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]

$$\frac{\partial P}{\partial t}=-{\tau}^{\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}\mathrm{(}P-\kappa E\mathrm{)}\mathrm{,}$$(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

$$\widehat{P}\mathrm{(}\omega \mathrm{)}=\chi \mathrm{(}\omega \mathrm{)}\widehat{E}\mathrm{(}\omega \mathrm{)}\mathrm{,}$$(2)

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

$$\chi \mathrm{(}\omega \mathrm{)}=\frac{\kappa}{1-i\omega \tau}\mathrm{.}$$(3)

The Fourier transform has been defined as

$$\widehat{a}\mathrm{(}\omega \mathrm{)}={\displaystyle {\int}_{-\text{\hspace{0.17em}}\infty}^{\infty}dt\text{\hspace{0.17em}}a\mathrm{(}t\mathrm{)}{e}^{i\omega t}}\mathrm{.}$$(4)

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

$$\epsilon \mathrm{(}\omega \mathrm{)}=1+4\pi \chi \mathrm{(}\omega \mathrm{}\mathrm{)}\mathrm{.}$$(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

$$\frac{ds}{dt}=-\frac{1}{T}\frac{\partial P}{\partial t}\mathrm{(}{E}_{eq}-E\mathrm{)}\mathrm{,}$$(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 (*E*−*E*_{eq})/*T*. The linear coupling of both quantities yields

$$\frac{\partial P}{\partial t}=\frac{L}{T}\mathrm{(}E-{E}_{eq}\mathrm{)}$$(7)

where *L* is an Onsager coefficient. Identifying *τ* with *Tκ*/*L* and using the relation *E*_{eq}=*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]

$$\frac{\partial P}{\partial t}=-{\displaystyle {\int}_{0}^{t}\varphi \mathrm{(}t-{t}^{\prime}\mathrm{)}[P\mathrm{(}{t}^{\prime}\mathrm{)}-\kappa E\mathrm{(}{t}^{\prime}\mathrm{)}]d{t}^{\prime}}$$(8)

where *ϕ(t*−*t*′) 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

$$\chi \mathrm{(}\omega \mathrm{)}=\frac{\kappa}{1-i\omega \tau \mathrm{(}\omega \mathrm{)}}\mathrm{,}$$(9)

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

$${\tau}^{\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}\mathrm{(}\omega \mathrm{)}={\displaystyle {\int}_{-\text{\hspace{0.17em}}\infty}^{\infty}dt\text{\hspace{0.17em}}\varphi \mathrm{(}t\mathrm{)}{e}^{i\omega t}}\mathrm{.}$$(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

$${\tau}^{\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}={\tau}_{1}^{\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}+\dots +{\tau}_{n}^{\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}\mathrm{,}$$(11)

where all the *τ*_{ℓ} 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*_{ℓ} whose effect on the relaxation time at stage *ℓ* is proportional to *ξ*_{ℓ} 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 *ℓ*−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.

$${\tau}_{\ell}^{\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}={\tau}_{\ell \text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}^{\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}+{\xi}_{\ell}{\tau}_{\ell \text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}^{\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}\mathrm{.}$$(12)

From (12), one can obtain

$$\sum _{i\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{n}{\xi}_{i}}={\displaystyle \sum _{i\text{\hspace{0.17em}}=\text{\hspace{0.17em}}1}^{n}\frac{{\tau}_{\ell}^{\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}-{\tau}_{\ell \text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}^{\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}}{{\tau}_{\ell \text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}^{-\text{\hspace{0.17em}}1}}\propto}{\displaystyle {\int}_{{\tau}_{0}^{\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}}^{{\tau}^{\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}}\frac{d\epsilon}{\epsilon}}=\mathrm{ln}\frac{{\tau}^{\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}}{{\tau}_{0}^{\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}}\mathrm{.$$(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,

$${\tau}^{\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}\mathrm{(}\omega \mathrm{)}=\frac{1}{\sqrt{2\pi}\sigma {\tau}_{0}}\mathrm{exp}[-{\mathrm{ln}}^{2}\mathrm{(}\omega \mathrm{/}{\omega}_{0}\mathrm{)}/2{\sigma}^{2}]\mathrm{.}$$(14)

Here, *τ*_{0} is a material-dependent parameter, which can be estimated from thermal diffusivity [13], *ω*_{0}=*k*_{B}T_{0}/*ħ* is the thermal frequency and *σ* the standard deviation that describes the deviations from *ω*_{0}. 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 $\mathrm{ln}\mathrm{(}{\tau}_{\ell}^{\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}\mathrm{/}{\tau}_{0}^{\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}\mathrm{)}\propto \mathrm{ln}\mathrm{(}{\omega}_{\ell}\mathrm{/}{\omega}_{0}\mathrm{}\mathrm{)}\mathrm{.}$ 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 *ω*_{0}=*k*_{B}*T*_{0}/*ħ*, at room temperature, *τ*_{0}=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.

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