Collapse of inelastic hard spheres in dimension $d \geq 2$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916996535287808 |
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| author | Dolmaire, Théophile Velázquez, Juan J. L. |
| author_facet | Dolmaire, Théophile Velázquez, Juan J. L. |
| contents | We investigate the collapse of three inelastic particles in dimension $d \geq 2$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13803 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Collapse of inelastic hard spheres in dimension $d \geq 2$ Dolmaire, Théophile Velázquez, Juan J. L. Mathematical Physics We investigate the collapse of three inelastic particles in dimension $d \geq 2$. 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. |
| title | Collapse of inelastic hard spheres in dimension $d \geq 2$ |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2402.13803 |