Heavy-tail properties of relaxation time distributions underlying the Havriliak–Negami and the Kohlrausch–Williams–Watts relaxation patterns

2007

Abstract A detailed discussion of asymptotic properties of the Havriliak–Negami and the Kohlrausch–Williams–Watts relaxation time distributions is presented. The heavy-tail property of the Havriliak–Negami relaxation time distribution, leading to the infinite mean relaxation time, is discussed. In contrast, the existence of the finite mean relaxation time for the Kohlrausch–Williams–Watts response is shown. The discussion of the Cole–Davidson and the Cole–Cole cases is also included. Using the Tauberian theorems we show that these properties are determined directly by the asymptotic behavior of the considered empirical functions.

Wait-and-switch stochastic model of the non-Debye relaxation. Derivation of the Burr survival probability

2006

Abstract Stochastic mechanism of relaxation, in which a dipole waits until a favourable condition for reorientation exists, is discussed. Assuming that an imposed direction of a dipole moment may be changed when a migrating defect reaches the dipole, we present a mathematically rigorous scheme relating the local random characteristics of a macroscopic system to its effective relaxation behaviour. We derive a relaxation function (the Burr survival probability) that is characterized by the stretched exponential or the power-law behaviour.

Wait-and-switch relaxation model: Relationship between nonexponential relaxation patterns and random local properties of a complex system

2006

The wait-and-switch stochastic model of relaxation is presented. Using the ``random-variable'' formalism of limit theorems of probability theory we explain the universality of the short- and long-time fractional-power laws in relaxation responses of complex systems. We show that the time evolution of the nonequilibrium state of a macroscopic system depends on two stochastic mechanisms: one, which determines the local statistical properties of the relaxing entities, and the other one, which determines the number (random or deterministic) of the microscopic and mesoscopic relaxation contributions. Within the proposed framework we derive the Havriliak-Negami and Kohlrausch-Williams-Watts funct…

Survival probability approach to the relaxation of a macroscopic system in the defect-diffusion framework

2004