Alternative adiabatic elimination schemes for fast variables: A case study

Abstract

Adiabatic elimination of fast variables from a stochastic process requires the determination of an effective asymptotic evolution operator for the slow variables and of a projection operator that extracts the initial distribution for the asymptotic, reduced, evolution from that for the underlying problem. For a Brownian particle in a harmonic potential, with the velocity as the fast variable and the position as the slow one, both these ingredients can be determined exactly; the projection operator is given here for the first time in compact form. We then compare the exact results with those of two perturbative schemes: one corresponding to expansion around a Maxwellian in velocity space and the other to expansion around a Maxwellian shifted over the local sedimentation velocity. The explicit numerical results for a special initial distribution confirm the a priori expectation that the second scheme is better in describing the first stages of the asymptotic decay towards equilibrium for a distribution originally not too close to equilibrium; in the final stages the first scheme gives a better description. The Two stages correspond to the decay via states with net macroscopic currents and the subsequent dissolution of these currents. The scheme based on the shifted Maxwellian performs somewhat better than expected on the basis of a priori esimates, at least for the special case considered, if we confine our attention to the lowest moments of the distribution in position space.