Geometric approach to multiple-time-scale kinetics: A nonlinear master equation describing vibration-to-vibration relaxation

A geometric approach to the study of multiple-time-scale kinetics is taken here. The approach to equilibrium for kinetic systems is studied via low-dimensional manifolds, with an application to a nonlinear master equation for vibrational relaxation. One of our main concerns is the asymptotic (in time) behavior of the system and whether there is a well-defined rate of approach to equilibrium. One-dimensional slow manifolds provide a good means for studying such behavior in nonlinear systems, because they are the analogue of the eigenvector with least negative eigenvalue for linear kinetics.