  In an attempt to extend the mode coupling theory (MCT) to lower temperatures,
an Unified theory was proposed which within the MCT framework incorporated the
activated dynamics via the random first order transition theory (RFOT). Here we
show that the theory although successful in describing other properties of
supercooled liquids is unable to capture the Stokes-Einstein breakdown. We then
show using continuous time random work (CTRW) formalism that the Unified theory
is 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 is
dominated by the fast motion. This explains the failure of the Unified theory
in predicting the SE breakdown as both the structural relaxation and the
diffusion process are described by the comparatively fast MCT like dynamics.
The study also predicts that other forms of extended MCT will face a similar
issue. We next modify the Unified theory by applying the concept of renewal
theory, 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 slower
than the subsequent jumps given by the exchange time. We first show that for
systems with two waiting time distributions even when both the distributions
are exponential the persistent time is larger than the exchange time. We also
identify the persistent time with the slower activated process. The extended
Unified theory can now explain the SE breakdown. In this extended theory at low
temperatures the structural relaxation is described by the activated dynamics
whereas the diffusion is primarily determined by the MCT like dynamics leading
to a decoupling between them. We also calculate a dynamic lengthscale from the
wavenumber dependence of the relaxation time. We find that this dynamic length
scale grows faster than the static length scale.
