  Generalized mode-coupling theory (GMCT) has recently emerged as a promising
first-principles theory to study the poorly understood dynamics of
glass-forming materials. Formulated as a hierarchical extension of standard
mode-coupling theory (MCT), it is able to systematically improve its
predictions by including the exact dynamics of higher-order correlation
functions into its hierarchy. However, in contrast to Newtonian dynamics, a
fully generalized version of the theory based on Brownian dynamics is still
lacking. To close this gap, we provide a detailed derivation of GMCT for
colloidal mixtures obeying a many-body Smoluchowski equation. We demonstrate
that a hierarchy of coupled equations can again be established and show that
these, consistent with standard MCT, are identical to the ones obtained from
Newtonian GMCT when taking the overdamped limit. Consequently, the non-trivial
similarity between Brownian and Newtonian MCT is maintained for our newly
developed multi-component GMCT. As a proof of principle, we also solve the
generalized mode-coupling equations for the binary Kob-Andersen Lennard-Jones
mixture undergoing Brownian dynamics, and confirm the improved predictive power
of the theory upon using more levels of the GMCT hierarchy of equations.
