Completely Monotone Solutions of the Mode-Coupling Theory for Mixtures

We establish that a mode-coupling approximation for the dynamics of multi-component systems obeying Smoluchowski dynamics preserves a subtle yet fundamental property: the partial density correlation functions are, considered as matrices, completely monotone, i.e., they can exactly be written as superpositions of decaying exponentials only. This statement holds, no matter what further approximations are needed to calculate the theory's coupling parameters. The long-time limit of these functions fulfills a maximum property, and an iteration scheme for its numerical determination is given. We also show the existence of a unique solution to the equations of motion for which power series both for short times and small frequencies exist, the latter except at special points where ergodic-to-nonergodic transitions occur. These transitions are bifurcations that are proven to be of the cuspoid family.