Stochastic transition in two-dimensional Lennard-Jones systems

We study by computer simulation the behavior at low energy of two-dimensional Lennard-Jones systems, with square or triangular cells and a number of degrees of freedom N up to 128. These systems exhibit a transition from ordered to stochastic motions, passing through a region of intermediate behavior. We thus find two stochasticity borders, which separate in the phase space the ordered, intermediate, and stochastic regions. The corresponding energy thresholds have been determined as functions of the frequency ω of the initially excited normal modes; they generally increase with ω and appear to be independent of N. Their values agree with those found by other authors for one-dimensional LJ systems. We computed also the maximal Lyapunov characteristic exponent χ* of our systems, which is a typical measure of stochasticity; this analysis shows that even in the ordered region certain stochastic features may persist. At higher energies, χ* increases linearly with the energy per degree of freedom e. The law χ*(e) has been determined in the thermodynamic limit by extrapolation. The values found for the stochasticity thresholds fall in a physically significant energy range. The behavior of the thresholds as a function of ω and N is compatible with the hypothesis on the existence of a classical zero-point energy, advanced by Cercignani, Galgani, and Scotti.