Systematic elimination of fast variables in linear systems

We consider linear systems in which some of the variables evolve on a much faster time scale than the remaining ones. By means of a systematic perturbation theory in the time-scale ratio we extract a reduced dynamics in terms of the slow variables only, valid after an initial transient period. The procedure provides a systematic extension of the usual adiabatic elimination scheme and gives corrections to it. We also find an expression for the long-time behavior of the correlation functions of the slow variables. These asymptotic expressions do not extrapolate back towards the equal-time correlations for t going to zero; the reason for this “initial slip” is given and its magnitude calculated. Method and results are illustrated with a simple example of coupled oscillators.