On Complete Monotonicity of Some Functions of the Mittag-Leffler Type in Non-Debye Relaxation Processes

Professor Francesco Mainardi
Department of Physics, University of Bologna, and INFN
Via Irnerio 46, I-40126 Bologna, Italy
E-mail: francesco.mainardi@bo.infn.it.it

Abstract: In this talk we discuss some interesting examples of relaxation occurring in viscoelastic and dielectric materials, which are described by special completely monotone functions of the Mittag-Leer type. This means that these response functions are represented by continuous distributions of elementary (i.e. exponential) relaxation processes via non-negative spectra of relaxation in frequency or time. In addition to the well known functions of Mittag-Leffler type in one and two parameters, we revisit two more general kinds of Mittag- Leffler functions in three parameters, that is the Prabhakar and the Kilbas-Saigo functions. For all these functions we prove the conditions on the parameters to ensure the complete monotonicity and compute the corresponding frequency spectra. For some study-cases we present numerical results with illustrative plots for the field variable and for the corresponding spectral distribution. We hope that our results can be adopted when the field variable is the response function associated with non-Debye relaxation processes found e.g. in dielectrics. In particular we have derived as noteworthy particular cases the classical models of non-Debye relaxation phenomena referred to as Cole-Cole, Davidson-Cole, Havriliak-Negami along with the so-called Kohlrausch-Williams-Watts (KWW) law based on the stretched exponential function.

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