A rigorous implementation of the Jeans--Landau--Teller approximation

Rigorous bounds on the rate of energy exchanges between vibrational and translational degrees of freedom are established in simple classical models of diatomic molecules. The results are in agreement with an elementary approximation introduced by Landau and Teller. The method is perturbative theory ``beyond all orders'', with diagrammatic techniques (tree expansions) to organize and manipulate terms, and look for compensations, like in recent studies on KAM theorem homoclinic splitting.Comment: 23 pages, postscrip

Adiabatic invariants and trapping of point charge in a strong non-uniform magnetic field

We consider here the classical problem of charge trapping by strong nonuniform magnetic fields (Van Allen belts, magnetic bottles), and study it by means of the averaging methods of classical perturbation theory. At variance with the usual cases, this problem has three (in place of two) different time scales, which are associated to the Larmor rotation around field lines (fast motion), to the motion along field lines (intermediate scale motion), and to the drift across field lines (slow motion). Such time scales get well separated for strong magnetic fields, so that Nekhoroshev-Neishtadt perturbation methods can be applied. As a result, one finds that the system admits two adiabatic invariants (namely, the magnetic moment and a second quantity, related to the energy of the motion along the field lines), which turn out to be preserved for time scales growing exponentially with the field intensity. In fact, the system can be given an integrable form, up to an exponentially small remainder

Exponentially long equilibrium times in a one dimensional collisional model of a classical gas.

Around 1900, J. H. Jeans suggested that the "abnormal" specific heats observed in diatomic gases, specifically the lack of contribution to the heat capacity from the internal vibrational degrees of freedom, in apparent violation of the equipartition theorem, might be caused by the large separation between the time scale for the vibration and the time scale associated with a typical binary collision in the gas. We consider here a simple 1D model and show how, when these time scales are well separated, the collisional dynamics is constrained by a many particle adiabatic invariant. The effect is that the collisional energy exchanges between the translational and the vibrational degrees of freedom are slowed down by an exponential factor (as Jeans conjectured). A metastable situation thus occurs, in which the Fast vibrational degrees of freedom effectively do not contribute to the specific heat. Hence, the observed "freezing out" of the vibrational degrees of freedom could in principle be explained in terms of classical mechanics. We discuss the phenomenon analytically, on the basis of an approximation introduced by Landau and Teller (1936) for a related phenomenon, and estimate the time scale for the evolution to statistical equilibrium. The theoretical analysis is supported by numerical examples

On the Landau--Teller approximation for the energy exchanges with fast degrees of freedom.

We revisit the Landau-Teller heuristic approach to adiabatic invariants and, following Rapp, use it to investigate the energy exchanges between the different degrees of freedom, in simple Hamiltonian systems describing the collision of fast rotating or vibrating molecules with a fixed wall. We critically compare the theoretical results with particularly accurate numerical computations (quite small energy exchanges, namely of one part over 10^{30}, are measured)