  We consider the problem of slow activation dynamics in glassy systems
undergoing a random first order phase transition. Using an effective potential
approach to supercooled liquids, we determine the spectrum of activation
barriers for entropic droplets. We demonstrate that fluctuations of the
configurational entropy and of the liquid glass surface tension are crucial to
achieve an understanding of the barrier fluctuations in glassy systems and thus
are ultimatively responsible for the broad spectrum of excitations and
heterogeneous dynamics in glasses. In particular we derive a relation between
the length scale for dynamic heterogeneity and the related barrier
fluctuations. Diluted entropic droplets are shown to have a Gaussian
distribution of barriers, strongly suggesting that non-Gaussian behavior
results from droplet-droplet interactions.
