A model for an heterogeneous glass-forming liquid was proposed, and the low-temperature behaviour of a tagged molecule moving within it was computed. This model exhibited stretched-exponential decay of the wave-number dependent self-intermediate scattering function in the limit of long times. At temperatures close to the glass transition, where the heterogeneities were much larger in extent than the molecular spacing, the time-dependence of the scattering function crossed over from stretched-exponential decay (with an index, b, of ½) at large wave-numbers, to normal diffusive behaviour (with b = 1) at small wave-numbers. There was a clear separation between early-stage cage-breaking β-relaxation and late-stage α-relaxation. The spatial representation of the scattering function exhibited an anomalously broad exponential (non-Gaussian) tail for sufficiently large values of the molecular displacement at all finite times.

Anomalous Diffusion and Stretched Exponentials in Heterogeneous Glass-Forming Liquids - Low-Temperature Behaviour. J.S.Langer, S.Mukhopadhyay: Physical Review E, 2008, 77[6], 061505