Comparison of some master equation descriptions of relaxation in complex systems

Several models of relaxation based on master equation approaches have obtained the Kohlrausch fractional exponential form φ(t) = exp − (tτ∗)1−n, 0 < n < 1, or its equivalent for the relaxation function in complex systems. Representative models include (i) the Cohen-Grest free-volume theory, (ii) the work of Dhar and Barma, and Skinner based on Glauber's kinetic Ising model, (iii) the theory of De Dominicis et al. based on a random energy model for the spin glass, (iv) the Ogielski-Stein theory based on dynamics in an ultrametric space, and (v) Ngai's theory of time-dependent transition rates. In view of the different nature of these models and because of the claims that they are applicable outside of their original contexts, it is useful to make an intercomparison of these models and their consequences. A presentation of these models is here given based on a unified master equation approach. By experiment, many real systems have been shown to exhibit not only the Kohlrausch form but two additional related properties which are not encompassed in model types (i)–(iv). Only models that include time-dependent transition rates have so far been shown to be consistent with the experimental observations of the three empirical relations.