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

A direct method for the Boltzmann equation based on a pseudo-spectral velocity space discretization

A deterministic method is proposed for solving the Boltzmann equation. The method employs a Galerkin discretization of the velocity space and adopts, as trial and test functions, the collocation basis functions based on weights and roots of a Gauss-Hermite quadrature. This is defined by means of half- and/or full-range Hermite polynomials depending whether or not the distribution function presents a discontinuity in the velocity space. The resulting semi-discrete Boltzmann equation is in the form of a system of hyperbolic partial differential equations whose solution can be obtained by standard numerical approaches. The spectral rate of convergence of the results in the velocity space is shown by solving the spatially uniform homogeneous relaxation to equilibrium of Maxwell molecules. As an application, the two-dimensional cavity flow of a gas composed by hard-sphere molecules is studied for different Knudsen and Mach numbers. Although computationally demanding, the proposed method turns out to be an effective tool for studying low-speed slightly rarefied gas flows

Behavioral Human Crowds and Society

This chapter provides an introduction to the contents of this edited volume. In keeping with the style of the previous edited volumes, we also consider research perspectives. The first part of this chapter contributes to the selection of some key perspectives that take into account not only the technical interest of modeling and simulation, but also the impact that this research activity can have on the well-being of society. The second part provides a brief introduction to the contents of the chapters that follow this editorial introduction. The contents of the chapter refer both to the aforementioned key topics and to the contents of the preceding edited volumes (Bellomo and Gibelli, Behavioural human crowds, recent results towards new research frontiers. In: Bellomo, Gibelli (eds) Crowd dynamics, Volume 3 - Modeling and social applications in the time of COVID 19. Birkhäuser, New York, pp 1–9, 2021; Bellomo et al., Behavioural human crowds. In: Gibelli (ed) Crowd dynamics, Volume 2 - Theory, models, and applications. Birkhäuser, New York, pp 1–10, 2020; Gibelli and Bellomo, Behavioral human crowds. In: Crowd dynamics, Volume 1 - Theory, models, and safety problems. Birkhäuser, New York, pp 1–14, 2018).</p

Direct simulation Monte Carlo applications to liquid-vapor flows

The paper aims at presenting Direct Simulation Monte Carlo (DSMC) extensions and applications to dense fluids. A succinct review of past and current research topics is presented, followed by a more detailed description of DSMC simulations for the numerical solution of the Enskog-Vlasov equation, applied to the study of liquid-vapor flows. Results about simulations of evaporation of a simple liquid in contact with a dense vapor are presented as an example

Molecular kinetic modelling of non-equilibrium transport of confined van der Waals fluids

A thermodynamically consistent kinetic model is proposed for the non-equilibrium transport of confined van der Waals fluids, where the long-range molecular attraction is considered by a mean-field term in the transport equation, and the transport coefficients are tuned to match the experimental data. The equation of state of the van der Waals fluids can be obtained from an appropriate choice of the pair correlation function. By contrast, the modified Enskog theory predicts non-physical negative transport coefficients near the critical temperature and may not be able to recover the Boltzmann equation in the dilute limit. In addition, the shear viscosity and thermal conductivity are predicted more accurately by taking gas molecular attraction into account, while the softened Enskog formula for hard-sphere molecules performs better in predicting the bulk viscosity. The present kinetic model agrees with the Boltzmann model in the dilute limit and with the Navier-Stokes equations in the continuum limit, indicating its capability in modelling dilute-to-dense and continuum-to-non-equilibrium flows. The new model is examined thoroughly and validated by comparing it with the molecular dynamics simulation results. In contrast to the previous studies, our simulation results reveal the importance of molecular attraction even for high temperatures, which holds the molecules to the bulk while the hard-sphere model significantly overestimates the density near the wall. Because the long-range molecular attraction is considered appropriately in the present model, the velocity slip and temperature jump at the surface for the more realistic van der Waals fluids can be predicted accurately