Composition law for the Cole-Cole relaxation and ensuing evolution equations
Introduction
The Cole-Cole (CC) relaxation model was introduced into dielectric physics by the Cole brothers [1] to fit experimental data obtained in the measurements of frequency dependence of the electric permittivity. The CC model provides us with an example of non-Debye relaxation for which the spectral function (frequency dependent normalized complex dielectric permittivity) is phenomenologically adjusted to In the above ω denotes the frequency, and ϵ are frequency dependent and static permittivities, respectively, while is the dielectric constant of induced polarization. Parameters appearing in the RHS of Eq. (1) come from purely phenomenological analysis: α is called the width parameter and ranges in the interval ; means an effective time constant related to the so-called loss-peak frequency [2]. The formalism of the relaxation phenomena theory, namely the rules which connect the frequency and time regimes, relates the spectral function to the time dependent pulse-response function through the Laplace transform Inversion of the Laplace transform (2) with the relation (1) inserted in is long-time known [3] where is the Mittag-Leffler (ML) function whose properties have been examined for many years. The ML function itself, as well as its generalizations, are widely used in many branches of mathematical analysis, first of all in fractional calculus and special functions theory [4], [5] and also in the probability theory [6].
The CC model, being among the oldest and the simplest examples of the non-Debye relaxation, frequently fits the experimental data of relaxation measurements unsatisfactorily. In contemporary experimental research it needs to be replaced by more sophisticated models for which it remains a particular case [7]. Nevertheless, being a training ground of various research concepts it still attracts theoreticians. Their efforts, rooted in the search of physical background of Jonscher's universal relaxation law [8], [9], are two-fold. The first approach is based on the analysis of stochastic processes supposed to underlie the relaxation phenomena [10], [11], [12] and after that the extensive use of generalized central limit theorems [6], [13], [14]. The alternative method starts from analytical properties of the phenomenologically determined spectral functions. This leads, using tools of the theory of completely monotonic functions [15], [16], to the time dependent relaxation functions uniquely determined as weighted sums of elementary Debye relaxations [17], [18], [19]. It should be noted here that in both approaches various Mittag-Leffler type functions appear and do play a very important, even crucial, role.
Despite of the limited applicability of the CC model in dielectric physics its usefulness goes beyond this branch of physically oriented research and concentrates on various aspects of material science. Examples of its geophysical applications are exhibited, e.g., in [20], [21] in which the authors have used it to describe the induced polarization of porous rocks. Applications oriented to the life sciences may be found, e.g., in [22] where the CC model has been employed to investigate processes of molecular recognition and also in studies how the age-dependent dielectric properties impact on the brain tissues and proportions of the skull [23]. Another examples are provided by the analysis of electric conductivity measured in tissues of the hepatic tumors [24] and fitting the CC parameters to dielectric data measured in biological tissues and organs [25]. A little striking are recently presented applications of the CC model in winery [26].
In what follows we will adopt a notation treated as one symbol parametrized by a real number α and providing us with the ratio of the number of some objects counted at an instant of time t and the number of the same kind of objects counted at . In relaxation processes means the relative number (i.e. calculated with respect to the initial number ) of objects which decay, e.g. depolarize, during the time interval . It bears the name of the relaxation function and is defined as a minus primitive of , i.e., . Thus we get where, and in what follows, 's denote dimensionless variables . It should be also recalled that the function counts objects which survive the decay in the period [10]. Obviously, for the relaxation function reduces to the exponential function describing the Debye relaxation.
Recall that the basic property of the exponential function is that of being the only solution to the functional equation . This implies that in the Debye case the relaxation function satisfies the composition law where the symbol ‘⋅’ denotes the usual multiplication. An “intermediate” instant of time splits the time interval into and . The Debye relaxations are evolution processes without memory [7], [27]. For other evolution patterns, in particular involving the memory effects like it happens in the case of non-Debye models, the composition law given in the form of Eq. (6) must not be valid any longer. Describing any non-Debye relaxations we have to adopt another (different from usual multiplication) composition rule if we want to describe correctly the evolution . Any proper composition rule must take into account the basic requirement that the evolution in (and also its functional form) must coincide with the evolution which starts from some initial condition, goes on to the “intermediate state” (taken at the instant of time freely chosen at ) and next evolves in reaching the final condition at . Illustrative example how to realize such a rule is provided by relaxation described in terms of the stretched exponential functions (the Kohlrausch-Williams-Watts (KWW) ones) for which the composition law is given by the Laplace convolution [28]. It is natural and absolutely justified to ask for the analogous rule to be obeyed by the CC relaxation. Looking for and finding suitable composition laws satisfied by any model of the time evolution is necessary (although not sufficient) condition for verification its consistency. Such a check is particularly meaningful when one investigates evolution problems coming out from phenomenology with not fully understood origin in fundamental physics. The aim of our paper is to fill this gap for the CC model and to present how realization of this basic condition, in what follows called evolution consistency requirement, is achieved in the framework of the CC model.
The paper is organized as follows. In Sec. 2 we recall our main postulate that the composition of two CC relaxation functions in the time domain leads to another CC relaxation function in the time domain and next we define the analytical form of such a composition. Its explicit form, if applied in the evolution consistency requirement, implies the structure of the integral evolution equation which the CC relaxation function holds. In Sec. 3, using the numerical calculation we show that our integral version of the evolution equation is equivalent to its differential version. Differential form of the evolution equation is the fractional Fokker-Planck equation, solvable in terms of the ML function. The solution of this equation is also given. The similarity between integral and differential forms of the evolution equation shows that it is the fractional kinetics which underlines the CC relaxation. The paper is concluded in Sec. 4.
Section snippets
“Convolution” of the CC relaxations
Let us consider a non-Markovian process for which the composition rule is written down as valid for any such that . The basic property of the “abstract” composition rule ‘∘’, namely its invariance with respect to the “intermediate” time , is graphically illustrated in Fig. 1.
The LHS of Eq. (7), if rewritten in terms of the series representation (cf. Eq. (4)), gives a double infinite sum. According to the so-called splitting formula for
The differential evolution equation
Eq. (12) represents the basic evolution law as the integro-differential relation derived from properties of the ML function and the composition rule Eq. (9) being assumed or, one may say, even guessed. From our derivation is clear that the ML function satisfies Eq. (12) but if we assume the latter as primarily given then we should ask for its solutions. Looking for them we will proceed analogously to the case B of the previous section. Evaluating in Eq. (12) the derivative over we get
Conclusion
We have studied the consequences of the physically natural composition rule required to be obeyed by the time evolution of the CC relaxation function. The condition which we have been proposing reflects the rule that the CC relaxation taking place at the time interval can be composed from CC relaxations at the times and , where . If such a composition rule is represented as the multiplication then we deal with the standard Debye relaxation which may be interpreted as
Acknowledgements
The authors were supported by the NCN research project OPUS 12 no. UMO-2016/23/B/ST3/01714. K. G. acknowledges the support of the MNiSW (Warsaw, Poland) Programme “Iuventus Plus 2015-2016”, project no IP2014 013073 and the NCN Programme Miniatura 1, project no. 2017/01/X/ST3/00130.
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