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# Journal of Non-Equilibrium Thermodynamics

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Volume 43, Issue 1

# Fractional Effects on the Light Scattering Properties of a Simple Binary Mixture

Rosalío F. Rodríguez
• Corresponding author
• Instituto de Física, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, 01000 CDMX, México
• Fellow of SNI, México. Also at FENOMEC, UNAM, México
• Email
• Other articles by this author:
/ Jorge Fujioka
• Instituto de Física, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, 01000 CDMX, México
• Fellow of SNI, México. Also at FENOMEC, UNAM, México
• Email
• Other articles by this author:
/ Elizabeth Salinas-Rodríguez
• Departamento I. P. H., Universidad Autónoma Metropolitana, Iztapalapa, Apdo. Postal 55-534, 09340 CDMX, México; Fellow of SNI, México
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Published Online: 2017-12-14 | DOI: https://doi.org/10.1515/jnet-2017-0036

## Abstract

A fractional generalized hydrodynamic (GH) model of the concentration fluctuations correlation function is analyzed. Our analysis is based on a previously proposed irreversible thermodynamics non-fractional model that describes the first GH deviations in a finite frequency and wavelength regime where the local equilibrium assumption is not valid. Our basic purpose is to generalize this theory to investigate the time fractional effects on the intermediate scattering function (ISF) of a model of a binary mixture. We discuss two different forms of introducing the fractional derivatives into the hydrodynamic time evolution equation of the ISF and examine their consistency in the non-fractional limit. We calculate analytically the fractional intermediate scattering function (FISF) and compare it with the non-fractional one. We find that although the FISFs are in general less than ISF, the FISF may be a significant fraction of the ISF (∼36%) and might be measurable by light or neutron scattering techniques. We show that fractional time derivatives provide a consistent description of this correlation function in the GH regime and reduce to its well-known behavior in the Navier–Stokes limit (NS) where local equilibrium is restored. We also suggest an experimental verification of our results for near equilibrium states. Finally, we summarize the main results of our work and make some further physical remarks.

## 1 Introduction

A successful account of many irreversible processes has been provided over the years by linear (classical) irreversible thermodynamics (LIT), a theory built upon the local equilibrium hypothesis and restricted to the linear regime [1, 2, 3, 4, 5, 6]. However, to describe more general processes where the local equilibrium assumption if no longer valid, a generalization of LIT is called for. This situation arises in the generalized hydrodynamic (GH) regime of intermediate wave vectors $\stackrel{⃗}{k}$ and frequencies ω, where local equilibrium breaks down [7]. These states may arise for systems like fractals or self-organizing structures where the microscopic conditions for the molecular chaos assumption may not be met and, therefore, the local equilibrium hypothesis is no longer valid [8, 9].

Among the several phenomenological approaches used for this purpose is extended irreversible thermodynamics (EIT). The central idea in EIT is to enlarge the space of state variables by assuming that the entropy depends not only on the conserved hydrodynamic densities but also on the dissipative fluxes or non-conserved variables as well. In this process the assumption of local equilibrium breaks down, but when the subspace spanned by the non-conserved variables reduces to that of the conserved ones, the local equilibrium assumption is restored and EIT reduces to the well-known LIT [10, 11, 12, 13]. In this way EIT provides for relaxation type equations which play the role of constitutive equations for the fast variables of the system in terms of the slow ones, and associates to each of them a specific time scale. In this way a set of characteristic relaxation times is established for nonequilibrium states beyond local equilibrium.

Distinctive features of the nature of the spectrum of characteristic relaxation times of nonequilibrium states may be such that a well-defined separation between the time scales associated with the macroscopic transport processes and the microscopic ones giving rise to them, no longer exist. Under these conditions the fluctuations of the state variables may exhibit strange dynamics where there is no preferred time scale and the central limit theorem (CLT) may not be applicable [14, 15, 16, 17]. Then the usual smoothing process of statistical mechanics is no longer feasible to describe the dynamics of fluctuations and the usual formalism based on ordinary calculus would no longer be adequate to describe the transport processes. One way to consider these effects of non-localizability in time (memory) and non-localizability in spatial coordinates (spatial correlations) is to develop new methodology based on fractional calculus. This leads to a slow relaxation of the correlation functions associated with many transport properties, such as the concentration fluctuations correlation function producing the scattered spectrum of light from complex fluids [18, 19]. In this sense, the violation of the principle of local equilibrium makes the traditional methods in transport theory to become unsuitable. Examples of macroscopic systems which may exhibit these features are the large variety of relaxation processes that occur in viscoelastic fluids, glassy materials, synthetic polymers or biopolymers, all of which have in common that their relaxation functions are non-exponential, due to the large number of highly coupled elementary units responsible for the relaxation. It is this requirement of high cooperation between these elements that calls for a fractional description [20, 21].

For these reasons, it appears worthwhile to investigate these issues on a simple, but not unrealistic model, which lends itself to explicit analytic mathematical manipulation. In previous work we have developed such a model using EIT [12]. In [12] we discuss how the behavior of the concentration correlation function is modified for nonequilibrium states in the GH regime, where additional time scales influence the dynamics of state variables fluctuations. In the present work, we show that the inclusion of fractional time derivatives in the transport equations provides a consistent description to include these scales and examine their effect on time correlation functions. For this purpose, we propose two different novel methods to introduce fractional time derivatives into the macroscopic equations for the fluctuations of a simple model for an isothermal-isobaric, heat insulating, inviscid and chemically inert binary mixture. Although at first sight this model system might seem to be somewhat restrictive, mixtures of this sort arise in real permeation systems where, for instance, an inert binary mixture of He-N₂ is produced in different regions of a tubular separator involving porous membranes used for gas–gas separation processes [22, 23] and may comply with the requirements of the present model. This type of mixtures also appears in other systems of engineering and physico-chemical interest like the steady-state mass transfer of non-reactive species [24].

More specifically, we calculate analytically the fractional intermediate scattering function FISF in the GH regime and show how when the fast variables relax and local equilibrium is restored, this correlation function recovers its NS behavior. We find that although the FISFs are in general less than the ISF, the FISF may be a significant fraction (∼36 %) of ISF and might be measurable by light or neutron scattering techniques. This model calculations show that fractional time derivatives provide a consistent description of this correlation function in the GH regime and reduces to its well-known behavior in the NS limit. Although we are not aware of the existence of light or neutron scattering experimental results for these systems, we suggest light scattering experiments which could shed some light on the confrontation of our theoretical model predictions and experimental measurements. This even partial verification could hopefully serve to test the nature of these processes in a fractional and non-classical hydrodynamic regime [7].

## 2 Model

Since the formal theory behind our hydrodynamic model has been developed in our previous work [12] using EIT, in this section we shall only briefly review the underlying theoretical ideas which lead to the final dynamic equations that we shall use later. The state variables for the binary mixture were chosen as one conserved variable, the relative concentration of one of the components $c\left(\stackrel{⃗}{r},\tau \right)$, and two non-conserved variables, the local mass flux $\stackrel{⃗}{J}\left(\stackrel{⃗}{r},\tau \right)$ and a second-order tensor $\frac{↔}{\mathrm{\Im }\left(\stackrel{⃗}{r},\tau \right)}$, which accounts for the spatial inhomogeneities in the space of state variables. The mass conservation equation reads $\frac{dc}{d\tau }=-\mathrm{\nabla }\bullet \stackrel{⃗}{J}\left(\stackrel{⃗}{r},\tau \right)$(1)

where d/ denotes the hydrodynamic derivative; the equation of motion is given by $\rho \frac{d\stackrel{⃗}{u}}{d\tau }=-\mathrm{\nabla }p$(2)

where ρ is the local mass density, p is the pressure and $\stackrel{⃗}{u}$ is the macroscopic flow velocity. If as explained in [12], we apply the usual procedure of EIT, restricting the calculations to lowest order in the non-conserved variables, we end up with the following coupled set of relaxation equations for the non-conserved variables ($\stackrel{⃗}{J}$,$\stackrel{↔}{\mathrm{\Im }}$) (see eqs. (2.8)–(2.9) in [12])

$-{\tau }_{\stackrel{⃗}{J}}\frac{d\stackrel{⃗}{J}}{d\tau }=\stackrel{⃗}{J}+\frac{1}{{\mu }_{10}}\mathrm{\nabla }\left(\frac{\mu }{T}\right)-{l}_{2}\mathrm{\nabla }\bullet \stackrel{↔}{\mathrm{\Im }},$(3)$-{\tau }_{\stackrel{↔}{\mathrm{\Im }}}\frac{d\stackrel{↔}{\mathrm{\Im }}}{d\tau }=\stackrel{↔}{\mathrm{\Im }}-{l}_{1}\mathrm{\nabla }\stackrel{⃗}{J},\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\text{\hspace{0.17em}}\phantom{\rule{thickmathspace}{0ex}}$(4)

where μ10, l1 and l2 are phenomenological coefficients. The relaxation times ${\tau }_{\stackrel{⃗}{J}}$ and ${\tau }_{\stackrel{↔}{\mathrm{\Im }}}$ of $\stackrel{⃗}{J}$ and $\stackrel{↔}{\mathrm{\Im }}$ are defined explicitly in [12] in terms of phenomenological coefficients (see eqs. (2.8)–(2.11) in [12]) which can only be determined either from experiment or from a microscopic theory. On intuitive grounds one would be tempted to expect that $\stackrel{↔}{\mathrm{\Im }}$ will vary on a shorter time scale than that of $\stackrel{⃗}{J}$, so we restrict our description to a time scale in which $\stackrel{↔}{\mathrm{\Im }}$ has already relaxed and ${\tau }_{\stackrel{↔}{\mathrm{\Im }}}$ is negligible so that eq. (3) becomes (see eq. (2.14) in [12]) $-{\tau }_{\stackrel{⃗}{J}}\frac{d\stackrel{⃗}{J}}{d\tau }=\stackrel{⃗}{J}+D\mathrm{\nabla }c-{l}_{1}{l}_{2}{\mathrm{\nabla }}^{2}\stackrel{⃗}{J}.$(5)

Since in most experimental situations involving mass diffusion the pressure is kept constant, D denotes the isothermal-isobaric diffusion coefficient $D=\left(T{\mu }_{10}{\right)}^{-1}\left(\mathrm{\partial }\mu /\mathrm{\partial }c{\right)}_{T,p}$, where μ is the chemical potential. Note that although this relaxing equation (generalized constitutive equation) is valid in the GH limit where local equilibrium is lost, in the NS regime with ${\tau }_{\stackrel{⃗}{J}}={l}_{1}{l}_{2}=0$, it reduces to Fick’s law restoring the validity of local equilibrium.

Equations (1)–(5) constitute our basic closed set of hydrodynamic equations and to derive transport properties from them, they should be linearized in the small deviations with respect to an initial reference state identified with the subscript 0, $\delta c\left(\stackrel{⃗}{r},t\right)=c\left(\stackrel{⃗}{r},t\right)-{c}_{0}\left(\stackrel{⃗}{r}\right)$, $\delta \stackrel{⃗}{J}\left(\stackrel{⃗}{r},t\right)=\stackrel{⃗}{J}\left(\stackrel{⃗}{r},t\right)-{\stackrel{⃗}{J}}_{0}\left(\stackrel{⃗}{r}\right)$. Since these deviations are random quantities, they will manifest themselves as the hydrodynamic fluctuations. The relevant correlation function to calculate the light scattering properties of the mixture is the equilibrium correlation function of concentration fluctuations [25]

$I\left(\stackrel{⃗}{r},\tau ;{\stackrel{⃗}{r}}^{{}^{\prime }},{\tau }^{{}^{\prime }}\right)=〈\delta c\left(\stackrel{⃗}{r},\tau \right)\delta c\left({\stackrel{⃗}{r}}^{{}^{\prime }},{\tau }^{{}^{\prime }}\right)〉;,$(6)

where the angular brackets denote an average over an equilibrium ensemble. The intensity distribution of the (isotropic) component of the Rayleigh–Brillouin scattering spectrum of a molecular liquid, i. e., the intermediate scattering function $I\left(\stackrel{⃗}{k},\tau \right)$ (ISF) or van Hove’s self-correlation function [7], is proportional to the spatial Fourier transform of $\delta c\left(\stackrel{⃗}{r},\tau \right)$ within the scattering volume $I\left(\stackrel{⃗}{k},\tau \right)=〈\delta c\ast \left(\stackrel{⃗}{k},\tau \right)\delta c\left(\stackrel{⃗}{k},0\right)〉;,$(7)

where the asterisk (*) indicates complex conjugate. Substitution of eq. (1) into eq. (5) yields the time evolution equation for $I\left(\stackrel{⃗}{k},\tau \right)$

$\frac{{d}^{2}I}{d{\tau }^{2}}+{A}_{1}\left(\stackrel{⃗}{k}\right)\frac{dI}{d\tau }+{A}_{2}\left(\stackrel{⃗}{k}\right)I=0,$(8)

where ${A}_{1}\left(\stackrel{⃗}{k}\right)\equiv \left(1+{l}_{1}{l}_{2}{k}^{2}\right)/{\tau }_{\stackrel{⃗}{J}}$ and ${A}_{2}\left(\stackrel{⃗}{k}\right)\equiv D{k}^{2}/{\tau }_{\stackrel{⃗}{J}}$ are two free parameters which are combinations of phenomenological coefficients ${\tau }_{\stackrel{⃗}{J}}$, l1, l2. For given $I\left(\stackrel{⃗}{k},0\right)$ with vanishing time derivative ${I}^{{}^{\prime }}\left(\stackrel{⃗}{k},0\right)=0$ its solution reads $\frac{I\left(\stackrel{\to }{k},\tau \right)}{I\left(\stackrel{\to }{k},0\right)}=\mathrm{exp}\left(-\frac{1}{2}{A}_{1}\tau \right)\left(\mathrm{cos}R\tau +\frac{{A}_{1}}{2R}\mathrm{sin}R\tau \right)$(9)

with $R\left(k\right)=\left[{A}_{2}\left(k\right)-{A}_{1}^{2}\left(k\right)/4{\right]}^{1/2}>0$, which imposes the condition ${A}_{2}\left(k\right)>{A}_{1}^{2}\left(k\right)/4$. It must be emphasized once again that the inclusion of spatial inhomogeneities through the non-conserved variable $\stackrel{↔}{\mathrm{\Im }}$ directly took us out of the NS regime and drove us into the domain of GH. However, the local equilibrium regime is easily regained by taking the limit ${\tau }_{\stackrel{↔}{\mathrm{\Im }}}\to 0$, ${l}_{1}{l}_{2}\to 0$, in eq. (9) which yields the classical exponential decay of $I\left(\stackrel{⃗}{k},\tau \right)$

$I\left(\stackrel{\to }{k},\tau \right)=I\left(\stackrel{\to }{k},0\right)\mathrm{exp}\left(-\frac{1}{2}{A}_{1}\tau \right).$(10)

Furthermore, $I\left(\stackrel{⃗}{k},\tau \right)$must satisfy certain restrictions, the so-called sum rules [7], which yield a relation between the static structure factor $S\left(\stackrel{⃗}{k}\right)$ and ${A}_{2}\left(\stackrel{⃗}{k}\right)$, namely (see eq. (3.11) in [12]) $S\left(\stackrel{⃗}{k}\right)=\frac{{k}_{B}T}{{A}_{2}\left(\stackrel{⃗}{k}\right)}{k}^{2}=\frac{{k}_{B}T}{D}{\tau }_{\stackrel{⃗}{J}}.$(11)

Since $S\left(\stackrel{⃗}{k}\right)$ may be determined from a microscopic theory in terms of the measurable quantities T and D at least in principle, this equation allows us to determine ${\tau }_{\stackrel{⃗}{J}}$. This would leave us with one free parameter ${A}_{1}\left(\stackrel{⃗}{k}\right),$ or equivalently l1l2. Usually, phenomenological approaches like EIT introduce a wealth of phenomenological coefficients in the equations of motion or the transport equations, which can only be determined either experimentally or from microscopic theories [10]. However, in the present model neither the values of A1 or A2 are known for the binary mixture, but since the condition ${A}_{2}\left(k\right)>{A}_{1}^{2}\left(k\right)/4$ must be fulfilled, it is convenient to reduce the number of phenomenological coefficients in eq. (8) by rescaling the equation through the following change of variables $I\equiv a{J}$, $\tau \equiv bt$. These yields $\frac{a}{{b}^{2}{A}_{2}}{{J}}_{tt}+\frac{a{A}_{1}}{b{A}_{2}}{{J}}_{t}+a{J}=0$(12)

where ${{J}}_{t}\equiv \mathrm{\partial }{J}\left(\stackrel{⃗}{k},t\right)/\mathrm{\partial }t$, ${{J}}_{tt}\equiv {\mathrm{\partial }}^{2}{J}\left(\stackrel{⃗}{k},t\right)/\mathrm{\partial }{t}^{2}$. If we now choose a and b in such a way that the coefficients of both derivatives equal unity, then $a={A}_{2}/{A}_{1}^{2},b={A}_{1}^{-1}$ and eq. (12) is rewritten in the rescaled simpler form ${{J}}_{tt}+{{J}}_{t}+aJ=0$(13)

with only one phenomenological coefficient a. For given ${J}\left(\stackrel{⃗}{k},0\right)$, ${{J}}_{t}\left(\stackrel{⃗}{k},0\right)=0$ its solution becomes $\frac{{J}\left(\stackrel{⃗}{k},t\right)}{{J}\left(\stackrel{⃗}{k},0\right)}={e}^{-t/2}\left(cos\text{\hspace{0.17em}}Mt+\frac{1}{2M}sin\text{\hspace{0.17em}}Mt\right),$(14)

where $M={\left(a-1/4\right)}^{1/2}$. By taking the Laplace transform of ${J}\left(\stackrel{⃗}{k},t\right)$ with respect to time and setting $s=i\omega$ gives the structure factor

$S\left(\stackrel{⃗}{k},\omega \right)=S\left(\stackrel{⃗}{k}\right)\frac{{A}_{1}\left(\stackrel{⃗}{k}\right){A}_{2}\left(\stackrel{⃗}{k}\right)}{\left({\omega }^{2}+{B}_{1}\right)\left({\omega }^{2}+{B}_{2}\right)},$(15)

where we have defined $S\left(\stackrel{⃗}{k}\right)={J}\left(\stackrel{⃗}{k},0\right)$ and used the abbreviations

$2{B}_{1,2}=\left({A}_{1}^{2}-2{A}_{2}\right)\mp {\left({A}_{1}^{4}-4{A}_{1}^{2}{A}_{2}\right)}^{1/2}.$(16)

## 3 Time fractional derivatives

Physical models where first-order conventional time derivatives have been formally replaced by fractional differential operators have been proposed for a variety of physical systems [17, 26]. It should be remarked, however, that there are several different, but not necessarily equivalent, ways of defining the time fractional derivative (TFD) of a function $f\left(t\right)$. Among these, the definitions of Riemann–Liouville [27], Grünwald–Letnikov [28], Caputo [29, 30], Hadamard [31], Erdelyi [32] or Ortigueira [33] can be found in the literature; however, despite their differences, all the TFDs share the common feature of being non-local operators. In other words, the TFD of $f\left(t\right)$ at a point to not only depends on the behavior of $f\left(t\right)$ in the neighborhood of t0, but also on the values of $f\left(t\right)$ over finite intervals of the form t1tt0 or t0tt2. Therefore, each of the above possible definitions of a FTD (except for Ortigueira’s centered derivative) provides an expression for the left-hand sided derivative, and a different expression for the right-handed derivative.

The selection of the appropriate FTD to be used in our analysis is a delicate matter which should be performed by considering criteria consistent with the physical situation at hand; we have used the following criteria: First, the FTDs should be left-handed because they can only be calculated in terms of their past values. Secondly, the solution of a fractional differential equation requires to know appropriate initial conditions to obtain a particular solution of the fractional version of eq. (13). We choose the left-handed Caputo time fractional derivative (LHCD) because then this equation can be solved by Laplace transformation, which incorporates the natural initial conditions, i. e., the integer order derivatives at t=0. Since these are the initial conditions that can be specified or controlled in the relaxation processes described in the present model, it is consistent to use a LHCD in eq. (13). In contrast, the Laplace transform of the Riemann–Liouville derivative demands to know fractional derivatives at t=0, which are unknown for the usual type of relaxation processes toward equilibrium in dissipative systems.

For an arbitrary function $f\left(\stackrel{⃗}{k},t\right)$ the LHCD is defined by [28, 29], ${}_{0}^{C}{D}_{t}^{\alpha }f\left(\stackrel{⃗}{k},t\right)=\frac{1}{\mathrm{\Gamma }\left(n-\alpha \right)}\underset{0}{\overset{t}{\int }}\frac{{f}^{\left(n\right)}\left(\stackrel{⃗}{k},\tau \right)}{{\left(t-\tau \right)}^{\alpha +1-n}}\text{\hspace{0.17em}}d\tau$(17)

Here ${f}^{\left(n\right)}\left(\stackrel{⃗}{k},t\right)\equiv {d}^{n}f/d{t}^{n}$ is the conventional n order time-derivative, being n the smallest integer greater than α, i. e., n-1 ≤ α<n, and where $\mathrm{\Gamma }\left(n-\alpha \right)$ is the Euler gamma function. An intuitive picture of the action of ${}_{0}^{C}{D}_{t}^{\alpha }f\left(\stackrel{⃗}{k},t\right)$ implies that the LHCD considers the past behavior of ${f}^{\left(n\right)}$ by adding to its initial values, the successive weighted increments over time. These increments per unit time are represented by ${f}^{\left(n\right)}\left(\stackrel{⃗}{k},t\right)$ in eq. (17), while the weights are indicated by the factors $1/\left(t-\tau {\right)}^{\alpha +1-n}$, whose value decreases with increasing time separation from time t.

To introduce TFD in eq. (8) we consider two different approaches in the two following subsections. The first one (Case A) is motivated by the description of the dispersion of ultra-short pulses in fiber optics and consists in replacing the first and second derivatives which appear in eq. (13) by a fractional derivative of order 1<α<2 [34]. In our second approach (Case B) instead of the rather abrupt replacement of the first and second integer derivatives by a unique fractional derivative, we shall gradually approximate both integer derivatives toward an intermediate value. That is, we shall replace the first (integer) order derivative by a fractional derivative of order $1+\epsilon$ and the second one by a fractional derivative of order $2-\epsilon$. Let us now examine in detail these two generalizations of eq. (8).

## 3.1 Case A: One centered fractional derivative

According to the first approach, we replace the two integer derivatives in eq. (13) by a single LHCD of order ε, 1<ε<2. Since in our numerical tests we shall use ε=1.5, eq. (13) becomes ${D}^{1.5}{{J}}^{FA}+a{{J}}^{FA}=0,$(18)

whose solution is

$\frac{{{J}}^{FA}\left(\stackrel{⃗}{k},t\right)}{{{J}}_{0}^{FA}}={E}_{1.5}\left(-a{t}^{1.5}\right),$(19)

where ${E}_{1.5}$ is the one-parameter Mittag–Leffler function. To simplify the notation here and in what follows we shall not write explicitly the vector character of $\stackrel{⃗}{k}$.

## 3.2 Case B: Two fractional derivatives

In our second approach, we replace the first and second derivatives by fractional derivatives of order $1+\epsilon$ and $2-\epsilon$, respectively, thus obtaining an equation of the form ${D}^{2-\epsilon }{{J}}^{F}+{D}^{1+\epsilon }{{J}}^{F}+a{{J}}^{F}=0.$(20)

It should be noted that ε could be positive or negative. However, we will only consider the case ε>0 so that eq. (20) approaches eq. (18) when $\epsilon \to 0.5$. In this equation the parameter ε only affects the first two terms of the equation, but has no effect whatsoever on the third term. This feature is unsatisfactory, but it can be improved by first introducing the fractional derivatives into eq. (8) and afterwards performing the scaling. Following this procedure, we obtain a fractional equation of the form ${c}_{2}\left(\epsilon ,k\right){D}^{2-\epsilon }{{J}}^{FB}+{c}_{1}\left(\epsilon ,k\right){D}^{1+\epsilon }{{J}}^{FB}+{A}_{2}\left(k\right){{J}}^{FB}=0$(21)

where ${c}_{1}\left(\epsilon ,k\right)={A}_{1}\left(k\right)\left(1-\epsilon \right)+\epsilon ,$(22) ${c}_{2}\left(\epsilon ,k\right)=1-\epsilon +\epsilon {A}_{1}\left(k\right).\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}$(23)

With this choice of c1 and c2, when ε=1/2 the two fractional derivatives in eq. (21) coalesce into a single derivative of order 1.5, as in eq. (18). If now we perform the change of variable ${J}^{FB}=\stackrel{ˆ}{a}{{J}}^{FB}$ in eq. (21) and if we also make the change of variables $T=\stackrel{ˆ}{b}t$ in eq. (13), we arrive at ${}_{0}^{C}{D}_{t}^{\alpha }{J}^{FB}\left(k,T\right)=\frac{1}{\mathrm{\Gamma }\left(n-\alpha \right)}\underset{0}{\overset{t}{\int }}{\stackrel{ˆ}{b}}^{n-\alpha -1}{\left(t-Y\right)}^{n-\alpha -1}\frac{1}{{\stackrel{ˆ}{b}}^{n}}\frac{{d}^{n}{J}^{FB}}{d{Y}^{n}}\text{\hspace{0.17em}}\stackrel{ˆ}{b}dY,$(24)

where $n-1\le \alpha . If we now set $y=\stackrel{ˆ}{b}dY$, eq. (24) takes de form ${}_{0}^{C}{D}_{t}^{\alpha }{J}^{FB}\left(k,T\right)=\frac{{\stackrel{ˆ}{b}}^{-\alpha }}{\mathrm{\Gamma }\left(n-\alpha \right)}\underset{0}{\overset{t}{\int }}{\left(t-Y\right)}^{n-\alpha -1}\frac{{d}^{n}{J}^{FB}}{d{Y}^{n}}dY=\frac{1}{{\stackrel{ˆ}{b}}^{\alpha }}{}_{0}^{C}{D}_{t}^{\alpha }{J}^{FB}\left(k,T\right).$(25)

On the other hand, by setting ${{J}}^{F}=\stackrel{ˆ}{a}{J}^{FB}$ in eq. (21) and then by introducing the change $T=\stackrel{ˆ}{b}t$ in the resulting equation, we obtain

$\frac{\stackrel{ˆ}{a}}{{A}_{2}{\stackrel{ˆ}{b}}^{2-\epsilon }}{D}_{t}^{2-\epsilon }{J}^{FB}\left(k,t\right)+\frac{\stackrel{ˆ}{a}{c}_{1}}{{A}_{2}{\stackrel{ˆ}{b}}^{1+\epsilon }}{D}_{t}^{1+\epsilon }{J}^{FB}\left(k,t\right)+\stackrel{ˆ}{a}{J}^{FB}\left(k,t\right)=0.$(26)

To determine the values of $\stackrel{ˆ}{a}$ and $\stackrel{ˆ}{b}$ we require that the coefficients of the two derivatives in this equation be equal to unity with the result

${\stackrel{ˆ}{b}}^{1-2\epsilon }=\frac{{c}_{2}}{{c}_{1}}$(27)

and

$\stackrel{ˆ}{a}=\frac{{A}_{2}}{{c}_{1}}{\stackrel{ˆ}{b}}^{1+\epsilon }=\frac{{A}_{2}}{{c}_{1}}{\left(\frac{{c}_{2}}{{c}_{1}}\right)}^{\frac{1+\epsilon }{1-2\epsilon }}.$(28)

Finally, substituting eqs. (22) and (23) in these equations yields ${}_{0}^{}{D}_{t}^{2-\epsilon }{J}^{FB}\left(t\right)+{}_{0}^{}{D}_{t}^{1+\epsilon }{J}^{FB}\left(t\right)+\stackrel{ˆ}{a}{J}^{FB}\left(t\right)=0,$(29)

where now $\stackrel{ˆ}{a}$ depends on A1, A2 and ε,

$\stackrel{ˆ}{a}={A}_{2}\frac{{\left[\epsilon \left({A}_{1}-1\right)+1\right]}^{\frac{1+\epsilon }{1-2\epsilon }}}{{\left[{A}_{1}\left(1-\epsilon \right)+\epsilon \right]}^{\frac{2-\epsilon }{1-2\epsilon }}},$(30)

instead of a=A2/A12 as used before regarding eq. (12).

The solution of eq. (29) for 0<ε<0.5 is obtained as follows. We first take the Laplace transform of eq. (29) with respect to t, recalling that the Laplace transform of the Caputo’s derivative is given by ${L}\left[{}_{0}^{}{D}_{t}^{\alpha }f\right]=\frac{1}{{s}^{\delta }}{L}\left[{f}^{\left(n\right)}\right],$(31)

where $n-1\le \alpha and $\delta =n-\alpha$. In this way, we arrive at the Laplace transform of the solution of eq. (29), $\frac{{\stackrel{ˆ}{J}}^{FB}\left(k,s\right)}{{J}_{0}^{FB}}=\stackrel{ˆ}{U}\left(s\right)+\stackrel{ˆ}{V}\left(s\right),$(32)

where

$\stackrel{ˆ}{U}\left(s\right)=\sum _{m}\sum _{n}\frac{{\left(-1\right)}^{m+n}\left(m+n\right)!}{m!n!}\frac{{\stackrel{ˆ}{b}}^{n}}{{\stackrel{ˆ}{a}}^{m+n+1}}\frac{1}{{s}^{2+m+2n-\left(2m+n+2\right)\epsilon }}$(33)

and

$\stackrel{ˆ}{V}\left(s\right)=\sum _{m}\sum _{n}\frac{{\left(-1\right)}^{m+n}\left(m+n\right)!}{m!n!}\frac{{A}_{1}^{m}{A}_{2}^{n}}{{\left(1-\epsilon \right)}^{n}{s}^{1+m+2n-\left(2m+n\right)\epsilon }}.$(34)

By inserting these expressions into eq. (32) and by taking the inverse Laplace transform of the resulting equation we obtain $\frac{{\stackrel{ˆ}{J}}^{FB}\left(k,t\right)}{{J}_{0}^{FB}}={t}^{1-2\epsilon }\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(-{t}^{1-2\epsilon }\right)}^{m}}{m!}{E}_{2-\epsilon ,2-m-\left(m+2\right)}^{\left(m\right)}\left(-\stackrel{ˆ}{a}{t}^{2-\epsilon }\right)+\sum _{m=0}^{\mathrm{\infty }}\frac{{\left(-{t}^{1-2\epsilon }\right)}^{m}}{m!}{E}_{2-\epsilon ,1-m\left(1+\epsilon \right)}^{\left(m\right)}\left(-\frac{\stackrel{ˆ}{a}}{1-\epsilon }{t}^{2-\epsilon }\right),$(35)

Here ${E}_{\alpha ,\beta }^{\left(m\right)}\left(z\right)$ denote the derivatives of the generalized Mittag–Leffler function (also known as Erdely’s function) which for negative and large values of z have the following asymptotic behavior [28] ${E}_{\alpha ,\beta }^{\left(m\right)}\left(z\right)\sim \sum _{m=0}^{\mathrm{\infty }}\frac{{\left(-1\right)}^{m+1}m!}{{z}^{m+1}\mathrm{\Gamma }\left(\alpha -\beta \right)}.$(36)

For future use we use this expression to obtain the following asymptotic (long-time) behavior of ${\stackrel{ˆ}{J}}^{FB}\left(k,t\right)/{J}_{0}^{FB}$ [19], $\frac{{\stackrel{ˆ}{J}}^{FB}\left(k,t\right)}{{J}_{0}^{FB}}=\sum _{m=0}^{\mathrm{\infty }}\frac{1}{{t}^{m\left(2-\epsilon -2\alpha \right)}}\left(\frac{1}{{\mathrm{\Gamma }}_{1}{t}^{2-\epsilon }}+\frac{1}{{\mathrm{\Gamma }}_{2}}\right),$(37)

where we have defined ${\mathrm{\Gamma }}_{1}=\mathrm{\Gamma }\left\{\left(2-\epsilon \right)m+\epsilon -1/2\right\},$(38) ${\mathrm{\Gamma }}_{2}=\mathrm{\Gamma }\left\{\left(2-\epsilon \right)m+1\right\}.\phantom{\rule{2em}{0ex}}\text{\hspace{0.17em}}$(39)

Taking the leading term of the sum in eq. (37) yields the power law behavior

$\frac{{\stackrel{ˆ}{J}}^{FB}\left(k,t\right)}{{J}_{0}^{FB}}\sim \frac{{A}_{2}}{\mathrm{\Gamma }\left(\epsilon +1\right){t}^{2-\epsilon }}.$(40)

## 4 Results

For the purpose of having a reference case to compare with, let us first examine the behavior of the non-fractional normalized correlation function ${M}\left(k,t;{A}_{2}\right)\equiv {J}\left(k,t\right)/{J}\left(k,0\right)$ defined by eq. (14) as a function of the dimensionless time t, for different values of ${A}_{2}\left(k\right)\equiv D{k}^{2}/{\tau }_{\stackrel{⃗}{J}}$ and setting A1=1. Figure 1 shows its damped oscillating behavior. For all the considered values A2=5, 10, 20, the correlation ${M}\left(t;{A}_{2}\right)$ shows the dominant oscillatory damped exponential decay leading to the exponential decay behavior of the NS regime given by eq. (10). Small values of A2 correspond to large relaxation time ${\tau }_{\stackrel{⃗}{J}}$, indicating that the fast variable $\stackrel{⃗}{J}$ has not yet relaxed and prevents the validity of local equilibrium. Thus, A2 controls the degree of departure of the state from the NS regime toward the GH regime. However, as A2 decreases the amplitudes of the GH correlations also decrease, but their range (wavelength) increases.

Figure 1:

Plot of the non-fractional correlation ${M}\left(t;{A}_{2}\right)={J}\left(t;{A}_{2}\right)/{J}\left(0;{A}_{2}\right)$, as given by eq. (17), for A1=1 and A2=5 (_ _), 10( ….), 20 (____).

Fractional effects are considered in Figure 2 which shows the time behavior of the fractional correlation ${N}\left(t;{A}_{2}\right)\equiv {{J}}^{FA}\left(t;{A}_{2}\right)/{{J}}^{FA}\left(0;{A}_{2}\right)$ for A2=1, 5, 10, 20, as calculated from eq. (19) for Case A. The curves do not oscillate in the same time interval considered in Figure 1, and their ranges and amplitudes in the GH regime (A2=1) increase with respect to the non-fractional correlation in Figure 1. The amplitudes become smaller as the NS limit is approached (A2=10, 20).

Figure 2:

Plot of the fractional correlation ${N}\left(t;{A}_{2}\right)\equiv {{J}}^{FA}\left(t;{A}_{2}\right)/{{J}}^{FA}\left(0;{A}_{2}\right)$ as defined by eq. (18) for A1=1 and A2=1 (-.-.-), 5 (_ _), 10 (…), 20 (____). In this case the curves do not oscillate and they vanish as t grows.

A more quantitative comparison between both behaviors is more clearly seen in Figure 3 where both, the non-fractional ${M}\left(t;{A}_{2}\right)$ and the fractional ${N}\left(t;{A}_{2}\right)$ correlation functions, are plotted vs. t for A2=1, 5, 10, 50. In the NS regime (A2=50) the fractional effects are almost negligible, however, as A2 decreases (${\tau }_{\stackrel{⃗}{J}}$ increases) and moves toward the GH limit, the amplitude of ${N}\left(t;{A}_{2}\right)$ increases. These variations can be quantified; for instance, if A2=1, in the interval 0.2<t<1.78 the amplitude of ${N}\left(t;{A}_{2}\right)$ varies between 0.69 and 0.25, which amounts to a maximum per cent difference (ΔC)${M}{N}$=[(${M}$(t;A2)-${N}$(t;A2))/${M}$(t;A2)]×100=29.6 %.

Figure 3:

The fractional correlation function ${N}\left(t;{A}_{2}\right)\equiv {{J}}^{FA}\left(t;{A}_{2}\right)/{{J}}^{FA}\left(0;{A}_{2}\right)$ plotted vs. t for A2=1, 5, 10, 50. The solid black line is the non-fractional correlation ${M}\left(t;{A}_{2}\right)$ and it is plotted as a reference.

An important feature of these behaviors should be pointed out. Notice that since EIT gradually departs from the NS limit, it is to be expected that the behavior of ${N}\left(t;{A}_{2}\right)$ should reproduce some of the oscillations observed in the non-fractional solution ${M}\left(t;{A}_{2}\right)$. However, this is not observed in Figure 3. This suggests that the fractional description of Case A might not be a sufficiently good fractional extension of the standard non-fractional model. This was the motivation to introduce the second fractional model (Case B) described in subsection 3.2. Indeed, if the method of Case B is used, Figure 4 shows the time behavior of the fractional correlation function ${P}\left(t;{A}_{2}\right)\equiv {J}^{FB}\left(t;{A}_{2}\right)/{J}^{FB}\left(0;{A}_{2}\right)$, as calculated from eq. (35). Notice that the most striking feature is that now it reproduces part of the oscillatory behavior of the non-fractional case for A2=10, thus correcting the inability of the first fractional model (Case A). Moreover, for A2=20 both, the fractional and the non-fractional correlations are very similar, in agreement with the analytic result given by eq. (35), since it can be proven that this equation reduces to the non-fractional solution given by eq. (14) when ε=0. Although for A2=10 this similarity is less satisfactory, still exist.

Figure 4:

The fractional intermediate scattering function ${P}\left(t;{A}_{2}\right)\equiv {J}^{FB}\left(t;{A}_{2}\right)/{J}^{FB}\left(0;{A}_{2}\right)$ for case B is plotted as a function of t and for A2=5, 10 and for ε=0.1. For reference the non-fractional (black solid) curve ${M}\left(t;{A}_{2}\right)$ for A2=20 is also shown.

The corresponding per cent differences (ΔC)${M}{P}$=[(${M}$(t;A2)-${P}$(t;A2))/${M}$(t;A2)]×100 can be quantified from these curves. For instance, for A2=10, t=1.8 and ε=0.4 we find that (ΔC)${}_{{M}}{P}$=−21.43 %, which means that the magnitude of the fractional correlation ${P}$(1.8;10) is greater than the non-fractional ${M}$ (1.8;10) by a factor of ∼22 %. Thus, the fractional effect is not small. Likewise, for A2=5, t=2.8 and ε=0.4, this difference is even larger and amounts to ∼36 %.

At this point it is convenient to explain how the infinite series appearing in eq. (35) were approximated to calculate the fractional solutions shown in Figure 4. When we take eight terms in each of the infinite series in eq. (35), we obtain the dashed curve shown in Figure 5 for A2=20 and ε=0.01.

Figure 5:

Plot of ${M}\left(t;{A}_{2}\right)={J}\left(t;{A}_{2}\right)/{J}\left(0;{A}_{2}\right)$ and ${P}\left(t;{A}_{2}\right)\equiv {J}^{FB}\left(t;{A}_{2}\right)/{J}^{FB}\left(0;{A}_{2}\right)$ for A2=20 and ε=0.01. ${P}$8 results from the first eight terms of the series in eq. (33), whereas ${P}$10 arises from the first ten terms.

In this figure, we can observe a spurious growth of the fractional solution beyond t∼8 and beyond t∼10. It can be observed that as we increase the number of terms in the sums of eq. (31), the decaying oscillations of the fractional solution do not change appreciably and show a similar structure. The observed difference is that the time where the spurious growth starts is shifted toward larger times. Therefore, only eight terms are sufficient if we are only interested in calculating the solution in the interval 0<t<8. It is worth mentioning that if the values of A2 were different, more terms in the sums in eq. (31) might be needed. Thus, the approximation of considering only a few terms in the infinite series seems to be consistent.

## 5 Discussion

In this work, we reported model calculations to analyze the effect of including fractional time derivatives in the hydrodynamic equation for the intermediate scattering function of an isothermal-isobaric inert, inviscid and incompressible binary mixture in the GH regime of intermediate wave vectors $\stackrel{⃗}{k}$ and frequencies ω. We summarize the results obtained and add some further comments to put them into a more proper context. We also suggest alternative ways to confront our theoretical results with experiment.

The expression for the non-fractional dynamics of the ISF was calculated analytically from an extended irreversible thermodynamic model. The effect produced by fractional derivatives on the dynamics of the ISF was also calculated analytically by considering two different ways of introducing the fractional derivatives. In the first approach, we replaced the two integer-order derivatives which appear in the standard model described by eq. (9), by a single fractional derivative of order 1<ε<2. In the second approach, the two derivatives appearing in eq. (9) were replaced by fractional derivatives of orders 2-ε and 1-ε, respectively. One outcome is that the second approach describes better the propagative character of the FISF because it reproduces the oscillatory behavior of ISF near the NS limit. We find that although the FISFs are in general less than the ISF, FISF may be a significant fraction (∼36 %) of ISF and might be measurable by light or neutron scattering techniques. As shown in Figure 5, this approach also shows quantitatively that the inclusion of fractional time scales into the dynamics of fluctuations may produce a FISF greater than the ISF showing that the FISF may be a measurable effect. These qualitative and quantitative behaviors for the model considered, are consistent with the ones previously reported for the behavior of other fractional correlation functions in complex fluids [19].

We have interpreted the fractional fluctuations as a superposition of conventional thermal fluctuations with successive weighted increments over time, but a deeper interpretation of the nature of fractional fluctuation is still being formulated [35].

Our results and predictions have been obtained for a realistic, although in several aspects a rather idealized model of a binary mixture in a nonequilibrium state. For this reason, it is difficult to know real experimental values for the phenomenological coefficients A1 and A2. We avoided this necessity by restricting our analysis to the time scale of the slowest non-conserved variable (A2) and by reformulating the transport equation only in terms of $\stackrel{ˆ}{a}$ (eq. (28)) which depends on the ratio of both coefficients. In this way by setting A1=1 and by changing the value of A2 we could vary the nonequilibrium states in the GH regime, where the local equilibrium assumption is not strictly valid, to near-equilibrium states where the system is not that far from the NS limit.

There are or course many open questions and further possibilities regarding the predicted behavior of the calculated FISF, but its eventual validity can only come from experiments. Yet, to our knowledge there are no available experimental measurements of the ISF, the FISF or the (non-fractional) dynamic structure factor $S\left(\stackrel{⃗}{k},\omega \right)$ defined by eq. (15). Usually ISF or $S\left(\stackrel{⃗}{k},\omega \right)$ are measured by neutron scattering [7], but other experimental techniques such as forced Rayleigh scattering has been also used to measure the ISF in simple fluids [36, 37, 38]. If such experiments were available, they would provide alternative ways of comparing our theoretical predictions with experiments. From our theoretical results in this work we now suggest how a quantitative and analytic estimation of the FISF can be made when the initial sate of the system starts in the GH regime and approaches and gets very close to the NS limit. In this regime the system will eventually be accessible to measurements of ISF and FISF by using conventional or forced Rayleigh scattering, but still containing information of the GH states.

The general expression for FISF is given by eq. (35), and in the asymptotic limit $t\to \mathrm{\infty }$ it reduces to eq. (40). As mentioned in the previous section, A2 controls the degree of departure of the state from the NS regime. Since ${A}_{2}\left(\stackrel{⃗}{k}\right)\equiv D{k}^{2}/{\tau }_{\stackrel{⃗}{J}}$, for given D and k, small values of A2 correspond to a large relaxation time ${\tau }_{\stackrel{⃗}{J}}$, indicating that the fast variable $\stackrel{⃗}{J}$ has not yet relaxed and prevent the validity of local equilibrium. In contrast, large values of A2 correspond to near local equilibrium states. Therefore, for given A2, k may be large or small depending on the value of D, which is unknown. Of course, since k is related to the scattering angle θ in a light scattering experiment, $k=2sin\left(\theta /2\right)$, a choice of A2 may correspond to very small θ. In this case the above-mentioned forced Rayleigh scattering technique is suitable. In Figure 6 we illustrate this approximated but quantitative estimation of FISF. Note that all curves decay to the exponential amplitude $I\left(\stackrel{⃗}{k},\tau \right)$ as given by eq. (10). Since we have approximated eq. (37) by its leading term, the oscillations of ${\stackrel{ˆ}{J}}^{FB}\left(k,t\right)$ are no longer perceptible. To observe them would require a more complicated approximation to eq. (37) involving more terms than the leading one. It should be pointed out that the decay rate of the different curves depends on the value of ε, which controls the order of fractional derivatives, and in this sense, reflects the extent of the memory or influence of other time scales.

Figure 6:

Plot of the FISF ${\stackrel{ˆ}{J}}^{FB}\left(k,t\to \mathrm{\infty }\right)$ vs t as given by eq. (37) for A2=20 and ε=0.01, 0.1, 0.5.

If such experiments would exist for our system, the comparison of our theoretical curves against experimental measurements would provide an additional confrontation between theory and experiment. However, we are not aware of any experiments of this sort and the experimental verification of our model predictions remains to be assessed.

## References

• [1]

L. Onsager, Reciprocal relations in irreversible processes. I., Phys. Rev. 37 (1931), 405, 2265. Google Scholar

• [2]

S. R. De Groot and P. Mazur, Nonequilibrium Thermodynamic, North Holland, Amsterdam, 1962. Google Scholar

• [3]

L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed., Pergamon, Oxford, 1987. Google Scholar

• [4]

H. J. M. Hanley editor, Transport Phenomena in Fluids, M. Dekker, New York, 1971. Google Scholar

• [5]

E. L. Cussler, Multicomponent Diffusion, Elsevier, New York, 1976. Google Scholar

• [6]

R. D. Mountain and J. M. Deutch, Light scattering from binary solutions, J. Chem. Phys. 50 (1969), 1103–1108.

• [7]

J. P. Boon and S. Yip, Molecular Hydrodynamics, McGraw-Hill, New York, 1980. Google Scholar

• [8]

R. P. Meilanov and R. A. Magomedov, Thermodynamics in fractional calculus, J. Eng. Phys. Thermophysics 87 (2014), 1521–1531.

• [9]

S. L. Sobolev, Locally nonequilibrium transfer processes, Usp. Fiz. Nauk. 167 (1997), 1095–1106. Google Scholar

• [10]

D. Jou, C. Pérez-García, L. S. García-Colín, M. López De Haro and R. F. Rodríguez, Generalized hydrodynamics and extended irreversible thermodynamics, Phys. Rev. A 31 (1985), 2502–2508.

• [11]

R. F. Rodríguez, L. S. García-Colín, M. López De Haro, D. Jou and C. Pérez-García, The underlying thermodynamic aspects of generalized hydrodynamics, Phys. Lett. A 107 (1985), 17–20.

• [12]

R. F. Rodríguez, L. S. García-Colín and M. López De Haro, Extended thermodynamic description of diffusion in an inert binary mixture, J. Chem. Phys. 83 (1985), 4099–4102.

• [13]

R. F. Rodríguez, M. López De Haro and L. S. García-Colín, Mutual diffusion in a binary mixture, Lect. Notes Phys. 253 (1986), 343–348.

• [14]

B. J. West and S. Picozzi, Fractional Langevin model of memory in financial time series, Phys. Rev. E 65 (2002), Article 037106.

• [15]

P. Pramukkul, A. Svenkeson, P. Grigolini, M. Bologna and B. J. West, Complexity and the fractional calculus, Adv. Math. Phys. 2013 (2013), Article 498789. Google Scholar

• [16]

B. J. West, Physiology, Promiscuity and Prophecy at the Millennium: A Tale of Tails, Studies of Nonlinear Phenomena in the Life Sciences, Vol. 7, World Scientific, Singapore, 1999. Google Scholar

• [17]

R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339 (2000), 1–77.

• [18]

R. F. Rodríguez, J. Fujioka and E. Salinas-Rodríguez, Fractional fluctuation effects on the light scattered by a viscoelastic suspension, Phys. Rev. E 88 (2013), Article 022154.

• [19]

R. F. Rodríguez, J. Fujioka and E. Salinas-Rodríguez, Fractional correlation functions in simple viscoelastic liquids, Physica A 427 (2015), 326–340.

• [20]

G. Stolovitzky and K. R. Sreenivasan, Kolmogorov’s refined similarity hypotheses for turbulence and general stochastic processes, Rev. Mod. Phys. 66 (1994), 229–240.

• [21]

R. Gorenflo, F. Mainardi and A. Vivoli, Continuous-time random walk and parametric subordination in fractional diffusion, Chaos Solitons & Fractals 34 (2007), 87–103.

• [22]

M. G. Hernández, E. Salinas-Rodríguez, S. A. Gómez, J. A. E. Roa-Neri, S. Alfaro and F. J. Valdés-Parada, Helium permeation through a silicalite-1 tubular membrane, Heat Mass Transfer 51 (2015), 847–857.

• [23]

F. J. Valdés-Parada, J. A. Ochoa-Tapia, E. Salinas-Rodríguez, S. Gómez-Torres and M. G. Hernández, Upscaled model for dispersive mass transfer in a tubular porous membrane separator, Rev. Mex. Ingen. Quím. 13 (2014), 237–257. Google Scholar

• [24]

J. M. Van De Graaf, F. Kapteijn and J. A. Moulijn, Permeation of weakly adsorbing components through a silicalite-1 membrane, Chem. Eng. Sci. 54 (1999), 1081–1092.

• [25]

B. J. Berne and R. Pecora, Dynamic Light Scattering, J. Wiley, New York, 1976. Google Scholar

• [26]

B. J. West, Colloquium: Fractional calculus view of complexity: A tutorial, Rev. Mod. Phys. 86 (2014), 1169–1189.

• [27]

L. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. Google Scholar

• [28]

A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. Google Scholar

• [29]

M. Caputo and C. Cametti, Diffusion with memory in two cases of biological interest, J. Theoret. Biol. 254 (2008), 697–703.

• [30]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Beach, Amsterdam, 1993. Google Scholar

• [31]

K. S. Miller and B. Ross, An Introduction to Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. Google Scholar

• [32]

K. B. Oldham and J. Spanier, The Fractional Calculus, Vol. 11, Academic Press, New York, 1974. Google Scholar

• [33]

M. D. Ortigueira, J. A. T. Machado and J. S. Da Costa, Which differ integration? IEE proceedings-vision, Image Signal Process 152 (2005), 846–850. Google Scholar

• [34]

J. Fujioka, A. Espinosa and R. F. Rodríguez, Fractional optical solitons, Phys. Lett. A 374 (2010), 1126–1134.

• [35]

P. Pramukkul, A. Svenkeson, P. Grigolini, M. Bologna and B. West, Complexity and the fractional calculus, Adv. Math. Phys. 2013 (2013), Article 498789. Google Scholar

• [36]

J. P. Boon, C. Allain and P. Lallemand, Propagating thermal modes in a fluid under thermal constraint, Phys. Rev. Lett. 43 (1979), 199–203.

• [37]

P. N. Segrè, R. W. Gammon, J. V. Sengers and B. M. Law, Rayleigh scattering in a liquid far from thermal equilibrium, Phys. Rev. A 45 (1992), 714–724.

• [38]

B. M. Law, P. N. Segrè, R. W. Gammon and J. V. Sengers, Light-scattering measurements of entropy and viscous fluctuations in a liquid far from thermal equilibrium, Phys. Rev. A 41 (1990), 816–824.

Accepted: 2017-10-27

Revised: 2017-10-20

Published Online: 2017-12-14

Published in Print: 2018-01-26

Citation Information: Journal of Non-Equilibrium Thermodynamics, Volume 43, Issue 1, Pages 43–55, ISSN (Online) 1437-4358, ISSN (Print) 0340-0204,

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