# Pure Math Seminar

Tuesday, January 17, 2023 at 10:00am to 11:00am

MIT-Math Dept., /Room 2-449

**Speaker**: Matthew Rosenzweig (MIT)

**Title**: Recent progress on mean-field limits for systems with Riesz interactions

**Abstract**:

In statistical physics, many particle models are described by an interaction energy determined by the Coulomb potential, or more generally an inverse power law called a Riesz potential. To this energy, one can associate a dynamics, either conservative or dissipative, which takes the form of a coupled system of nonlinear differential equations. In principle, one could solve this system of differential equations directly and perfectly describe the behavior of every particle in the system. But in practice, the number of particles (e.g., 10^{23 } in a gas) is too large for this to be feasible. Instead, one can focus on the "average" behavior of a particle, which is encoded by the empirical measure of the system. Formally, this measure converges to a solution of a certain nonlinear partial differential equation, called the mean-field limit, as the number of particles tends to infinity; but proving this convergence is a highly nontrivial matter. We will review results over the past few years on mean-field limits for Riesz systems, including important questions such as how fast this limit occurs and how it deteriorates with time, and discuss open questions that still remain.

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**https://mit.zoom.us/j/94994298254**

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