The Cole Cole model  is a generalization of the Debye dielectric relaxation model which fits measurements in many applications including the bioimpedance field, [2, Sec. 9.2.7]. One interpretation is that it represents a distribution of relaxation processes, each described by the Debye model. Since the Debye model has a simple time domain interpretation and both the current and charge responses to a voltage step are exponential, the Cole-Cole responses can therefore be expressed as sums of exponential functions. In practice, however, this result is often too complex to lend itself to interpretation.
In recent years, there has been a development in understanding of the responses of the Cole-Cole model found in a direct way. These results depend on the Mittag-Leffler function, a generalization of the exponential which is named after Gösta Mittag-Leffler (1846–1927). This function is rightly called the “queen function of fractional calculus”  showing the close link between non-integer derivatives and the Cole-Cole model. The asymptotes of the Mittag-Leffler function are just as important as the function itself and is what will be emphasized here.
The purpose of this paper is to increase awareness of the time domain properties of the Cole-Cole model by collecting and interpreting some results from in particular [5, 7, 8]. The paper starts with the Debye model in order to define the relevant current and charge responses, called the response function and the relaxation function respectively. It will also be shown that both the Curie-von Schweidler power law and the Kohlrausch-Williams-Watt stretched exponential response are approximations to those of the Cole-Cole model. Finally, it is also shown that just as the Debye model corresponds to an ordinary partial differential equation for the constitutive law between the displacement field and the electric field, the Cole-Cole model corresponds to a similar equation but with non-integer, i.e. fractional derivatives.
Characterization of general models
The definitions of φ(t) and ψ(t) are such that they both are non-negative and non-increasing functions of time, meaning that the zero order derivative is positive and the first order derivative is negative. This pattern of sign changes repeats infinitely with the second order derivative positive and so on. This is what characterizes a completely monotone function and it ensures that it can be expressed as a continuous distribution of exponential functions and that the Laplace transform is non-negative . In this way the physical realizability of the considered system is guaranteed .
Time domain characterization
Approximation of the response function
Approximation of the relaxation function
Beyond the Cole-Cole model
The familiar frequency domain expression for the Cole-Cole model of order α can also be expressed in the time domain. The response function, which is related to the current response to a voltage step excitation, is expressed with a two-parameters Mittag-Leffler function. Its asymptote for small time is a power-law function which corresponds to the Curie-von Schweidler law. The relaxation function, which describes the charge response, is given by a one-parameter Mittag-Leffler function where the asymptote for small time is the stretched exponential or Kohlrausch-Williams-Watt function. For the Debye model, both these responses are given by exponential functions in time. The Debye model is also equivalent to a first-order differential equation between electric field, E, and the displacement field, D. This generalizes to a fractional differential equation with non-integer derivatives of order α for the Cole-Cole model.
I want to thank Thomas Holm for having read and commented the final manuscript.
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