Two forms of self-similarity as a fundamental feature of the power-law dielectric response

A new mathematical representation of the cluster model for dielectric relaxation, in a bound dipole case, is established by employing the extremal value theory. Two distinct probabilistic mechanisms, which drive the dielectric response function to acquire the power-law form, are presented. Consequently, two forms of self-similarity, one of which dominates the response at short times and the other at long times, leading to a general relaxation equation, are identified. Finally, the conditions under which the derived response function takes the well known empirical forms (Williams-Watts, Cole-Cole, Cole-Davidson, 'broadened' Debye, and 'flat loss' responses) are recognized.