Pinned Billiard Balls in Half-Space and on a Flat Torus

Systems of pinned billiard balls serve as simplified models of collisions, where all particles remain fixed in their positions while their (pseudo-)velocities evolve in accordance with the laws of conservation of energy and momentum. Under free-boundary conditions, Athreya, Burdzy, and Duarte have established the maximum upper bound for the number of pseudo-collisions, thereby demonstrating that the number of collisions is finite. In this project, we do extensive numerical simulations to study two alternative environments. First, we consider the balls arranged in half-space with a single ball with inward (pseudo-)velocity. Numerical simulations suggest that in equilibrium, most of the energy is concentrated near the boundary. Second, when the balls are arranged on a flat torus, we find that in the stationary regime, the distributions of the velocity components are i.i.d. normal, and therefore the energies of the balls are exponentially distributed. Additionally, we find that the components of the velocities in the direction of impact between two touching balls are uncorrelated.