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 pores, close to a surface, or 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 in particular 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 also derive inhomogeneous MCT equations valid to second order in gradients.