On the Convergence of Soft Potential Dynamics to Hard Sphere Dynamics

TL;DR

This paper proves that soft potential particle dynamics converge to hard sphere dynamics in a specific mathematical topology, providing a new method to construct solutions for hard sphere motion.

Contribution

It introduces a topological approach to demonstrate the convergence of soft potential to hard sphere dynamics, filling a gap in the mathematical understanding of these systems.

Findings

Hard sphere dynamics is the limit of soft sphere dynamics in the weak-star BV topology.

The result provides a new topological method for constructing weak solutions to hard sphere ODEs.

The convergence is established specifically for the two-particle case.

Abstract

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.