Atomic clusters: Laboratories for studying chaos and ergodicity

Small clusters of atoms, consisting of 3 to 7 particles, exhibit complexities characteristic of large systems yet can be analyzed at a level of detail approaching that used for the simplest models for chaotic behavior. In a process in 3-atom clusters that mimics melting, the system becomes less chaotic, less ergodic and more regular as its energy is increased upward into the ''melting'' range. This is a consequence of the system spending long intervals in flat, tubular saddle regions where its potential energy is high, its kinetic energy is low and its trajectories are locally like regular trajectories. Behavior of this kind led to a useful characterization of the local contributions to chaotic behavior on a multidimensional potential surface, a characterization in terms of a ''local Liapunov function,'' whose energy dependence is a moderately good mimic of the energy dependence of the largest Liapunov exponent. With increasing numbers of particles, this onset of regularity disappears, partly because h is hidden by the chaotic character of al the degrees of freedom that play no role in the development of a soft mode and whose Liapunov exponents and K-entropy increase monotonically with energy. Another factor contributes to the disappearance of the nonmonotonic, s-shaped behavior of the K-entropy, largest Liapunov exponent and fractal dimension of the phase space trajectory: the saddles of the potential surface become sharper, less tubular and more chaos-inducing with increasing dimensionality. The greater the number of particles comprising the cluster, the less the saddle regions differ from the other parts of the potential surface. The local Liapunov function can reflect this; so do the extents of mode-mode coupling in saddle regions, compared with the extent of coupling in other parts of the configuration space. More can be derived from incompletely averaged Liapunov exponents: by examining the distributions of sample values of approximate Liapunov exponents as functions of the length or duration of the trajectory over which the averages are made, one can make inferences regarding the time evolution of the extent of ergodicity of a system