  It is widely believed that the breakdown of the Stokes-Einstein (SE) relation
between the translational diffusivity and the shear viscosity in supercooled
liquids is due to the development of dynamic heterogeneity i.e. the presence of
both slow and fast moving particles in the system. In this study we
\emph{directly} calculate the distribution of the diffusivity for a model
system for different temperatures in the supercooled regime. We find that with
decreasing temperature, the distribution evolves from Gaussian to bimodal
indicating that on the time scale of the typical relaxation time, mobile (fluid
like) and less mobile (solid like) particles in the system can be
\emph{unambiguously} identified. We also show that less mobile particles obey
the Stokes-Einstein relation even in the supercooled regime and it is the
mobile particles which show strong violation of the Stokes-Einstein relation in
agreement with the previous studies on different model glass forming systems.
Motivated by some of the recent studies where an ideal glass transition is
proposed by randomly pinning some fraction of particles, we then studied the SE
breakdown as a function of random pinning concentration in our model system. We
showed that degree of SE breakdown increases quite dramatically with increasing
pinning concentration, thereby providing a new way to unravel the puzzles of SE
violation in supercooled liquids in greater details.
