The collective motion of individuals (correlated motion of ants or migration of bacteria, flocks of birds, just to mention but a few) is a fascinating phenomenon capturing our attention. Besides the aesthetic aspects induced by such an expression of collective behavior, there are many crucial aspects of practical nature that attracted the interest of various scientific communities ranging from logistics, theoretical biology, and ecology to statistical physics and mathematics. On one hand, we wish to better understand, for instance, the formation of swimming patterns in large communities of fish to improve large scale fishing strategies. On the other hand, in congested flows, pedestrians display significantly different behaviors from those typical to situations when they are walking in free conditions. If panic situations occur, then small microscopic (individual-level) interactions can lead to disastrous macroscopic patterns (e.g. shock-like waves) leading to the jamming of a desired evacuation option or even to losses of human lives.
The aim of this school is to present, by means of 6 mini-courses, the state-of-the-art of the theoretical (statistical mechanics and mathematics) understanding of collective motions of crowds. The topics we include here are:
• Kinetic models for self-organized collective motion.
• Discrete and continuum dynamics of reacting and interacting
• Finite-speed propagation models of chemo-tactic movements.
• Modeling with measures:
(i) Multiscale modeling of pedestrian motions by time-evolving measures;
(ii) Motions and interactions in heterogeneous domains.
• Handling contacts in pedestrian dynamics: On the concept of pressure.
Multiscale models in social (networks) applications, eventually combining mean-field and kinetic equations with either microscopic or macroscopic objects, are approaches of strongly increasing importance and high potential for future quantitative research. Typically, individual-based models need to be intelligently coarse-grained to translate the relevant microstructure information to a mesoscopic (Boltzmann-level) or to a macroscopic (continuum) level.
Relevant questions include: What is the natural scaling for the averaging? How much microstructure information needs to be kept to capture the specific individual-level interaction responsible for the formation and propagation of the macroscopically-observed pattern (for instance, lane formation in pedestrian counterflow). What are the main microscopic interactions responsible for the macroscopic cross-diffusion transport mechanism sometimes arising in pedestrian’s motion?
Within the frame of this school, we emphasize on one hand the role played by measure theory in deriving averaged equations, while on the other hand we show how measure theory can be used to prove rigorously the mean-field derivation of chemo-tactic movements, e.g. Numerical simulations of generic collective motions as well as experimental findings and simulations of pedestrian flows hosting macroscopic patterns will be pointed out.
The target audience of this summer school are graduate students, PhD candidates and young faculty members in mathematics, applied theoretical physics and biology, as well as (chemical, transportation, mechanical, ...) engineering having a strong research interest in understanding the multiscale complexity of the collective motion behavior. The participants are expected to have a good mathematical background. We hope that everybody will be willing to actively participate in both discussion and poster sessions.
KEYWORDS: Lanes and Flocks Formation, Social and Behavioral Sciences, Conservation laws, Micro and Macro Models, Mass Measures, Averaging, Social Networks, Initial and Boundary Value Problems.