Diffusion of a particle on a static rugged energy landscape with spatial correlations

Despite the broad applicability of the problem, we have limited knowledge about the effect of ruggedness on diffusion at a quantitative level. Every study seems to use the expression of Zwanzig [Proc. Natl. Acad. U.S.A, 85, 2029 (1988)] who derived the effective diffusion coefficient, $D_{eff} =D_{0} \exp \left( {-\beta^{2}\varepsilon^{2}} \right)$. We introduce and study two models of Gaussian random energy surface; a discrete lattice and a continuous field. Our simulations show that Zwanzig's expression overestimates diffusion in the uncorrelated Gaussian random lattice. The disparity originates from the presence of ``three-site traps'' (TST) on the energy landscape -- which are formed by the presence of deep minima flanked by high barriers on either side. Using mean first passage time (MFPT) formalism, we derive a general expression for the effective diffusion coefficient, $D_{eff} =D_{0} \exp \left( {-\beta^{2}\varepsilon^{2}} \right)\left[ {1+\mbox{erf}\left( {{\beta \varepsilon } \mathord{\left/ {\vphantom {{\beta \varepsilon } 2}} \right. \kern-\nulldelimiterspace} 2} \right)} \right]^{-1}$ in the presence of TST. In presence of spatial correlation we derive a more general form of the expression, which reduces to Zwanzig's form in certain limits. We characterize the same using non-Gaussian order parameter, and show that this ``breakdown'' scales with ruggedness following an asymptotic power law. The breakdown of Zwanzig's elegant expression was perhaps anticipated but was not clearly demonstrated earlier.