Collapse of inelastic hard spheres in dimension d ≥ 2

2024/02/21 by Dolmaire, Théophile, Velázquez, Juan J. L.
#FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2402.13803

Abstract

We investigate the collapse of three inelastic particles in dimension d ≥ 2. We obtain general results of convergence and asymptotics concerning the variables of the dynamical system describing a collapsing system of particles. We prove a complete classification of the singularities when a collapse of three particles takes place, obtaining only two possible orders of collisions between the particles. In the first case we recover that the particles arrange in a nearly-linear chain, already studied by Zhou and Kadanoff, and in the second case we obtain that the particles arrange in a triangle, and we show that, after sufficiently many collisions, the particles collide according to a unique order of collisions, which is periodic. Finally, we construct an initial configuration leading to a nearly-linear collapse, stable under perturbations, and such that the angle between the particles at the time of collapse can be chosen a priori, with an arbitrary precision.

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