Soft chaos in a Hamiltonian system with step potential. I: Statistical properties

In part I we first present the results of molecular dynamics simulations which were performed for a large number of initial data, most of them belonging to low energies E. Poincaré sections strongly suggest the following configurational ergodicity: If P is one of the observed momenta, then the set of all configurations X such that the potential energy V(X) = E−P2/2 is filled uniformly, as time proceeds, by those sections of X(t) for which X˙(t)=P. Based on this assumption we decompose the energy surface into ergodic components and infer the related invariant density. The velocity distribution functions calculated as ensemble averages over these components are found to be in excellent agreement with the corresponding time averages from the simulation data. Then it is shown that each trajectory travels in only finitely many directions in the lowest energy range E ∈ (−2, −1/2), whereas for all E ⩾ −1/2 the number of different momenta in the ergodic components is infinite.