New exact approach to models of diffusion in one-dimensional random systems and optimal network kinetics in spin glass models

It is shown that in one-dimensional stochastic models with gaussian random energy levels along a quantum reaction coordinate the dominant, rate-determining time-scale does not follow the conventional Arrhenius law, but rather has a much stronger temperature dependence, of the form τ ～ exp(B/k_BT)²], where B is proportional to the width of the energy distribution. The new activation law can be ascribed to the large number of energy barriers of varying heights which exist in the random structure, as distinct from the conventional case of a single barrier, leading to the Arrhenius form τ ～ T^p × exp(A/k_BT). In systems with random structure and configuration space which are not strictly one-dimensional it is discussed if the thermal energy bias of detailed balance may lead to a kinetics that is essentially restricted to an energetically optimal network at low temperatures, thus leading to an essentially one-dimensional diffusion. Several recent studies of spin glass models appear to support the relevance of this principle, and include the observation of the new activation law in Monte Carlo experiments.