Pinned Ball, Foldings and Particle Collisions 

In the foldings part, we estimated the convergence rate for the folding model on the unit sphere in \(R^n\). This is a toy model for the coupling approach to the convergence rate in densely packed pinned billiard balls on a torus. In the particle collisions part, we gave the asymptotic formula for the collision location of two sample particles with fixed initial locations and velocities distributed according to the microcanonical ensemble formula.