Free particle scattering off two oscillating disks

We investigate the two-dimensional classical dynamics of the scattering of point particles by two periodically oscillating disks. The dynamics exhibits regular and chaotic scattering properties, as a function of the initial conditions and parameter values of the system. The energy is not conserved since the particles can gain and loose energy from the collisions with the disks. We find that for incident particles whose velocity is on the order of the oscillating disk velocity, the energy of the exiting particles displays non-monotonic gaps of allowed energies, and the distribution of exiting particle velocities shows significant fluctuations in the low energy regime. We also considered the case when the initial velocity distribution is Gaussian, and found that for high energies the exit velocity distribution is Gaussian with the same mean and variance. When the initial particle velocities are in the irregular regime the exit velocity distribution is Gaussian but with a smaller mean and variance. The latter result can be understood as an example of stochastic cooling. In the intermediate regime the exit velocity distribution differs significantly from Gaussian. A comparison of the results presented in this paper to previous chaotic static scattering problems is also discussed.Comment: 9 doble sided pages 13 Postscript figures, REVTEX style. To appear in Phys. Rev.

An O(1) Time Algorithm for Generating Multiset Permutations

Abstract. We design an algorithm that generates multiset permutations in O(1) time from permutation to permutations, using only data structures of arrays. The previous O(1) time algorithm used pointers, causing O(n) time to access an element in a permutation, where n is the size of permutations. The central idea in our algorithm is tree traversal. We associate permutations with the leaves of a tree. By traversing this tree, going up and down and making changes when necessary, we spend O(1) time from permutation to permutation. Permutations are generated in a one-dimensional array.