Ergodic features of harmonic-oscillator systems. II. asymptotic ergodicity

Abstract

We continue the study, started in an earlier paper 1), of the ways in which certain quantities in harmonic-oscillator systems may show ergodic behaviour. In this paper we discuss in particular the ergodic features that emerge when the system under consideration becomes large. For some nanergodic phase functions the deviation from ergodic behaviour goes to zero in this limit. We propose to call those phase functions asymptotically ergodic. In a regular harmonic lattice all local phase functions turn out to be asymptotically ergodic, provided the frequency spectrum shows a gap around zero. When arbitrarily low frequencies do occur we have to impose some additional condition in order to exclude local functions that exhibit very large fluctuations.