Micro-macro issues in crowd motion modeling

Vortrag von Prof. Dr. Bertrand Maury

Datum: 04.10.17  Zeit: 16.15 - 17.15  Raum: ETH HG E 1.2

Modeling collections of living entities like human crowds has become a major challenge among physicists and mathematicians in the last two decades. This domain of investigation shares some features with the modeling of "classical" particle systems, but it also raises new issues from the mathematical standpoint. We shall focus here on very crude models (in terms of individual behavior), assuming everyone tends to selfishly realize a given goal (like exiting a room in fire). The resulting motion is then built as a trade-off between individual tendencies and congestion constraints. This standpoint can be straightforwardly instantiated at both microscopic and macroscopic scales. We shall present how the Wasserstein setting of Optimal Transportation makes it possible to transpose the microscopic framework (differential inclusions in a Hilbert space) at the macroscopic level. Beyond this formal analogy, we shall emphasize the deep differences between the two levels of description. In particular, while the macroscopic model exhibits a good old Laplacian, its microscopic counterpart relies on a bizarre discrete operator which in particular does not verify the maximum principle. We shall explain how the emergence of static jams upstream exits, as well as the so-called "faster is slower effect", can be explained by the very pathological properties of this operator.