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Anomalous dimensionality dependence of diffusion in a rugged energy landscape : How pathological is one dimension ?

Rugged (or, rough) energy landscape (REL) with spatially distributed maxima and minima are often employed in applications of physics, chemistry and biology (enzyme kinetics, protein folding, diffusion in disordered solids, transport in organic semiconductors, relaxation in random spin systems, in supercooled liquids and glasses). Sometimes the system needs to be modeled as a random walker in high dimensions (like in protein folding/unfolding) where dimensions could be the distances between different amino acid residues (as in unfolding of HP-36). Nevertheless, most of the theoretical studies of these phenomena still employ a one dimensional description. This is despite the prediction that in a rough (or, rugged) energy landscape (REL), diffusion in one dimension (1d) is predicted to be pathologically different from any higher dimension with the increased chance of encountering broken ergodicity (Stein and Newman, 2012). We explore the dimensionality dependent diffusion on REL by carrying out an effective medium approximation based analytical calculations and compare them with the available computer simulation results. We find that at intermediate level of ruggedness (assumed to have a Gaussian distribution), where diffusion is well-defined, the value of the effective diffusion coefficient depends on dimensionality and changes (increases) by several factors (~5-10) in going from 1d to 2d. In contrast, the changes in subsequent transitions (like 2d to 3d and 3d to 4d and so on) are far more modest, of the order of 10-20% only. When ruggedness is given by random traps with an exponential distribution of barrier heights, the mean square displacement is sub-diffusive (a well-known result), but the growth of MSD is described by different exponents in one and higher dimensions. The exponent of growth is larger in higher dimensions than in 1d.