This paper presents a review and critical analysis on the modeling of the dynamics of vehicular traffic, human crowds and swarms seen as living and, hence, complex systems. It contains a survey of the kinetic models developed in the last 10 years on the aforementioned topics so that overlapping with previous reviews can be avoided. Although the main focus of this paper lies on the mesoscopic models for collective dynamics, we provide a brief overview on the corresponding micro and macroscopic models, and discuss intermediate role of mesoscopic model between them. Moreover, we provide a number of selected challenging research perspectives for readers’ attention.

Kinetic theory methods are applied in this paper to model the dynamics of vehicular traffic. The basic idea is to consider each vehicular-driver system as a single part, or micro-system, of a large complex system, in order to capture the heterogeneous behavior of all the micro-systems that compose the overall system. The evolution of the system is ruled by nonlinearly additive interactions described by stochastic games. A qualitative analysis for the proposed model with discrete states is developed, showing well-posedness of the related Cauchy problem for the spatially homogeneous case and for the spatially nonhomogeneous case, the latter with periodic boundary conditions. Numerical simulations are also performed, with the aim to show how the model proposed is able to reproduce empirical data and to show emerging behavior as the formation of clusters.

This paper presents a new mathematical model of vehicular traffic, based on the methods of the generalized kinetic theory, in which the space of microscopic states (position and velocity) of the vehicles is genuinely discrete. While in the recent literature discrete-velocity kinetic models of car traffic have already been successfully proposed, this is, to our knowledge, the first attempt to account for all aspects of the physical granularity of car flow within the formalism of the aforesaid mathematical theory. Thanks to a rich but handy structure, the resulting model allows one to easily implement and simulate various realistic scenarios giving rise to characteristic traffic phenomena of practical interest (e.g., queue formation due to roadworks or to a traffic light). Moreover, it is analytically tractable under quite general assumptions, whereby fundamental properties of the solutions can be rigorously proved.Classically, the kinetic representation of vehicular traffic along a one-way road is provided by a distribution function f over the mechanical microscopic state of the vehicles. The latter is identified by the scalar position x ∈ D x and speed v ∈ D v , where D x , D v ⊆ R are the spatial and speed domains, respectively. While the former may either be a bounded interval, such as [0, L], L > 0 being the length of the road, or coincide with the whole real axis, the latter is normally of the form D v = [0, V max ], where V max > 0 is the maximum speed allowed along the road, or possibly D v = [0, V ′ max ], with V ′ max ≥ V max corresponding to the maximum average speed attainable by a single vehicle in free flow conditions. The distribution function, is then such that f (t, x, v)dxdv is the (infinitesimal) number of vehicles that at time t are located between x and x + dx and travel with a speed between v and v + dv.In the above presentation of the kinetic approach, the spatial position and speed of the vehicles are tacitly assumed to be continuously distributed over D x × D v . However, this does not reflect correctly the physical reality of vehicular flow. Indeed, the number of vehicles along a road is normally not large enough for the continuity of the distribution function over the microscopic states to be an acceptable approximation (like in the classical kinetic theory of gases). Vehicles do not span continuously the whole set of admissible speeds; likewise, one cannot expect that they are so spread in space that at every point x of the road there may be some. In other words, the actual distribution of vehicles in space, as well as that of their speeds, is strongly granular. It is reasonable to expect this fact to have a nontrivial impact on the resulting dynamics. It is hence worth being taken into account in a mathematical model.Recently discrete velocity models have been introduced [3, 7, 10]. The idea is to relax the hypothesis that the speed distribution is continuous by introducing in the domain D v a lattice of discrete speeds {v 1

In this paper we investigate the ability of some recently introduced discrete kinetic models of vehicular traffic to catch, in their large time behavior, typical features of theoretical fundamental diagrams. Specifically, we address the so-called "spatially homogeneous problem" and, in the representative case of an exploratory model, we study the qualitative properties of its solutions for a generic number of discrete microstates. This includes, in particular, asymptotic trends and equilibria, whence fundamental diagrams originate.Comment: 14 page

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