  Supercooled water exhibits a breakdown of the Stokes-Einstein relation
between the diffusion constant $D$ and the alpha relaxation time
$\tau_{\alpha}$. For water simulated with the TIP5P and ST2 potentials, we find
that the temperature of the decoupling of diffusion and alpha relaxation
correlates with the temperature of the maximum in specific heat that
corresponds to crossing the Widom line $T_W(P)$. Specifically, we find that our
results for $D\tau_{\alpha}/T$ collapse onto a single master curve if
temperature is replaced by $T-T_W(P)$, where $T_W(P)$ is the temperature where
the constant-pressure specific heat achieves a maximum. Also, we find agreement
between our ST2 simulations and experimental values of $D\tau_{\alpha}/T$. We
further find that the size of the mobile molecule clusters (dynamical
heterogeneities) increases sharply near $T_W(P)$. Moreover, our calculations of
mobile particle cluster size $<n(t^*)>_w$ for different pressures, where $t^*$
is the time for which the mobile particle cluster size is largest, also
collapse onto a single master curve if $T$ is replaced by $T-T_W(P)$. The
crossover to a more locally structured low density liquid (LDL) environment as
$T\to T_W(P)$ appears to be well correlated with both the breakdown of the
Stokes-Einstein relation and the growth of dynamic heterogeneities.
