An Out-of-Equilibrium 1D Particle System Undergoing Perfectly Plastic Collisions

Abstract

At time zero, there are $N$ identical point particles in the line (1D) which are characterized by their positions and velocities. Both values are given randomly and independently from each other, with arbitrary probability densities. Each particle evolves at constant velocity until eventually they meet. When this happens, a perfectly-plastic collision is produced, resulting in a new particle composed by the sum of their masses and the weighted average velocity. The merged particles evolve indistinguishably from the non-merged ones, i.e. they move at constant velocity until a new plastic collision eventually happens. As in any open system, the particles are not confined to any region or reservoir, so as time progresses, they go on to infinity. From this non-equilibrium process, the number of (now, non-identical) final particles, $\tilde{X}_N$, the distribution of masses of these final particles and the kinetic energy loss from all plastic collisions, is studied. The principal findings shown in this paper are outlined as follows: (1) A method has been developed to determine the number and mass of the final particles based solely on the initial conditions, eliminating the need to evolve the particle system. (2) A similar model of merging particles, with a universal number of final particles, $\tilde{Z}_N$, is introduced. (3) Strong evidence that $\tilde{X}_N$ is also universal and has the same law of probability as $\tilde{Z}_N$ is presented. (4) An accurate approximation of the energy loss is presented. (5) Results for $\tilde{X}_N$ for an explosive-like initial condition are analyzed.