  Dynamical heterogeneity and the decoupling of diffusion and structural
relaxation in a supercooled liquid is investigated in terms of a
time-dependent, fourth-order density correlation function. The generalized
susceptibility $\chi_4(t)$ corresponding to this correlation function has a
maximum at an intermediate time $t_4^*$, and both $t_4^*$ and $\chi_4(t_4^*)$
grow strongly with decreasing temperature. We show that the main contribution
to $\chi_4(t)$ arises from spatial correlations between temporarily localized
(``caged'') particles. We compare $\chi_4(t)$ with a generalized susceptibility
$\chi_M(t)$ related to a correlation function of squared particle
displacements, and show that while $t_4^*$ is roughly proportional to the
$\alpha$-relaxation time, $t_M^*$ is proportional to the inverse of the
self-diffusion coefficient.
