UNIVERSAL TIME-DEPENDENCE OF THE MEAN-SQUARE DISPLACEMENT IN EXTREMELY RUGGED ENERGY LANDSCAPES WITH EQUAL MINIMA

This paper presents a calculation of the time dependence of the mean-square displacement for symmetric random energy barrier hopping models at low temperatures, where the frequency dependence of the normalized diffusion constant ($) over tilde D becomes universal, i.e., independent of the energy barrier probability distribution [J. C. Dyre, Phys. Rev. B 49, 11 709 (1994)]. The universal time dependence of the mean-square displacement is calculated from the effective medium approximation (EMA) universality equation, ($) over tilde D ln ($) over tilde D = ($) over tilde s, where ($) over tilde s is the dimensionless imaginary frequency, as well as for the approximation to the EMA universality equation ($) over tilde D congruent to ($) over tilde s/ln(1+($) over tilde s). At long times the universal mean-square displacement is linear in time, corresponding to ordinary diffusion, whereas the mean-square displacement at short times t in dimensionless units varies as 2/ln(t(-1)).