  In a recent study by some of us, we have proposed a new measure of the
structure of a liquid, the softness of the mean-field caging potential, and
shown that it can describe the temperature dependence of the dynamics. In this
work, we put this parameter through a stringent test and study the causal
relationship between this softness parameter at the local level and the local
dynamics. We first extend the earlier study to describe the local softness. We
find a distribution of the softness value where the distribution has reasonably
strong temperature dependence. This shows that the parameter can identify the
local structural heterogeneity. We then study the lifetime of the softness
parameter and show that its lifetime is connected to the well-known cage
structure in the supercooled liquids. Finally, our theory predicts that the
local softness and the local dynamics is causal below the onset temperature
where there is a decoupling between the short and long time dynamics, thus
allowing a static description of the cage. We find that at lower temperatures,
the structural heterogeneity increases, and also, the structure becomes a
better predictor of the dynamics, thus giving rise to dynamic heterogeneity. We
also find that the softness of a hard, immobile region evolves with time and
becomes soft and eventually mobile due to the rearrangements in the
neighbourhood, confirming the well-known facilitation effect.
