17. April 2019
Robert Patterson (WIAS Berlin)
Kinetic Interaction Clusters
Abstract:
When trying to arrive at the Boltzmann equation by a limiting process starting from a collection of perfectly elastic billiard balls one major challenge is the web of dependency that arises as the billiards collide with each other and exchange momentum. This is a difficult problem which I shall not address! However, Kac introduced a stochastic model that is easier to analyse than the deterministic billiard model. In the Kac setting I will explain what interaction clusters are and show why they satisfy a generalisation of the Smoluchowski coagulation equation. I will also present a representation as an inhomogeneous random graph and show that the interaction clusters undergo a phase transition at a critical time given in terms of the norm of a convolution operator on a space of square integrable functions.
This is joint work with Wolfgang Wagner, Sergio Simonella and Daniel Heydecker.