Statistical mechanics of Hénon-Heiles oscillators.

Statistical mechanics is an asymptotic theory valid in the limit of an infinite number of degrees of freedom. The study of small-dimensional chaos, starting from the papers by Lorenz [1]and Henon and Heiles [2], raises the question: What is essential for the laws of statistical mechanics to be true —a large number of degrees of freedom or chaos? For ergodic Hamiltonian systems the answer was given in Ref. [3]: ergodicity (chaos) provides the validity of the laws of equilibrium thermodynamics and statistical mechanics, which ought to be slightly modified for finite numbers of degrees of freedom. Unfortunately, most small-dimensional Hamiltonian systems encountered in physics are not ergodic. Nevertheless, it seems plausible that statistical mechanics could be applied to describe systems of such kinds if the motion is "chaotic enough. " Our aim is to test this hypothesis for the case of the Henon-Heiles oscillators. We show that the motion of the Henon-Heiles oscillators matches very accurately the predictions of small-dimensional statistical mechanics for high-energy vibrations and we also study the changes that occur when the energy level decreases. First we outline, according to Ref. [3], thermodynamics and statistical mechanics of finite-dimensional ergodic Hamiltonian systems.