Non-Debye relaxations: two types of memories and their Stieltjes character

We show that spectral functions relevant for commonly used models of the non-Debye relaxation are related to the Stieltjes functions supported on the positive semiaxis. Using only this property it can be shown that the response and relaxation functions are nonnegative. They are connected to each other and obey the time evolution provided by integral equations involving the memory function $M(t)$ which is the Stieltjes function as well. This fact is also due to the Stieltjes character of the spectral function. Stochastic processes based approach to the relaxation phenomena gives possibility to identify the memory function $M(t)$ with the Laplace (L\'evy) exponent of some infinitely divisible stochastic process and to introduce its partner memory $k(t)$. Both memories are related by the Sonine equation and lead to equivalent evolution equations which may be freely interchanged in dependence of our knowledge on memories governing the process.


Introduction
The main information on the nature of dielectric relaxation phenomena comes from broadband dielectric spectroscopy [1] which provides us with data concerning dispersive and absorptive properties of dielectric materials. These properties are encoded in the complex dielectric permittivityε(iω) and its dependence on the frequency of external fields. In particular data of the spectroscopy experiments enable one to determine the behavior of permittivitŷ ε(iω) for asymptotic values of the frequency ω scaled with respect to some relaxation time τ which characterizes the material under investigation. Pioneer of this research was A. K. Jonscher who in 60's and 70's of the previous century studied, with his collaborators, a vast majority of available data and observed that they share universal asymptotic propertiesε (iω) ∝ (iωτ) a−1 , ωτ ≪ 1 nowadays known as the Universal Relaxation Law (URL) [2]. In the above the static permittivity ε 0 denotes the limit ofε(iω) for ω → 0 and the parameters 1 −a and b belong to the range (0, 1). The Jonscher's URL agrees with the most commonly used phenomenological models of the relaxation phenomena, namely with the Havrilak-Negami (HN) and the Jurlewicz-Weron-Stanislavski (JWS) models, both depending on a single characteristic time. For the HN model b ≥ a − 1 which means that the exponent governing asymptotics at infinity is larger than its counterpart governing asymptotics at zero. The opposite situation occurs for the JWS model for which b < a − 1.
Recall that the normalized ratio of permittivities [ε(iω) − ε ∞ ]/[ε 0 − ε ∞ ] (here ε ∞ means the infinite frequency limit ofε(iω)) is named the spectral functionφ(iω). Through the Laplace transform it is related to the time domain response and relaxation functions, denoted as φ(t) and n(t), respectively, φ(iω) = L[φ(t); iω] and [1 −φ(iω)]/(iω) = L[n(t); iω], (2) which simply correspond to each other Actually, the transform in Eq. (2) is the Fourier transform restricted to the semiaxis but following physicists' customs we call it the Laplace transform [3].) Nevertheless, except the Subsec. 3.1 of the manuscript, we do not need to use the complex variables will show that it is enough to make considerations for functions supported on the positive semiaxis. This is the reason why we will restrict ourselves to the Laplace integralf (s) = ∞ 0 e −st f (t) dt for s > 0. Its difference with respect to the Laplace transformf (z) = ∞ 0 e −zt f (t) dt is that s is real while z is complex. For s > 0 and z ∈ C \ R − they can be linked to each other through [4,Theorem 2.6], repeatedly quoted in [5,Theorem 1]. Because of the latter we feel free to use the same notationf for real and complex functions as well as to call interchangeably the s or Laplace domain.
The time evolution of the response function φ(t), for physical reasons requriments to be continuous and vanishing for t < 0, is governed by the equation where B is a nonnegative transition rate constant and M(t) is an integral kernel which plays the role of memory. For consistency of its further physical and mathematical interpretation we assume that M ∈ L 1 loc (R + ). Due to the conditions given by [13,18] there exists another function k(t) ∈ L 1 loc (R + ) which together with M(t) satisfies the Sonine equation [14,15] In addition, assuming the conditions (*) in [13,18] be fulfilled we can find that The authors of Refs. [16,17] proposed to employ Eq. (6) to justify using an integro-differential equation as governing the time evolution of the response function. We emphasize that mathematical structure of the Eq. (7) was the subject of the seminal paper [18] whose results gave the conditions under which the Cauchy problem for the Eq. (7) is uniquely solved.
As it was shown in [6,7,8],M(s) can be expressed by the algebraic inverse of the Laplace (Lévy) exponent Ψ(s), i.e.,M(s) = [Ψ(s)] −1 , of some infinitely divisible stochastic process U underlying the relaxation. Mathematically, the Laplace (Lévy) exponent corresponds to the characteristic function of the process U [6,7]. For instance, the characteristic function of the Lévy stable process X is given by the Lévy-Khintchine formula which naturally introduces Ψ(s) as a complete Bernstein function (CBF). Thus, Eq. (6) illustrates relation between CBF and Sonine pair noticed only a few years ago [18]. Completely Bernstein character of Ψ(s) leads to the crucial observation concerningM(s) -because the algebraic inverse of a CBF is a Stieltjes function (SF) thenM(s) does share this property. Following [11,18] we introduce where a, b ≥ 0 and σ is a measure (0, ∞) such that We remark that if a = 0 then the definition 2 is the same as given in [19] and [12] but for both cases considered for complex numbers and a complex Stieltjes functions. Thus, SF coming from the definition 2 is restriction of a complex SF to the positive semiaxis. As an alternative definition of SF, justified for its convenience for futher considerations, we will use also Theorem 7.3 of [11] which say that f : (0, ∞) → [0, ∞) is SF if, and only if, 1/ f is CBF. Note that SFs form a subclass of completely monotonic functions (CMF) defined as.
The above implies thatM(s), as a reciprocal of a CBF, is CMF as well and, according to Refs. [9,10], we can call such completely monotonic integral kernels as fading memories. Physical interpretation of Eqs. (4) and (7) yields that they should lead to the same physical results. To endowe this property mathematical meaning note that the first of them is the integral equation whereas the next one is an integro-differential equation. Recall that in the theory of integral equations the homogeneous integral equation can be transformed to its differential analogue -in what follows we are going to demonstrate that analogical procedure may be performed for Eqs. (4) and (7). It means that from Eq. (4) we should derive Eq. (7). Doing that we will see the importance of the condition (5) or its analogue in the Laplace domain Eq. (6). Moreover, to name the integral kernels M(t) and k(t) the fading memories it will be essential to prove that they are given by SFs being the subclass of CMFs. In the paper we will show all these properties using the fact that the spectral function in the complex domain can be rewritten as SF.
Our presentation goes as follows. Sec. 2 shows that for commonly used models of non-Debye relaxations their spectral functions are given by SFs. Thus it appears that the response and relaxation functions are nonnegative. In Sec. 3 we show that assuming the Stieltjes character of the spectral functions is enough to obtain two types of integral kernels which govern the evolution of the response functions and which also are SFs. Hence, we can call them the memory functions. We will give the exact and explicite forms of these memories. Requiring that both equations give the same result we conclude that the memories have to satisfy the Sonine equation. The paper is summarized in Sec. 5.

Basic models of non-Debye relaxations
The spectral functions of HN and JWS models, in physical literature called also relaxation patterns, from construction depend on a single characteristic time τ and are given bŷ . If the frequency of applied electric field is of the order 10 5 − 10 10 Hz then fitting the experimental data by a single standard relaxation pattern is not satisfactory any longer and it is much more effective to fit them by a (linear) combination of above mentioned non-Debye relaxation models or to use models which belong to the excess wing model class (EW). The latter involve the extended number of parameters, in particular introduce more than one characteristic time [20]. Thus, in the high frequency domain we deal with more than one characteristic time scale and the Jonscher's URL is not satisfied any longer. For instance, in the simplest version of the EW model we have two characteristic times, τ 1 and τ 2 , built into the spectral function as follows: The HN, JWS, and EW spectral functions, in common denoted asφ (·) (iω) are the (complex) Stieltjes functions which form the subclass of the Nevanlinna-Pick functions P(z), analytic in the upper half plane and satisfying ImP(z) ≥ 0 for Im z > 0 [19,12,11]. Analyticity in the upper half plane guarantees that the Kramers-Kronig relations are satisfied [3]. The link between the Nevanlinna-Pick functions and the CMFs is presented by [4,Theorem 2.6], repeatedly quoted in [5,Theorem 1]. As is signalized in Sec. 1 we will provide our studies in the real domain.
With the help of just mentioned theorem Eqs. (9) and (10) can be rewritten aŝ where s > 0 and 0 ≤ α, β ≤ 1. All these functions are SFs and their Stieltjes character can be shown using the power function s µ which due to the value of exponent µ is either CMF, or SF, or CBF. Namely, for µ ≤ 0 it is CMF, for µ ∈ [−1, 0] it is SF, and for µ ∈ [0, 1] it is CBF. Let us now check if the considered spectral functions really belong to SFs.

Assumption 7. The spectral functionφ (·) (s) for s > 0 is given by a bounded SF.
At this point of our considerations we make a comment concerning physical meaning of our work. Studying measured experimental data we are able to determine only the relaxation function, its first time derivative and eventually the second one. Therefore the requirement that all derivatives exist and alternate for s > 0, as it is needed by the definition of CMF, is impossible to be verified in practice. Nevertheless, from just listed Facts 4-6 we learn that the spectral functions used to fit the data are modelled by SFs, and thus CMFs. Moreover, Assumption 7 simplify many calculations because we know that any CMF is uniquely represented by a Laplace integral of a nonnegative function. Indeed, the Bernstein theorem (called also the Bernstein-Widder theorem) according to the classical D. V. Widder's book reads [21, Theorem 12a]:

Theorem 8. A necessary and sufficient condition that f (x) should be completely monotonic in
where α(ξ) is bounded and non-decreasing and the integral converges for 0 ≤ x < ∞.
The proof of the Bernstein theorem can be found in [21,22]. We notice that the formulation of the Bernstein theorem can confuse the reader because some of the authors use the Laplace transform (e.g. [18]) but in Widder's and Pollard's approach the theorem is formulated just in terms of the real valued integral. For majority of physicists the name "integral transform" means that we can invert Eq. (14) which usually demands using methods of the complex analysis.
Here, such considerations are not needed. Hence we will use Widder's and his student, H. Pollard, definition [21,22]. Thanks to the Bernstein theorem we can claim that Proposition 9. The response function φ(t) and the relaxation function n(t) are nonnegative.
Proof. The proof of nonnegativity of φ(t) flows immediately from the Bernstein theorem applied to the first formula of Eqs. (2) which written in the s domain readŝ To show that the relaxation function n(t) is given by a nonnegative function we use the second formula of Eqs.
(2) in the s domain, this is Note that the function 1 −φ(s) can be rewritten asφ(0) −φ(s) whereφ(0) is bounded andφ(0) = 1 is the maximum of φ(s). Then, the property (cb4) implies that it is CBF and, then, from definition of CBF it appears that [1 −φ(s)]/s is SF (property (cb6)). Furthermore, Eq. (16) means that the Laplace integral of n(t) is equal to [1 −φ(s)]/s which is SF and thus CMF. Then, the Bernstein theorem implies that n(t) is nonnegative.
The exact forms of the response function φ(t) and the relaxation function n(t) for the HN and JWS models can be found in, e.g., [24,23]. Relevant formulae are and For the EW model only the relaxation function n EW; α (t) is known. It can be expressed through the binomial Mittag-Leffler function [20] n EW; Analogically as in the derivation of Eq. (19) we can obtain the first derivative of the binomial Mittag-Leffler function E (ν 1 , ν 2 ), 1 (ax ν 1 , bx ν 2 ). It reads Eq. (22) can be also derived by employing the representation of the binomial Mittag-Leffler function in terms of the series of three parameter Mittag-Leffler functions given by Eq. (B.6) and, next, separate from these series the zero term, i.e. E 1 ν 1 ,1 (bx ν 2 ) or E 1 ν 2 ,1 (ax ν 1 ). The calculation goes as follows Now, applying Eq. (19) and Eq. (B.4) we get which restores Eq. (22). The same way can be repeated for the representation of E (ν 1 ,ν 2 ),1 (ax ν 1 , bx ν 2 ) through the second series in Eq. (B.6).
The series form of the three parameter Mittag-Leffler function E λ ν,µ (x) for x ∈ R and the binomial Mittag-Leffler function E (ν 1 ,ν 2 ),µ (x, y) for x, y ∈ R as well as their properties are recalled in Appendix Appendix B.

3.M(s) andk(s) and the Laplace (Lévy) exponents Ψ(s) related to them
In this section we shall show that to determine the Stieltjes character ofM(s) andk(s) as well as to explain the CBF nature of the Laplace (Lévy) exponent Ψ(s) it is enough to demand the Assumption 1 be fulfilled.
Eqs. (4) and (16) allows one to express the memory M(t) in terms of the spectral functionφ(s). From them we haveM which is SF.
Proof. The proof of the Stieltjes character ofM(s) goes as follows. We begin with n r=0 [φ(s)] r+1 which is SF as a linear combination of SFs. Then, taking the pointwise limit n → ∞ we obtain the series ∞ r=0 [φ(s)] r+1 which tends to BM(s) for allφ(s) (we remind thatφ(s) is a bounded SF which maximum isφ(0) = 1 and which vanishes for s → ∞). Due to property (s1) we end up the proof.
The algebraic inverse ofM(s) gives the Laplace (Lévy) exponent Ψ(s): The Stieltjes character ofk(s) flows out from the fact that Ψ(s) is CBF and the property (cb6). Moreover, from the property (cb5) it occurs that s/Ψ(s) = Φ(s) is CBF such that it can be treated as another Laplace (Lévy) exponent.

Examples of memories M(t) and k(t)
The examples ofM(s) andk(s) in the s domain for the HN, JWS, and EW models are listed in Table 1, whereas in the time t domain we have Table 2. Relations between functions listed in Tables 1 and 2  Proof. To show that Lemma 13 is true we integrate both sides of Eq. (4) taking T 0 k(T − t) · · · dt of it where instead of "· · · " we substitute all terms of Eq. (4). In this way we obtain Afterwards, due to the Dirichlet formula, the double integral Setting where the passage from upper to lower formulas goes by using Eq. (5) twice. Next, we differentiate it with respect to T . That ends the proof.
subclass of CMFs and in fact it can be deduced from the survey paper [24]. Our really new results are to pay the readers attention to the fact that we use SFs instead of CMFs and connect the Laplace (Lévy) exponent Ψ(s) with the integral kernels M(t) and k(t), basic objects which through the evolution equations govern the behavior of the response and relaxation functions. This observation is grace to the fact that Ψ(s) is CBF hence s/Ψ(s) is also CBF. That allows us to join these functions with M(t) and k(t). Namely, M(t) = [Ψ(s)] −1 and k(t) = Ψ(s)/s. Hence, one stochastic process underlying the relaxation can be described in two-fold -either by M(t) or by k(t). Throughout the paper we were able to reconstruct the previously known form of M(t) for the HN relaxation model [27,28] and to find M(t) for the JWS and EW models as well as k(t) for all models investigated models. Relevant kernels are itemized in Table 2 whereas their shapes in s domain are presented in Table 1. We have also shown that M(t) and k(t) are SFs so they can be called fading memories. Moreover, M(t) and k(t) satisfy the classical Sonine equation and have integrable singularities at zero, thus form the Sonine pairs. We provided three examples of them: k HN (t), M HN (t) , k JWS (t), M JWS (t) , and k EW (t), M EW (t) which, each other, lead to two equations: the integral and the integro-differential one which are mutually coupled by the Sonine equation for the memories M(t) and k(t).
A byproduct of our considerations is providing explicit expressions for functions belonging to the Mittag-Leffler family, namely for the first derivative of E λ ν,1 (x) and E (ν 1 ,ν 2 ),1 (x). We emphasize that these formulae were derived using physical arguments -relation between the response and relaxation functions.