A propagator theory of non-Gaussian diffusion and power-law relaxation in multidimensional disordered media

Disordered energy environments generate spatially heterogeneous mobility and persistent deviations from Gaussian diffusion. Such systems are common, and dynamics often provide a route to understand the heterogeneity, as in solution spectroscopy. We develop a propagator theory of tagged-particle motion in multidimensional media described by spatially correlated, quenched Gaussian site-energy disorder and detailed-balance hopping dynamics. The disorder-averaged self-propagator is formulated for a specified initial ensemble, and its Fourier transform, the self-dynamic structure factor, is analyzed through a small-wavevector cumulant expansion. To describe the approach to Gaussian diffusion, we introduce a coarse-grained local-mobility field and relate the non-Gaussian parameter to the time correlation of the mobility sampled along the particle trajectory. In the homogenized diffusive regime, convolution of the spatial mobility correlation with the long-time propagator produces an algebraically decaying mobility memory. Consequently, the non-Gaussian parameter exhibits dimension-dependent long-time relaxation: (t−1/2) in one dimension, [(ln t)/t] in two dimensions, and (t−1) for (d > 2). The amplitudes retain information about energetic disorder, spatial correlations, dimensionality, and initial preparation. The theory thus connects microscopic site-energy disorder, propagator dynamics, mobility memory, and experimentally measurable non-Gaussian corrections to the self-dynamic structure factor.