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Description
Many macroscopic systems can be justified under a microscopic local equilibrium condition. In the case of deterministic interacting particles, this corresponds to a statistical independence condition—known as molecular chaos—which is propagated over time in the limit of a large number of particles.
In mean-field models, substantial progress has been made when the microscopic interaction is regular. However, when the interaction becomes singular—i.e., diverges as two particles come close—the validity of such independence is far less clear.
After outlining some of the main challenges related to this issue, we will introduce the modulated energy method, which allows for a quantitative justification of molecular chaos, provided the solution to the mean-field PDE has sufficient regularity. When the interaction is more singular than the Coulomb potential (as is the case in some models of porous media), such regularity can be obtained by introducing viscosity. In the repulsive case, analyzing the asymptotic behavior of the equation then allows us to demonstrate that molecular chaos is propagated uniformly in time.
We will then extend the discussion to nonlinear mobilities. When the mobility is degenerate—typically governed by an inverse power law—it is expected that the solutions exhibit infinite speed of propagation. We will show how such solutions can indeed be constructed using simple maximum principles for the fractional Laplacian.
