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, European Mathematical Society (EMS), Zürich, 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.