# Rupert L. Frank

## Caltech & LMU Munich

### A ‘liquid-solid’ phase transition in a simple model for swarming

We consider a non-local optimization problem, which is motivated by a simple model for swarming and other self-assembly/aggregation models, and prove the existence of different phases. In particular, we show that in the large mass regime the ground state density profile is the characteristic function of a round ball. An essential ingredient in our proof is a strict rearrangement inequality with a quantitative error estimate. We formulate several open problems which might be amenable to PDE techniques.

The talk is based on joint work with E. Lieb.