Eigenvalue Distributions from Impacts on a Ring

We consider the collision dynamics produced by three beads with masses ($m_1$, $m_2$, $m_3$) sliding without friction on a ring, where the masses are scaled so that $m_1 = 1/ \epsilon$, $m_2 = 1$, $m_3 = 1- \epsilon$, for $0 \leqslant \epsilon \leqslant 1$. The singular limits $\epsilon = 0$ and $\epsilon = 1$ correspond to two equal mass beads colliding on the ring with a wall, and without a wall respectively. In both these cases, all solutions are periodic and the eigenvalue distributions (around the unit circle) associated with the products of collision matrices are discrete. We then numerically examine the regime which parametrically connects these two states, i.e. $0 < \epsilon < 1$, and show that the eigenvalue distribution is generically uniform around the unit circle, which implies that the dynamics are no longer periodic. By a sequence of careful numerical experiments, we characterize how the uniform spectrum collapses from continuous to discrete in the two singular limits $\epsilon \to 0$ and $\epsilon \to 1$ for an ensemble of initial velocities sampled uniformly on a fixed energy surface. For the limit $\epsilon \to 0$, the distribution forms Gaussian peaks around the discrete limiting values $\pm1$, $\pm i$, with variances that scale in power law form as $\sigma^2 \sim \alpha \epsilon^\beta$. By contrast, the convergence in the limit $\epsilon \to 1$ to the discrete values $\pm 1$ is shown to follow a logarithmic power-law $\sigma^2 \sim \log (\epsilon^\beta)$.