What is it about?

Modelling the behaviour and response of dielectric materials is important for various reasons. Namely, prediction of standard and anomalous phenomena is possible only by model analysis and model-based simulation. Moreover, in many applications, the models allow to evaluate performance indexes and to adjust the structure or parameters of dielectrics so that responses obey design specifications, reference behaviours, etc. Simple exponential models are often not satisfactory, while advanced non-exponential models are required to better explain experimental observations of complex systems. The “universal” response typically decays as a “stretched exponential” function at short to intermediate times or asymptotically as a power law at long times. Fractional calculus is then used to capture the anomalous relaxation process of dielectrics.

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Why is it important?

In the present paper, we try to survey the most common models existing in the literature to our best knowledge and describe their main properties under a mathematical point of view. All the models have the common feature that for large times the relaxation and response functions generally decay with a power law that indeed is found in most experiments. There is however a further feature which ties all these models and which we would like to highlight: relaxation and response are completely monotonic functions of time, which means that they are expressed as a continuous distributions of simple exponential functions with a nonnegative spectrum of relaxation times. To our knowledge, the property of complete monotonicity has not been sufficiently outlined in the existing literature at variance with the power law asymptotic behaviour.


This survey focuses on some of the most famous models based on fractional power laws, mainly when the relaxation can be characterized in terms of completely monotone functions and, in particular, of Mittag-Leffler type. Over the years, most of these models have gained a great importance and today they represent the basis for studying dielectric properties. In particular, this survey concerns with the main functions and operators involved in the time-domain analysis of the selected dielectric models, with the aim of providing a clear characterization under a mathematical point of view. I hope the survey can be helpful for studies involving anomalous relaxation and diffusion processes, biological tissues (fitting the experimental data from various tissues and from different samples of the same tissue) or dispersive biological materials having dielectric properties characterized by power-law frequency dependency.

Dr. Guido Maione
Politecnico di Bari

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This page is a summary of: Models of dielectric relaxation based on completely monotone functions, Fractional Calculus and Applied Analysis, January 2016, De Gruyter, DOI: 10.1515/fca-2016-0060.
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