Beyond the classical mean-field scaling: Moderately interacting particles and their macroscopic limit

In this talk, I will start by giving an introduction to mean-field limits in general and their limitations when deriving Partial Differential Equations (PDE) by using classical mean-field scalings. In the classical regime, also called weakly interacting particles, the influence of each particle on any other particle scales like 1/N, where N is the number of particles. In the large population limit -- under suitable assumptions -- the particles become independent with a common density function which solves a certain type of PDE, the McKeanVlasov equation. However, this scaling regime does not recover many important partial differential equations from physics and biology, like the well-known porous media equation.

Hence, I will explain how changing the strength of interaction from 1/N to a factor which is still vanishing in the limit for N large, but slightly stronger than 1/N, changes the limiting dynamics. This regime is called moderate interaction since the strength of interaction is stronger than in the weak regime (1/N), but weaker than in the hydrodynamic limit (strongly interacting particles with strength O(1)) and attracted a lot of attention recently.

In addition — if time allows --- I will give insights on recent results on the fluctuations around the mean-field limit for the moderate regime, which can be seen as a Central Limit Theorem for interacting particles.