Mode-coupling theory of the glass transition for polydisperse systems

We develop a covariant formulation of the mode-coupling theory (MCT) for the glass transition in polydisperse suspensions. We show that the Mori-Zwanzig equations of motion and the MCT memory functional are invariant under arbitrary changes of basis in the space of species-resolved density fluctuations, provided the vertices are transformed consistently. Exploiting this symmetry, we construct a mode basis via a diameter-moment expansion and an orthonormalization with respect to the structure factor. In this basis, the leading mode is the total density, and higher modes encode systematic size-contrast corrections. Truncating to a few modes yields a controlled, numerically stable reduction of the multicomponent MCT while retaining quantitative accuracy for the total intermediate scattering function. Benchmarking against a ten-component discretization of continuously polydisperse hard-sphere-like fluids (Schulz-Zimm and inverse-cubic distributions), we find that three to six modes reproduce the critical packing fraction and the full shape of the total correlator with large computational savings. This reformulation provides an efficient framework for the MCT of polydisperse systems with controlled discretization and basis-truncation errors.