Power-Law Relaxation of Non-Gaussian Parameter and Self-Dynamic Structure Factor in Multidimensional Rugged Energy Landscapes

Ruggedness of the underlying energy landscape gives rise to heterogeneous mobility and non-Gaussian diffusion. We develop a theoretical framework for tagged-particle diffusion in multidimensional rugged energy landscapes modeled as correlated quenched Gaussian random fields. Using the self-propagator and self-dynamic structure factor, we characterize finite-time diffusion beyond the effective diffusion coefficient. We determine the effects of dimensionality, spatial correlations, and initial preparation. By introducing a coarse-grained mobility field and a mobility-memory approximation, we relate the non-Gaussian parameter to the time correlation of the mobility sampled by the particle. In the homogenized diffusive regime, the mobility correlation decays algebraically, leading to long-time relaxation of the non-Gaussian parameter as $t^{-1/2}$ in one dimension, $(\ln t)/t$ in two dimensions, and $t^{-1}$ for $d>2$, with amplitudes that depend on dimensionality and the initial ensemble. Our results show that rugged energy landscapes leave distinct signatures in the effective diffusion coefficient, self-dynamic structure factor, and relaxation of non-Gaussian fluctuations.