Mode-coupling theory of the glass transition for a liquid in a periodic potential

We derive a microscopic theory for the structural dynamics in the vicinity of the glass transition for a liquid exposed to a one-dimensional periodic potential. The periodic potential breaks translational invariance; in particular, the density exhibits a periodic modulation. Using techniques familiar from solid-state theory, we define generalized intermediate scattering functions from fluctuating densities in wave-vector space. Exact equations of motion are derived within the Mori-Zwanzig projection-operator formalism reflecting the residual lattice symmetries. Due to the lack of rotational symmetry it is necessary to split the currents into components parallel and perpendicular to the modulation. We provide a closure of the equations in terms of a mode-coupling approximation for the force kernel. The theory reflects the usual analytic properties of correlation functions and encodes all phenomena known for mode-coupling theories. We prove that the theory reduces to the conventional mode-coupling theory in the case of vanishing amplitude of the modulation.