Melting and phase space transitions in small clusters: Spectral characteristics, dimensions, and K entropy

The Grassberger-Procaccia method has been employed to study the transitions which occur as a classical Ar₃ cluster, modeled by pairwise Lennard-Jones potentials, passes from a rigid, solid-like form to a nonrigid, liquid-like form with increasing energy. Power spectra and lower bounds on the fractal dimensions and K entropies are presented at several energies along the caloric curve for the Ar₃ cluster. In addition, the full spectrum of Liapunov exponents has been computed at these same energies to get an accurate value of the K entropy. Chaotic behavior, though relatively small, is observed even at low energies where the power spectrum displays largely normal-mode structure. The degree of chaotic behavior increases with energy at energies where some degree of regularity is observed in the spectrum. However, at energies that just allow the system to pass into and across saddle regions separating local potential minima, the phase space appears to be separable into a region within the equilateral triangle potential well where the behavior is highly chaotic, and a region of lower dimensions and less chaos around the saddle of the linear configuration. Dimensions from approximately three to eight are observed. A clear separability of time scales for establishment of different extents of ergodicity permits the determination of fractal dimensions of the manifold on which the phase points moves, for time scales of physical, i.e., observable significance. We believe this to be the first evaluation of the dimensionality of the space on which the phase point moves, for a Hamiltonian system displaying this range of dimensions.