# Vehicular trafic, human crowds and social systems

This research line deals with the characterisation of the macroscopic collective behaviour of systems of agents (“particles”) seen as a quality which emerges spontaneously from the microscopic interactions among the single individuals.

- VEHICULAR TRAFFIC

Vehicles along a road produce a one-dimensional particle flow governed, at the microscopic scale, mainly by their mutual interactions which cause speed changes through acceleration and braking. The link between such microscopic dynamics and the time evolution of the macroscopic distribution of the vehicles can be investigated by the methods of the kinetic theory, taking inspiration from the Boltzmann-type binary interaction algorithms and the corresponding Fokker-Planck asymptotic approximation (mean field). In particular, this allows one to characterise both analytically and numerically the so-called fundamental diagrams of traffic, i.e. relationships between the vehicle density and their mean speed and average flux at equilibrium. These diagrams, which are a typical emergent property of the system due to the collective evolution of the vehicles, can be compared with the numerous experimental measurements to validate and calibrate the models. They can also be used as closure relationships in the macroscopic models, thereby keeping track of the microscopic interactions.

- CROWD DYNAMICS

Pedestrian traffic shares some analogies with vehicular traffic. For instance, its phenomenology is intrinsically multiscale, because the observable flow results from the superposition of simple microscopic interactions among few individuals who predict and avoid mutual collisions via local corrections of their velocity (speed and orientation). Nevertheless, there are also considerable differences. Pedestrian flows are mainly two-dimensional and multi-directional, which gives rise to a greater variety of observable phenomena and collective configurations. Passing from the microscopic interactions to the macroscopic dynamics is possible also in this case by applying the methods of the kinetic theory, in particular the Boltzmann-type binary interaction schemes and the Fokker-Planck asymptotic limits. This is especially useful to characterise the emergence in time of self-organised group patterns, such as e.g. the synchronised movement of the individuals walking in the same direction or the stripe formation in crossing flows.

However, two-dimensional kinetic models are computationally demanding. This suggests to consider also different types of models for simulating crowd flows in two-dimensional domains with a complex structure (presence of boundaries and of the corresponding conditions of either impenetrability or inflow or outflow, presence of inner obstacles). Therefore in this research line we also study multiscale crowd models based on measure-valued conservation laws. The basic idea is to describe the space distribution of the crowd by means of a measure, which contains both a regular part with density (with respect to the Lebesgue measure) and a singular part with Dirac deltas. In this way it is possible to integrate the microscopic and the macroscopic descriptions, allowing for joint dynamical effects which could not be obtained at a single scale.

- SOCIAL SYSTEMS

The main objective of this research line is to contribute to the development of a mathematical theory for the analysis of complex social phenomena, such as, for example, the formation of opinions in specific "interest groups" in the absence / presence of "opinion leaders" and external actions of conditioning, the selection of dominant opinions and the emergence of consensus, phenomena of imitation (herding) in financial events and other aspects of behavioral economics. In general modeling aspects of complexity in social dynamics are object of study.

To this end, a formalism is mainly based on integro-differential equations, typical of populations with structure in which the playing entities (typically individuals) are characterized by a microscopic state relative to their social position or other variables, such as, for example, initial political opinions. In addition to the construction of the model is performed a qualitative analysis of the computational and mathematical problems generated by the application, aimed at highlighting the adherence to the reality of the model, exploring emergent behaviors.