Appears in collection : 2025 IHES Summer School – Statistical Aspects of Nonlinear Physics

Collective motions commonly refer to swarming phenomena, mimetism tendencies, collaboration behavior … We propose here to adopt a different standpoint, and to consider situations where interactions mainly stem from congestion principles. We shall in particular present some crowd motion models based on the assumption that entities have individual tendencies, which are counteracted by the presence of others, and investigate according to which principles « information » is propagated though the population. We will pay a special attention to models of the gradient flow type, which are built on the assumption that a given population tends to minimize some sort of global dissatisfaction (sum of individual dissatisfactions), together with an additional term which encodes the congestion contraints. We will present in details a microscopic version of this approach, together with macroscopic extensions, based on Optimal transportation principles. We shall also investigate how this framework, that is based on « classical mechanics » principles, and which carries in particular an Action-and-Reaction principle (symmetric interactions), can be distorted to elaborate more realistic interaction models, with asymmetric interactions between individuals.