Curvature, torsion, microcanonical density and stochastic transition

We introduce geometrical indicators (Frenet-Serret curvature and torsion) together with microcanonical density to give evidence to the stochastic transition of classical Hamiltonian models (Fermi-Pasta-Ulam and Lennard-Jones systems) when the specific energy grows. The transition is clearly detected through the breakdown of the harmonic-like behaviour, in combination with the vanishing of the dependence on the initial conditions. This method of analysis presents both experimental and theoretical advantages: it is fast and gives relatively sharp answers for the transition; moreover, a new insight is allowed on the deformations and the destruction of invariant surfaces in the ordered regime. Among the results, it is noteworthy that going from 32 to 4096 degrees of freedom the stochastic transition depends only on the specific energy and not on the number of degrees of freedom.