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Replica theory for fluctuations of the activation barriers in glassy systems

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.