On the Convergence of Soft Potential Dynamics to Hard Sphere Dynamics

We address a question raised in the work of Gallagher, Saint-Raymond and Texier (From Newton to Boltzmann: Hard Spheres and Short-range Potentials, Z"urich Lectures in Advanced Mathematics, EMS, 2013) that concerns the convergence of soft-potential dynamics to hard sphere dynamics. In the case of two particles, we establish that hard sphere dynamics is the limit of soft sphere dynamics in the weak-star topology of BV. We view our result as establishing a topological method by which to construct weak solutions to the ODE of hard sphere motion.