Maximal Codimension Collisions and Invariant Measures for Hard Spheres on a Line

TL;DR

This paper characterizes invariant measures for the dynamics of line-constrained hard spheres, identifying conditions under which the Liouville measure is invariant, and proves its uniqueness for linear scattering maps conserving momentum and energy.

Contribution

It provides a complete characterization of invariant measures for hard sphere collisions on a line, including a PDE-based criterion and the uniqueness of the Liouville measure for linear scattering.

Findings

Invariant measures are characterized via a nonlinear PDE boundary-value problem.

The Liouville measure is shown to be invariant under the unique linear scattering map.

The paper establishes conditions for measures to be invariant under momentum- and energy-conserving flows.

Abstract

For any $N\geq 3$, we study invariant measures of the dynamics of $N$ hard spheres whose centres are constrained to lie on a line. In particular, we study the invariant submanifold $\mathcal{M}$ of the tangent bundle of the hard sphere billiard table comprising initial data that lead to the simultaneous collision of all $N$ hard spheres. Firstly, we obtain a characterisation of those continuously-differentiable $N$-body scattering maps which generate a billiard dynamics on $\mathcal{M}$ admitting a canonical weighted Hausdorff measure on $\mathcal{M}$ (that we term the Liouville measure on $\mathcal{M}$) as an invariant measure. We do this by deriving a second boundary-value problem for a fully nonlinear PDE that all such scattering maps satisfy by necessity. Secondly, by solving a family of functional equations, we find sufficient conditions on measures which are absolutely continuous with respect to the Hausdorff measure in order that they be invariant for billiard flows that conserve momentum and energy. Finally, we show that the unique momentum- and energy-conserving linear $N$-body scattering map yields a billiard dynamics which admits the Liouville measure on $\mathcal{M}$ as an invariant measure.