A stationary isotropic gas of freely moving non-colliding disks
Consider a nonempty locally finite set $S \subset \mathbb{R}^2$ of particles with velocity marks $v : S \to \mathbb{R}^2$, each particle moving freely, $x \mapsto x + t v(x)$ for $t \in \mathbb{R}$. Benjamini, Shamov and Weiss gave, for the initial positions $\mathbb{Z}^2$, bounded pairwise distinct velocities with speeds bounded away from zero for which all closest-approach distances are bounded below by a positive constant, so that disks of small enough radius centred at these points never collide. They also proved that no bounded continuous velocity field on the whole plane gives every pair of points at distance greater than one distinct velocities and a closest approach above a fixed positive constant.
Problem. Does there exist a probability law for such marked sets $(S, v)$ with the following properties? Almost surely, for deterministic constants $0 < m < M < \infty$, every speed lies in $[m, M]$, distinct particles have different velocities, and $$|(x + t v(x)) - (y + t v(y))| \ge 1 \qquad \text{for all } x \ne y \text{ in } S \text{ and all } t \in \mathbb{R}.$$ The law is invariant under translations, under simultaneous rotations of positions and velocities, and under the evolution $(x, v(x)) \mapsto (x + t v(x), v(x))$, and it is ergodic under translations: every translation-invariant event has probability zero or one.
Source: Benjamini and Shamov (2026), the open problem of §1 of the first version of their preprint; the revision with Weiss (2026) splits it into an ergodic and an isotropic version.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- DS, MG
- Keywords
- AIM Open Applied Problems, hard-disk systems, stationary point processes
References
@misc{benjamini2026,
author = {Benjamini, Itai and Shamov, Alexander and Weiss, Barak},
title = {Non-colliding billiards in the plane},
year = {2026},
eprint = {2605.23575},
archivePrefix = {arXiv},
note = {Version 1 (22 May 2026) by Benjamini and Shamov; version 2 (27 May 2026) adds Weiss},
}
@misc{aim-stationary-noncolliding-billiards,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A stationary isotropic gas of freely moving non-colliding disks}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/165-stationary-noncolliding-billiards.md},
note = {Entry 165 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00904, "A stationary isotropic gas of freely moving non-colliding disks", clopen.solutions.