2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/160550We estimate numerically the normal modes of the free energy in a glass of hard discs. We observe that, near the glass transition or after a rapid quench deep in the glass phase, the density of states (i) is characteristic of a marginally stable structure, in particular it di splays a frequency scale $ω^*\sim p^{1/2}$, where $p$ is the pressure and (ii) gives a faithful representation of the short-time dyn amics. This brings further evidences that the boson peak near the glass transition corresponds to the relaxation of marginal modes of a we akly-coordinated structure, and implies that the mean square displacement in the glass phase is anomalously large and goes as $< δR^2 > \sim p^{-3/2}$, a prediction that we check numerically.5 pages, 4 figuresSoft Condensed MatterStatistical Mechanicstext