Bryan Cooley

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 $\sigma^2 \sim \log (\epsilon^\beta)$.

The collision sequence produced by N-beads sliding on a frictionless hoop gives rise to a dynamical system that can be formulated as a string of matrix products [4, 5]. The matrices that form the product are written in the order in which the collisions unfold and their corresponding eigenvalues on the unit circle are treated as a non-autonomous rotation map on $S^1 $. The problem of three beads of masses $m$, $m - \epsilon $, $m + \epsilon $ is studied computationally for splitting parameter values $ \epsilon \geqslant 0$. When $ \epsilon = 0$ (three equal masses), the spectrum is discrete on the unit circle underlying the fact that the dynamics are regular [5]. For $ \epsilon > 0$, the eigenvalue spectrum produced by a deterministically chaotic trajectory is compared to spectra produced by two different stochastic problems. The first is the spectrum associated with the sequence of matrix products in which a random number algorithm is used to produce the matrix orderings. The second is the spectrum generated from a random walk process on the unit circle. We describe how to use the chaotic collision sequences as the basis for a random number generating algorithm and we conclude, by an examination of both the runs and reverse arrangement tests, that the degree of randomness produced by these sequences is equivalent to Matlab's rand() routine for generating random numbers.