Toward a Mathematical Theory of Behavioral-Social Dynamics for Pedestrian Crowds

This paper presents a new approach to behavioral-social dynamics of pedestrian crowds by suitable development of methods of the kinetic theory. It is shown how heterogeneous individual behaviors can modify the collective dynamics, as well as how local unusual behaviors can propagate in the crowd. The main feature of this approach is a detailed analysis of the interactions between dynamics and social behaviors.Comment: 22 pages, 5 figure

Superfluid hydrodynamics of polytropic gases:dimensional reduction and sound velocity

Motivated by the fact that two-component confined fermionic gases in Bardeen-Cooper-Schrieffer-Bose-Einstein condensate (BCS-BEC) crossover can be described through an hydrodynamical approach, we study these systems - both in the cigar-shaped configuration and in the disk-shaped one - by using a polytropic Lagrangian density. We start from the Popov Lagrangian density and obtain, after a dimensional reduction process, the equations that control the dynamics of such systems. By solving these equations we study the sound velocity as a function of the density by analyzing how the dimensionality affects this velocityComment: Accepted for publication in J. Phys. A: Math. Theo

Kinetic Theory and Swarming Tools to Modeling Complex Systems—Symmetry problems in the Science of Living Systems

This MPDI book comprises a number of selected contributions to a Special Issue devoted to the modeling and simulation of living systems based on developments in kinetic mathematical tools. The focus is on a fascinating research field which cannot be tackled by the approach of the so-called hard sciences—specifically mathematics—without the invention of new methods in view of a new mathematical theory. The contents proposed by eight contributions witness the growing interest of scientists this field. The first contribution is an editorial paper which presents the motivations for studying the mathematics and physics of living systems within the framework an interdisciplinary approach, where mathematics and physics interact with specific fields of the class of systems object of modeling and simulations. The different contributions refer to economy, collective learning, cell motion, vehicular traffic, crowd dynamics, and social swarms. The key problem towards modeling consists in capturing the complexity features of living systems. All articles refer to large systems of interaction living entities and follow, towards modeling, a common rationale which consists firstly in representing the system by a probability distribution over the microscopic state of the said entities, secondly, in deriving a general mathematical structure deemed to provide the conceptual basis for the derivation of models and, finally, in implementing the said structure by models of interactions at the microscopic scale. Therefore, the modeling approach transfers the dynamics at the low scale to collective behaviors. Interactions are modeled by theoretical tools of stochastic game theory. Overall, the interested reader will find, in the contents, a forward look comprising various research perspectives and issues, followed by hints on to tackle these

On the dynamics of social conflicts: looking for the Black Swan

This paper deals with the modeling of social competition, possibly resulting in the onset of extreme conflicts. More precisely, we discuss models describing the interplay between individual competition for wealth distribution that, when coupled with political stances coming from support or opposition to a government, may give rise to strongly self-enhanced effects. The latter may be thought of as the early stages of massive, unpredictable events known as Black Swans, although no analysis of any fully-developed Black Swan is provided here. Our approach makes use of the framework of the kinetic theory for active particles, where nonlinear interactions among subjects are modeled according to game-theoretical tools.Comment: 26 pages, 7 figure

On the mathematical theory of behavioral swarms emerging collective dynamics

This paper considers a system consisting of a number of interacting living entities whose state at the microscopic scale is heterogeneously distributed among the said entities. This state includes, in addition, the classical mechanical variables, such as position and velocity, also a behavioral variable which is modi ed by interactions. It is shown how the pioneering ideas proposed in Bellomo et al. [Towards a mathematical theory of behavioral swarms, ESAIM Control Optim. Calc. Var. 26 (2020) 125] can be developed towards modeling behavioral swarms within a quest towards a mathematical theory of living systems. The rst part of the paper presents a qualitative analysis of the emerging behaviors predicted by the model in aforementioned work. Some simulations follow to depict the said emerging behaviors. The last part of the paper is devoted to derive a new, more general theory in view of applications to model living systems.University of GranadaNational Research Foundation of Korea NRF-2020R1A2C3A0100388

Cosmological implications of Primordial Black Holes

The possibility that a relevant fraction of the dark matter might be comprised of Primordial Black Holes (PBHs) has been seriously reconsidered after LIGO's detection of a $\sim 30 M_{\odot}$ binary black holes merger. Despite the strong interest in the model, there is a lack of studies on possible cosmological implications and effects on cosmological parameters inference. We investigate correlations with the other standard cosmological parameters using cosmic microwave background observations, finding significant degeneracies, especially with the tilt of the primordial power spectrum and the sound horizon at radiation drag. However, these degeneracies can be greatly reduced with the inclusion of small scale polarization data. We also explore if PBHs as dark matter in simple extensions of the standard $\Lambda$CDM cosmological model induces extra degeneracies, especially between the additional parameters and the PBH's ones. Finally, we present cosmic microwave background constraints on the fraction of dark matter in PBHs, not only for monochromatic PBH mass distributions but also for popular extended mass distributions. Our results show that extended mass distribution's constraints are tighter, but also that a considerable amount of constraining power comes from the high-$\ell$ polarization data. Moreover, we constrain the shape of such mass distributions in terms of the correspondent constraints on the PBH mass fraction.Comment: 20 pages, 9 figures. Matches version accepted to publish in JCA