Inelastic Particle Clusters from Cumulative Momenta

We consider a physical system comprising discrete massive particles on the real line whose trajectories interact via perfectly inelastic collision, also known as sticky particles. It turns out that polygons formed in a convex “cumulative momentum diagram” of the initial conditions allow us to easily predict how many particle clusters form as time $t\to \mathrm{\infty }$. We explore an application of this to a unit mass system with $\pm 1$ velocities, which has ties to simple symmetric random walks and lattice path counting.