One-dimensional inelastic collapse of four particles: asymmetric collision sequences and spherical billiard reduction

TL;DR

This work analyzes a one-dimensional four-p particle system with a fixed restitution coefficient $r\in(0,1)$, focusing on inelastic collapse under asymmetric collision orders and introducing a spherical reduction that encodes collision sequences on a reduced state space. It proves the feasibility of the asymmetric $ababcb$ pattern and identifies a critical threshold $r_{crit,\mathrm{ababcb}}=5-2\sqrt{6}$ governing self-similar realizations, though these are unstable; it then develops a spherical reduction, yielding a dynamical system on $\{1,2,3\}\times S^2$ whose orbit structure determines collision orders. The authors present two representations of the spherical reduction (trigonometric and vectorial) and establish a theorem ensuring invariance of the planes $\mathcal{P}(k)$ across collisions. Numerical simulations of the reduced system reveal regimes with quasi-periodic behavior and stability windows consistent with known results, while highlighting rich dynamics and open questions for higher-particle extensions and the full landscape of stable collapse patterns.

Abstract

We consider a one-dimensional system of four inelastic hard spheres, colliding with a fixed restitution coefficient $r$, and we study the inelastic collapse phenomenon for such a particle system. We study a periodic, asymmetric collision pattern, proving that it can be realized, despite its instability. We prove that we can associate to the four-particle dynamical system another dynamical system of smaller dimension, acting on $\{1,2,3\} \times \mathbb{S}^2$, and that encodes the collision orders of each trajectory. We provide different representations of this new dynamical system, and study numerically its $ω$-limit sets. In particular, the numerical simulations suggest that the orbits of such a system might be quasi-periodic.

Figures (11)

• Figure 1: The spectrum of $M$.
• Figure 2: The spectrum of $M$ for $0<r<0.13$, and $0 < \lambda < 0.05$.
• Figure 3: Sign changes of $a-d$.
• Figure 4: Sign changes of $b$.
• Figure 5: Representation in the first octant of $\mathbb{R}^3$ of a trajectory assuming that $p(0)$ and $q(0)$ are normalized, and that the initial configuration corresponds to a collision of type $\mathfrak{a}$. The two vectors $p(0)$ and $q(0)$ span the plane $\mathcal{P}(0)$ (in purple on the figure). The intersection between $\mathcal{P}(0)$ and $y=0$ (because $\varphi \in\ ]0,\pi/2[$ on the figure, so the next collision is of type $\mathfrak{b}$) defines the direction of $p(1)$. To determine $\mathcal{P}(1)$, one computes $Bq(0)$, then projects this vector on $p(1)^\perp$ and finally normalizes it to obtain $q(1)$, so that $\mathcal{P}(1) = \text{Span}\left(p(1),q(1)\right)$. One can then repeat the process forever to define the sequence of planes $\left(\mathcal{P}(k)\right)_k$.

Theorems & Definitions (16)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Theorem 3.1
• Remark 3.2
• proof
• Remark 3.3
• Theorem 4.1: Spherical billiard reduction
• proof