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In an attempt to extend the mode coupling theory (MCT) to lower temperatures,an Unified theory was proposed which within the MCT framework incorporated theactivated dynamics via the random first order transition theory (RFOT). Here weshow that the theory although successful in describing other properties ofsupercooled liquids is unable to capture the Stokes-Einstein breakdown. We thenshow using continuous time random work (CTRW) formalism that the Unified theoryis equivalent to a CTRW dynamics in presence of two waiting time distributions.It is known from earlier work on CTRW that in such cases the total dynamics isdominated by the fast motion. This explains the failure of the Unified theoryin predicting the SE breakdown as both the structural relaxation and thediffusion process are described by the comparatively fast MCT like dynamics.The study also predicts that other forms of extended MCT will face a similarissue. We next modify the Unified theory by applying the concept of renewaltheory, usually used in CTRW models where the distribution has a long tail.According to this theory the first jump given by the persistent time is slowerthan the subsequent jumps given by the exchange time. We first show that forsystems with two waiting time distributions even when both the distributionsare exponential the persistent time is larger than the exchange time. We alsoidentify the persistent time with the slower activated process. The extendedUnified theory can now explain the SE breakdown. In this extended theory at lowtemperatures the structural relaxation is described by the activated dynamicswhereas the diffusion is primarily determined by the MCT like dynamics leadingto a decoupling between them. We also calculate a dynamic lengthscale from thewavenumber dependence of the relaxation time. We find that this dynamic lengthscale grows faster than the static length scale.

Continuous time random walk concepts applied to extended mode coupling theory: A study of the Stokes-Einstein breakdown