Abstract
Many macroscopic systems are justified by a condition of local microscopic equilibrium. In the case of particles in deterministic interactions, this is a condition of statistical independence between the particles, known as "molecular chaos", propagated in time in the limit of their large number.
For mean-field models, immense progress has been made when the microscopic interaction is regular. However, if the interaction diverges when two particles move closer together, it is much less clear that such independence occurs.
After outlining some of the issues involved in this question, we shall present the modulated energy method, which makes it possible to quantitatively justify this property of chaos when the solution to the mean-field PDE has sufficient regularity. When the interaction becomes more singular than the Coulomb potential (which is the case for porous media models), this regularity is obtained with viscosity. In the case of a repulsive interaction, the study of the asymptotic behaviour of the equation allows us to demonstrate that molecular chaos is propagated uniformly in time.