Inhomogeneous Mode-Coupling Theory and Growing Dynamic Length in Supercooled Liquids

We extend mode-coupling theory (MCT) to inhomogeneous situations, relevant for supercooled liquid in an external field. We compute the response of the dynamical structure factor to a static inhomogeneous external potential and provide the first direct evidence that the standard formulation of MCT is associated with a diverging length scale. We find that the so-called cages are, in fact, extended objects. Although close to the transition the dynamic length grows as $|T-T_c{|}^{-1/4}$ in both the $\beta$ and $\alpha$ regimes, our results suggest that the fractal dimension of correlated clusters is larger in the $\alpha$ regime. We derive inhomogeneous MCT equations valid to second order in gradients.