2024/01/02 by Daniel Fraiman, Fraiman, Daniel
Engineering · Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #FOS: Mathematics #FOS: Physical sciences #Probability (math.PR) #Sports Dynamics and Biomechanics #Statistical Mechanics (cond-mat.stat-mech) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2401.01279
openalex publication_date 2024/01/02 · openalex created_date 2024/01/04 · openalex updated_date 2026/07/28
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, XN, 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, ZN, is introduced. (3) Strong evidence that XN is also universal and has the same law of probability as ZN is presented. (4) An accurate approximation of the energy loss is presented. (5) Results for XN for an explosive-like initial condition are analyzed.