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# Zeitschrift für Naturforschung A

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# Multiscale Model for the Dielectric Permittivity

• Statistical and Interdisciplinary Physics Section, Departament de Física de la Matèria Condensada, Facultat de Física, Universitat de Barcelona, Mart i Franquès 1, 08028 Barcelona, Spain
• Other articles by this author:
/ Luciano C. Lapas
• Corresponding author
• Interdisciplinary Centre for Natural Sciences, Universidade Federal da Integração Latino-Americana, P.O. Box 2067, 85867-970 Foz do Iguaçu, Brazil
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• Other articles by this author:
/ J. Miguel Rubí
• Statistical and Interdisciplinary Physics Section, Departament de Física de la Matèria Condensada, Facultat de Física, Universitat de Barcelona, Mart i Franquès 1, 08028 Barcelona, Spain
• Other articles by this author:
Published Online: 2017-01-18 | DOI: https://doi.org/10.1515/zna-2016-0453

## Abstract

We present a generalisation of the Debye relaxation model for the dielectric permittivity in the case in which the global relaxation process is the result of many elementary excitations. The relaxation dynamics is in this case non-Markovian. In the case of many events, for which the central limit theorem holds and Gaussianity as well as the assumption of independency are both plausible, the global relaxation time is given by a log-normal function. The hierarchy of relaxation times leads to a generalised expression of the dielectric permittivity.

PACS: 05.70.Ln; 65.60.+a; 68.65.-k; 71.45.-d

## 1 Introduction

Debye theory describes the relaxation of the polarisation in an isotropic and homogeneous material. Although very successful in many instances, the theory is limited to the case in which the relaxation dynamics is governed by a single relaxation time [1]. This situation is not observed in complex materials in which the global evolution results from the existence of a wide variety of time and length scales which leads to a nonexponential decay of the response function. To characterise the relaxation dynamics of these systems, it is one of the most challenging problems of nonequilibrium statistical mechanics nowadays.

Our purpose in this article is to generalise the single-time Debye relaxation model to the case in which many collective excitations take place in the system. The relaxation process can then be characterised by a hierarchy of relaxation times [2], [3], [4]. We will consider that the global relaxation consists of a cascade of elementary independent relaxation processes whose relaxation times follow a recurrence law in which each time differs form the previous one by factor that follows a Gaussian distribution. Under this condition, one can show that the global relaxation time is given by a log-normal function. The relaxation process is then non-Markovian. We will show that the global relaxation time of the process corresponds to the Fourier transform of the memory function of the non-Markovian description. The correlation of the polarisation random source is then proportional to the computed relaxation time.

This general behaviour will be applied to the case of relaxation of the polarisation. We will show that this quantity follows a generalised Langevin equation that accounts for its stochastic dynamics. In this equation, the random term satisfies a fluctuation-dissipation theorem (FDT) that depends on the memory function. We have proposed a generalised Debye relation for the dielectric permittivity that includes the hierarchy of relaxation times.

The article is organised as follows. In Section 2, we extend Debye theory to the case in which the dynamics is non-Markovian. We present the relaxation model and the way to calculate the memory function. In Section 3, we propose a generalised Debye relation for the dielectric permittivity proper of that non-Markovian dynamics. In Section 4, we formulate a generalised Langevin equation for the polarisation and discuss the FDT. Finally, in Section 5, we summarise our main results.

## 2 Non-Markovian Dielectric Relaxation

In the framework of Debye theory for dielectric relaxation, the polarisation P of an isotropic and homogeneous material relaxes according to the equation [5], [6]

$∂P∂t=−τ − 1(P−κE),$(1)

where τ is the relaxation time, κ the static electric susceptibility, and E the local electric field. Equation (1) describes the relaxation of the polarisation of an assembly of noninteracting dipoles. In Fourier space, it is given by

$P^(ω)=χ(ω)E^(ω),$(2)

where the dynamic susceptibility χ(ω) is given by through

$χ(ω)=κ1−iωτ.$(3)

The Fourier transform has been defined as

$a^(ω)=∫− ∞∞dt a(t)eiωt.$(4)

Since the electric displacement is defined as D(ω)=E+4πP, one concludes that

$ε(ω)=1+4πχ(ω).$(5)

The relaxation of the polarisation can also be studied in the framework of nonequilibrium thermodynamics [5], [7]. The relaxation process is dissipative and therefore entails an entropy production. This quantity is given by

$dsdt=−1T∂P∂t(Eeq−E),$(6)

The entropy production of nonequilibrium processes is in general written in terms of flux-force pairs. In the case of dielectric relaxation, the flux is ∂P/∂t and the force (EEeq)/T. The linear coupling of both quantities yields

$∂P∂t=LT(E−Eeq)$(7)

where L is an Onsager coefficient. Identifying τ with /L and using the relation Eeq=P/κ lead to the Debye relaxation equation (1).

When the relaxation dynamics results from the intervention of several collective excitations or vibration modes, a generalisation of the Debye equation is given by [6], [8]

$∂P∂t=−∫0tϕ(t−t′)[P(t′)−κE(t′)]dt′$(8)

where ϕ(tt′) is a memory function resulting from the existence of different time scales.

The formal solution of the previous equation is again (2) but with a susceptibility given by

$χ(ω)=κ1−iωτ(ω),$(9)

where τ(ω) is related to the memory function through the equation

$τ − 1(ω)=∫− ∞∞dt ϕ(t)eiωt.$(10)

To know the polarisation relaxation dynamics, we have to obtain τ(ω). To this purpose, according to the Matthiessen rule, an overall relaxation time τ can be obtained through

$τ − 1=τ1 − 1+…+τn − 1,$(11)

where all the τ correspond to scattering events. Our model is completed by assuming a system having a multiple relaxation scenario triggered by cascade processes (see Figure 1). Examples can be found in systems such as metallic glasses [9], graphene [10], and semiconductors [11], [12]. In this context, let us assume a small random elementary input ξ whose effect on the relaxation time at stage is proportional to ξ and to the cumulative effect ${\tau }_{\ell \text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}^{\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}$ of the −1 previous inputs,

Figure 1:

Schematic diagram for a cascade of relaxation times defined through (11) and (12). The relaxation time function is a log-normal distribution.

$τℓ − 1=τℓ − 1 − 1+ξℓτℓ − 1 − 1.$(12)

From (12), one can obtain

$∑i = 1nξi=∑i = 1nτℓ − 1−τℓ − 1 − 1τℓ − 1− 1∝∫τ0 − 1τ − 1dεε=lnτ − 1τ0 − 1.$(13)

Since random variation can be understood as a product of several random effects, the Gaussianity condition can be applied to the multiplication of log-normal random variables, which gives a log-normal distribution,

$τ − 1(ω)=12πστ0exp[−ln2(ω/ω0)/2σ2].$(14)

Here, τ0 is a material-dependent parameter, which can be estimated from thermal diffusivity [13], ω0=kBT0/ħ is the thermal frequency and σ the standard deviation that describes the deviations from ω0. The thermal diffusivity is associated with the effect of temperature on the lattice parameters and affecting the mechanical vibration of the arrangement of atoms. This implies that a characteristic time relaxation resulting from the thermal diffusivity will also be a function of a characteristic frequency (real ω>0) and temperature. Thus, we consider a general approach in which the relaxation times are written in a frequency-dependent [14], [15], [16] form such that $\mathrm{ln}\left({\tau }_{\ell }^{\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}/{\tau }_{0}^{\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}\right)\propto \mathrm{ln}\left({\omega }_{\ell }/{\omega }_{0}\right).$ In Figure 2, we have represented the memory function as a function of frequency. When the standard deviation becomes very large, the memory function tends to a delta function. One then recovers in this limit the results of Debye theory.

Figure 2:

Representation of the Fourier transform of the memory function for ω0=kBT0/ħ, at room temperature, τ0=40 fs and different values of the standard deviation. When σ goes to zero, the function tends to a delta function proper of Markov processes.

## 3 Generalised Debye Relation for the Dielectric Permittivity

As a good approximation, it has been known that for most condensed systems in time-dependent fields [17], [18], the orientation polarisation behaviour can be characterised by a relaxation time function, τ(t), which is generally meant as dielectric relaxation. In harmonic fields, this implies that the complex dielectric permittivity in the frequency range corresponding to the characteristic times for the molecular reorientation can be written as

$ϵ(ω)=ϵ∞+(ϵs−ϵ∞)∫0∞ζ(τ)1+iωτdτ,$(15)

where ϵs is the static dielectric constant and ϵ the permittivity at infinite frequency. Since the normalisation of (14) is given through $\int {\tau }^{\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}\left(\omega \right)d\omega ={\omega }_{o}\mathrm{exp}\left({\sigma }^{4}\right)/{\tau }_{o},$ one can assume that the relaxation time distribution ζ(t) satisfies the normalisation condition ${\int }_{0}^{\infty }\zeta \left(\tau \right)d\tau =1$ for the sake of simplicity. For a time-independent time scale τ, after integration in (15), one can recover the Debye relaxation equation [1]

$ϵ(ω)=ϵ∞+(ϵs−ϵ∞)11+iωτ.$(16)

This relation is also obtained after Fourier-transforming the relaxation function φ(t)=exp(−t/τ); the relaxation function φ(t) is obtained from the Smoluchowski kinetic equation when a electric field, being constant in the time interval [− ∞, 0], is instantly switched off at t=0. Let us now consider a time-dependent relaxation time, τ(t), which includes a particular way of extracting the response on different time scales and can be associated with the normalised dielectric function (ϵsϵ(t))/(ϵsϵ) [4]. Hence, we can understand τ(ω) as defined through the relation

$∫0∞ζ(τ)1+iωτdτ≡11+iωτ(ω),$(17)

going beyond the scope of Markovian stochastic processes by considering memory effects. When applied to linear and nonlinear system, in equilibrium limit, this model is equivalent to previous approaches [19], [20], [21].

The hierarchy of relaxation times given through (14) accounts for dissipation in the system. In fact, τ − 1(ω) represents the asymptotic of the diffusion coefficient. This circumstance points out the close relation between the interaction forces and dissipation via the FDT [6]. In addition, the introduction of τ − 1(ω) leads to a new model for the dielectric permittivity [2], [4] from an analytical continuation

$ϵ(ω)=ϵ∞+ϵs−ϵ∞1+iωτ(ω).$(18)

Equation (18) can be interpreted as a generalised Drude relation that accurately incorporates the dissipative effects, in contrast to the plasma model, which does not include dissipation. The real and imaginary parts of the permittivity have been shown in Figure 3. In the limit of small standard deviations in the inverse of the relaxation time, they tend to the corresponding ones of Drude model.

Figure 3:

Imaginary (top) and real (bottom) parts of complex dielectric permittivity for ϵs=10 (a.u.) and ϵ=1 (a.u.). The parameters ω0 and τ0 are the same as in Figure 2. In the limit of large standard deviations, both parts tend to the corresponding ones in Drude model.

Since the tails (when (ω/ω0) → 0 or ∞) of the log-normal (14) are negligible, τ − 1(ω) ≥ 0 always, thus guaranteeing that the complex susceptibility (9) as well as the dielectric permittivity (5) satisfy the Kramers–Kroning relations. To make our arguement more clear, we must show that the complex susceptibility is analytic in the upper-half complex plane. In this sense, note that the zeros of the denominator in (9) satisfy the relation

$ω=− iτ− 1(ω);$(19)

Let ω* be one of these zeros, since τ − 1(ω*) is real, ω* is situated in the lower-half complex plane. The previous discussion demonstrate the validity of the causality principle.

## 4 The Fluctuation-Dissipation Theorem

For near-field interactions, the thermal conductance in the case of two nanoparticles interchanging photons was calculated under the assumption that they behave as effective dipoles at different temperatures [22]. Since these dipoles undergo thermal fluctuations, the FDT [5], [6], [19], [23] should account for the energy that dissipates into heat in each particle. Near contact, the conductance can deviate dramatically from the prediction of the dipole model as consequence of the fact that when particles become very close the positions of the atoms are highly correlated, consequently the charge distributions become non-symmetric and cannot be described merely as two interacting dipoles. To account for this distortion in the distribution of charges, whenever the charge distribution of each particle in the presence of mutual interactions has reached equilibrium with the heat bath, a more general formalism involving higher order multipoles aside from the dipoles has been proposed in [24]. However, when the particles become even closer, correlations drive the system into an aging regime [5], [7].

The memory function as well as the hierarchy of inverse relaxation times that corresponds to the Fourier transform of the memory function is related to the dissipation associated with the entropy production in the system. The dissipation is caused by the noncompensated heat generated during the decay of the fluctuations that is transferred to the thermal bath. The FDT establishes precisely this fact. For the case of Debye relaxation, when the process is described by means of a single relaxation time, the Langevin equation for the polarisation is given by

$∂P∂t=− τ− 1(P−κE)+J(t),$(20)

where the random current satisfies the FDT

$〈J(t)J(t′)〉=〈P2〉eqδ(t−t′)$(21)

or equivalently

$〈J(t)J(t′)〉=κ2〈E2〉eqδ(t−t′)$(22)

The FDT, (22), assumes that the random sources are delta correlated. It was formulated in the framework of fluctuating electrodynamics [25].

We have seen that in the presence of multiple time scales, the relaxation of the polarisation is non-Markovian. This quantity then relaxes according to the generalised Langevin equation [6], [8], [26]

$∂P∂t=−∫0∞dt′ϕ(t−t′)[P(t′)−κE(t′)]+J(t),$(23)

where the different time scales are contained in the memory function; see (10). The random current satisfies the FDT

$〈J(t)J(t′)〉=〈P2〉eqϕ(t−t′)$(24)

By Fourier transforming, (24) becomes

$〈J^(ω)J^(ω′)〉=2π〈P2〉eqτ − 1(ω)δ(ω+ω′)$(25)

or

$〈\stackrel{^}{J}\left(\omega \right)\stackrel{^}{J}\left({\omega }^{\prime }\right)〉=2\pi {\kappa }^{2}{〈{E}^{2}〉}_{eq}{\tau }^{\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}\left(\omega \right)\delta \left(\omega +{\omega }^{\prime }\right).$

Here, we have used the Fourier integral expression of the delta function

$δ(s)=12π∫− ∞∞dseiωs.$(26)

The scheme presented generalises the one proposed in fluctuating electrodynamics [25]. One has to notice that for large variances, τ − 1, tends to a delta function (see Figure 2). This limit describes the Markovian case in which the FDT is the one given in [25]. In addition, it is important to stress that (25) is completely general and applicable either to near-field or far-field thermal radiation processes. In this sense, our work differs from previous results [20], which only considers a single relaxation time.

## 5 Conclusions

We have presented a model that allows us to describe the dielectric properties of a material when a multiplicity of relaxation time scales is present. This relaxation process cannot be described by a conventional Debye equation due to the appearance of memory effects. The reason is that the relaxation dynamics is non-Markovian. To obtain the memory kernel, we have proposed a model that is non-Markovian and according to this the polarisation of the system satisfies an integrodifferential equation. In the framework of this model, we think that the dynamics of the system arises upon adding collective excitations or vibration modes. Each of these modes relaxes to a stationary state in its own time scale and all these decays occur in cascade. Under the assumption of many events, Gaussianity leads to a hierarchy of relaxation times given by a frequency-dependent log-normal function. This multiplicity of time scales brings in itself a generalisation of the Debye or Drude models containing a single time scale. Consequently, we have obtained a new expression of the dielectric permittivity in terms of the hierarchy of relaxation times. It has recently been show that a hierarchy of relaxation times as the one used in this article describes the dissipation inherent to the presence of Casimir forces [2].

The memory function as well as the inverse of the hierarchy of relaxation times is related to the dissipation and therefore to the entropy production in the system. On the other hand, the dissipation can be understood as the noncompensated heat arising in the decay of the fluctuations and transferred to the isothermal bath. Accordingly, the multiplicity of time scales inherent to the non-Markovian nature of the system must be implicitly contained in the FDT through the memory function or the hierarchy of relaxation times. This fact leads us to a reformulation of an FDT that generalises the one proposed in fluctuating electrodynamics.

## Acknowledgement

This work has been supported by MINECO of the Spanish Government and Consejo Superior de Investigaciones Científicas under Grant No. FIS2015-67837-P, CNPq of the Brazilian Government under Grant No. 476196/2012-4, and PRPPG-UNILA.

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Accepted: 2016-12-16

Published Online: 2017-01-18

Published in Print: 2017-02-01

Funding Source: Consejo Superior de Investigaciones Científicas

Award identifier / Grant number: FIS2015-67837-P

This work has been supported by MINECO of the Spanish Government and Consejo Superior de Investigaciones Científicas under Grant No. FIS2015-67837-P, CNPq of the Brazilian Government under Grant No. 476196/2012-4, and PRPPG-UNILA.

Citation Information: Zeitschrift für Naturforschung A, ISSN (Online) 1865-7109, ISSN (Print) 0932-0784,

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