Classical “freezing” of plane rotations: A proof of the Boltzmann-Jeans conjecture

Abstract

Using simple known methods and results of classical perturbation theory, especially those due to Nekhoroshev and Neishtadt, we study the energy exchanges between the rotational and the translational degrees of freedom in a particular model representing the planar motion of a rigid body in a bounded analytic potential. We prove that, if the angular velocityω is initially large, then the energy exchanges are small,O(ω ⁻¹), for times growing exponentially withω, |t|∼expω. We also deduce that in a scattering process from a (smooth) potential barrier, the overall change in the rotational energy of the incoming body is exponentially small inω, ℰ∼exp(−ω. The results are interpreted in the light of an old conjecture by Boltzmann and Jeans on the existence of very large time scales for equilibrium in statistical systems containing high-frequency degrees of freedom (purely classical “freezing” of the high-frequency degrees of freedom); the rotating object is, in this interpretation, a (classical) molecule, which moves in an external field, or collides with the wall of a container. Two different limits of largeω are considered, namely the limit of large rotational energy, and (as is interesting for the molecular interpretation) the limit of point mass, at finite rotational energy.