On The Jeans-Landau-Teller Approximation for Adiabatic Invariants

Abstract

The purpose of this paper is to present some results, both numeric and analytic, obtained in collaboration with A. Carati, F. Fasso and G. Gallavotti, see [1] and [2], concerning the so-called Jeans-Landau-Teller approximation for adiabatic invariants. The physical motivation, which unfortunately we cannot discuss here (see [3] for comments) is the very classical physical problem of the energy exchanges among the different degrees of freedom in molecular collisions. Just to make an example, consider the collinear collision of n diatomic molecules on a line: if the frequency of the internal vibration is large, while the translational kinetic energy (the temperature) is not too large, and the interaction potential is short range and smooth, then one has a Hamiltonian system with n fast and n slow degrees of freedom which, asymptotically for t → ±∞ (far from the collision), gets decoupled. So, one is indeed confronted with a problem of adiabatic invariance. The idea is that, in agreement with the original intuition by Boltzmann [4] and Jeans [5], the energy exchanges per collision are extremely small for large frequencies, and correspondingly equilibrium times are very large.