Models of Dielectric Relaxation Based on Completely Monotone Functions

• Published in: Fractional calculus & applied analysis Vol. 19; no. 5; pp. 1105 - 1160
• Main Authors: Garrappa, Roberto, Mainardi, Francesco, Maione, Guido
• Format: Journal Article
• Language: English
• Published: Warsaw Versita 01.10.2016; De Gruyter; Nature Publishing Group
• Subjects: 26A48; 34A08; 44A10; 91B74; Abstract Harmonic Analysis; Analysis; Calculus; complete monotonicity; dielectric models; Dielectric relaxation; Dielectrics; Differential equations; differential operators; fractional calculus; Functional Analysis; Integral Transforms; Mathematical analysis; Mathematical functions; Mathematical models; Mathematics; Mittag-Leffler functions; Monotone functions; Operational Calculus; Operators; Operators (mathematics); Primary 26A33; Response functions; Secondary 33E12; Stress relaxation; Survey Paper; Time domain; Time domain analysis; fractional calculus; Mittag-Leffler functions; Primary 26A33; 44A10; Secondary 33E12; differential operators; 26A48; dielectric models; 91B74; 34A08; complete monotonicity
• ISSN: 1311-0454; 1314-2224
• DOI: 10.1515/fca-2016-0060

The relaxation properties of dielectric materials are described, in the frequency domain, according to one of the several models proposed over the years: Kohlrausch-Williams-Watts, Cole-Cole, Cole-Davidson, Havriliak- Negami (with its modified version) and Excess wing model are among the most famous. Their description in the time domain involves some mathematical functions whose knowledge is of fundamental importance for a full understanding of the models. In this work, we survey the main dielectric models and we illustrate the corresponding time-domain functions. In particular, we stress the attention on the completely monotone character of the relaxation and response functions. We also provide a characterization of the models in terms of differential operators of fractional order.