Non-linear dynamics and energy transfer

Abstract

Non-linear dynamics is mostly concerned with coupled anharmonic oscillators, a common situation in the classical description of molecule dynamics. Non-linear systems are in general not integrable, but for small enough energy or couplings most trajectories are quasiperiodic (KAM theorem). As energy (or perturbation) of the N-dimensional system increases, and increasing fraction of trajectories becomes chaotic, i.e. the motion is not restricted to an N-dimensional torus, because the number of action variables is less than N. The onset of chaos is produced by the overlap of resonances, and is increasingly widespread when the energy increases, until finally the only constant of motion is energy. Chaos is characterized by a grassy power spectrum of the trajectory frequencies, and by the exponential rate of divergency of nearby trajectories. In the quasiperiodic regime time evolution of the system can be followed by Fourier expansion of the perturbation in terms of the fundamental frequencies, plus overtones and combinations, while this is not possible in the chaotic regime, although allways one can perform numerical integration of trajectories. Energy transfer in the quasiperiodic regime is caused by few low order resonances, and it is reversible process. Irreversible energy transfer (Relaxation) needs not only chaotic behavior, but also ergodic and mixing. From the chemical point of wiew the interest lies in the study of the resonance conditions and the specific nature of the vibration-rotation couplings that produce V-V and V-R energy transfers. Centrifugal interactions play in general a major role than Coriolis force; vibrations promote chaotic motion more effectively than rotation; resonance conditions are present not only between vibrations, but also with the rotational frequency. These studies can be connected with the unimolecular Chemical Kinetics: The rate of trajectory divergency is an upper limit to the microscopic rate constant when RRKM theory can be applied. Specific examples of those effects are presented for illustrative purposes.