Soft matter remains an active field that bridges physics, chemistry, biology, and materials science. Prominent areas of study are transport properties in biological tissues, the interplay between elasticity and relaxation in viscoelastic materials, and the onset of mechanical rigidity in topological disorder. Beyond this intrinsic diversity, a common feature of many soft matter systems is the emergence of different time scales. Examples are segmental versus chain relaxation in polymer melts, and rattling particles inside so-called cages versus their escape in supercooled liquids.
The Zwanzig-Mori projection operator formalism is a successful and widely used tool that builds on this separation of scales. Here, a few relevant variables capture the slow dynamics, while the fast degrees of freedom are integrated into a self-energy or memory kernel. Mode-Coupling Theory (MCT) relates the memory term again to the slow dynamics. This theoretical framework has successfully explained several characteristics of supercooled liquids, like the two-step relaxation of the density correlator [1]. However, applying the MCT approximation to the Lorentz gas violates a symmetry constraint, which ultimately leads to erroneous low-frequency divergences [2]. While this investigation followed the dynamics of a single-point tracer, here we show that an analogous discussion applies to colloidal systems. Lifting the discussion to the level of fields, we demonstrate that the standard MCT approximation again violates a symmetry constraint, as the kernel is not invariant under the permutation of statistically equivalent positions. This implies that a second topological class of scattering events must be taken into account. Hence, we argue that vertex renormalization and the inclusion of non-planar terms are necessary to capture the low-frequency divergences correctly and to predict Rayleigh sound attenuation [3].
[1] W. Götze, Complex Dynamics of Glass-Forming Liquid (Oxford University Press, 2009)
[2] E. Leutheusser, Self-consistent kinetic theory for the Lorentz gas, Phys. Rev. A 28, 1762 (1983)
[3] F. Vogel, P. Baumgärtel, and M. Fuchs, Self-consistent current response theory of unjamming and vibrational modes in low- temperature amorphous solids, Phys. Rev. X 15, 011030 (2025).