference mathematical framework for some discrete velocity kinetic models recently introduced in the literature, which proved capable to reproduce interesting traffic phenomena without using experimental information as modeling assumptions. To this aim, we firstly derive a general discrete velocity kinetic framework with binary nonlocal interactions. Then, resorting to some ideas on the stochastic game theory, we outline specific modeling guidelines for vehicular traffic, and finally we discuss the derivation of the above-mentioned vehicular traffic models from these mathematical structures
Teoria cinetica discreta e teoria dei giochi stocastica per il traffico veicolare: modellistica e problemi matematici
No full textUn approccio multiscala alla dinamica delle folle mediante misure che evolvono nel tempo
Full text linkQuesto articolo riguarda la modellizzazione matematica di sistemi complessi viventi, in particolare le folle, mediante leggi di conservazione e metodi della teoria della misura. Introdurremo un quadro modellistico che permette di trattare sistemi dinamici discreti e continui mediante idee fenomenologiche e strumenti matematici co-muni, nonché di accoppiare le due descrizioni in un’ottica multiscala. Inoltre presenteremo una teoria qualitativa di buona positura e approssimazione numerica dei problemi ai valori iniziali e discuteremo le sue implicazioni sulla modellistica
Discrete kinetic and stochastic game theory for vehicular traffic: Modeling and mathematical problems
Full text linkn this thesis we are concerned with the mathematical modeling of vehicular traffic atthe kinetic scale. In more detail, starting from the general structures proposed by Arlottiet al. and by Bellomo, we develop a discrete kinetic framework in which thevelocity of the vehicles is not regarded as a continuous variable but can take a finite number of values only. Discrete kinetic models have historically been conceived in connection with the celebrated Boltzmann equation, primarily as mathematical tools to reduce the analytical complexity of the latter (see e.g., Bellomo and Gatignol, Gatignol): The Boltzmann's integro-differential equation is converted into a set of partial differential equations in time and space, which share with the former some good mathematical properties being at the same time easier to deal with. In the present context, however, the discretization of the velocity plays a specific role in modeling the system rather than being simply a mathematical simplification, because it allows one to relax the continuum hypothesis for the velocity variable and to include, at least partially, the strongly granular nature of the flow of cars in the kinetic theory of vehicular traffic. The discrete velocity framework also gives rise to an interesting structure of the interaction terms of the kinetic equations, which are inspired by the stochastic game theor
Proposal of a risk model for vehicular traffic: A Boltzmann-type kinetic approach
Full text linkThis paper deals with a Boltzmann-type kinetic model describing the interplay between vehicle dynamics and safety aspects in vehicular traffic. Sticking to the idea that the macroscopic characteristics of traffic flow, including the distribution of the driving risk along a road, are ultimately generated by one-to-one interactions among drivers, the model links the personal (i.e., individual) risk to the changes of speeds of single vehicles and implements a probabilistic description of such microscopic interactions in a Boltzmann-type collisional operator. By means of suitable statistical moments of the kinetic distribution function, it is finally possible to recover macroscopic relationships between the average risk and the road congestion, which show an interesting and reasonable correlation with the well-known free and congested phases of the flow of vehicles
The Boltzmann legacy revisited: kinetic models of social interactions
Full text linkThe application of classical methods of statistical mechanics, originally developed by Ludwig Boltzmann in gas dynamics, to the description of social phenomena is a success story that we try to outline in this paper. On one hand, it is nowadays a flourishing research line, which is more and more permeating different contexts such as the Econophysics, Sociophysics, Biomathematics, Transportation Engineering to name just a few of them. On the other hand, it is a fascinating mathematical challenge, because it requires the interplay of various complementary expertises: modelling, model analysis, numerics. In this paper, we try to give a taste of all of this using the social phenomenon of opinion formation as a motivating example
Markov jump processes and collision-like models in the kinetic description of multi-agent systems
Full text linkMulti-agent systems can be successfully described by kinetic models, which allow one to explore the large scale aggregate trends resulting from elementary microscopic interactions. The latter may be formalised as collision-like rules, in the spirit of the classical kinetic approach in gas dynamics, but also as Markov jump processes, which assume that every agent is stimulated by the other agents to change state according to a certain transition probability distribution. In this paper we establish a parallelism between these two descriptions, whereby we show how the understanding of the kinetic jump process models may be improved taking advantage of techniques typical of the collisional approach
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