# Student Probability Seminar

#### Rigorous Derivation of Brownian Motion from Deterministic Hard Spheres

**Speaker:**
Ryan Denlinger

**Location:**
Warren Weaver Hall 1314

**Date:**
Thursday, December 10, 2015, 2 p.m.

**Synopsis:**

The problem we consider is N identical hard spheres colliding in a periodic box. We will confine our attention to the Boltzmann-Grad scaling; this is a low-density limit which should imply the mutual independence of all particles, asymptotically as N tends to infinity, under very weak conditions on the initial data. The main difficulty, however, is that we want to consider the full deterministic evolution of all N particles, and deduce independence from first principles. We will describe a proof of asymptotic independence in the case of a very small perturbation of equilibrium; in particular, we only perturb the initial distribution of one "tagged" particle, or a finite number of particles. The diffusive limit was proven in this case, and the resulting evolution is a Brownian motion for the tagged particle. All results discussed in this talk are from Bodineau, Gallagher, Saint-Raymond (Invent. math. 2015).