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 $c(\stackrel{\u20d7}{r},\tau )$, and two non-conserved variables, the local mass flux $\stackrel{\u20d7}{J}(\stackrel{\u20d7}{r},\tau )$ and a second-order tensor $\frac{\leftrightarrow}{\mathrm{\Im}(\stackrel{\u20d7}{r},\tau )}$, which accounts for the spatial inhomogeneities in the space of state variables. The mass conservation equation reads
$\frac{dc}{d\tau}=-\mathrm{\nabla}\bullet \stackrel{\u20d7}{J}(\stackrel{\u20d7}{r},\tau )$(1)

where *d*/*dτ* denotes the hydrodynamic derivative; the equation of motion is given by
$\rho \frac{d\stackrel{\u20d7}{u}}{d\tau}=-\mathrm{\nabla}p$(2)

where *ρ* is the local mass density, *p* is the pressure and $\stackrel{\u20d7}{u}$ 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 ($\stackrel{\u20d7}{J}$,$\overleftrightarrow{\mathrm{\Im}}$) (see eqs. (2.8)–(2.9) in [12])

$-{\tau}_{\stackrel{\u20d7}{J}}\frac{d\stackrel{\u20d7}{J}}{d\tau}=\stackrel{\u20d7}{J}+\frac{1}{{\mu}_{10}}\mathrm{\nabla}\left(\frac{\mu}{T}\right)-{l}_{2}\mathrm{\nabla}\bullet \overleftrightarrow{\mathrm{\Im}},$(3)$-{\tau}_{\overleftrightarrow{\mathrm{\Im}}}\frac{d\overleftrightarrow{\mathrm{\Im}}}{d\tau}=\overleftrightarrow{\mathrm{\Im}}-{l}_{1}\mathrm{\nabla}\stackrel{\u20d7}{J},\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\text{\hspace{0.17em}}\phantom{\rule{thickmathspace}{0ex}}$(4)where *μ*_{10}, *l*_{1} and *l*_{2} are phenomenological coefficients. The relaxation times ${\tau}_{\stackrel{\u20d7}{J}}$ and ${\tau}_{\overleftrightarrow{\mathrm{\Im}}}$ of $\stackrel{\u20d7}{J}$ and $\overleftrightarrow{\mathrm{\Im}}$ 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 $\overleftrightarrow{\mathrm{\Im}}$ will vary on a shorter time scale than that of $\stackrel{\u20d7}{J}$, so we restrict our description to a time scale in which $\overleftrightarrow{\mathrm{\Im}}$ has already relaxed and ${\tau}_{\overleftrightarrow{\mathrm{\Im}}}$ is negligible so that eq. (3) becomes (see eq. (2.14) in [12])
$-{\tau}_{\stackrel{\u20d7}{J}}\frac{d\stackrel{\u20d7}{J}}{d\tau}=\stackrel{\u20d7}{J}+D\mathrm{\nabla}c-{l}_{1}{l}_{2}{\mathrm{\nabla}}^{2}\stackrel{\u20d7}{J}.$(5)

Since in most experimental situations involving mass diffusion the pressure is kept constant, *D* denotes the isothermal-isobaric diffusion coefficient $D=(T{\mu}_{10}{)}^{-1}(\mathrm{\partial}\mu /\mathrm{\partial}c{)}_{T,p}$, 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 ${\tau}_{\stackrel{\u20d7}{J}}={l}_{1}{l}_{2}=0$, 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, $\delta c(\stackrel{\u20d7}{r},t)=c(\stackrel{\u20d7}{r},t)-{c}_{0}(\stackrel{\u20d7}{r})$, $\delta \stackrel{\u20d7}{J}(\stackrel{\u20d7}{r},t)=\stackrel{\u20d7}{J}(\stackrel{\u20d7}{r},t)-{\stackrel{\u20d7}{J}}_{0}(\stackrel{\u20d7}{r})$. 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]

$I(\stackrel{\u20d7}{r},\tau ;{\stackrel{\u20d7}{r}}^{{}^{\prime}},{\tau}^{{}^{\prime}})=\u3008\delta c(\stackrel{\u20d7}{r},\tau )\delta c({\stackrel{\u20d7}{r}}^{{}^{\prime}},{\tau}^{{}^{\prime}})\u3009;,$(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 $I(\stackrel{\u20d7}{k},\tau )$ (ISF) or van Hove’s self-correlation function [7], is proportional to the spatial Fourier transform of $\delta c(\stackrel{\u20d7}{r},\tau )$ within the scattering volume
$I(\stackrel{\u20d7}{k},\tau )=\u3008\delta c\ast (\stackrel{\u20d7}{k},\tau )\delta c(\stackrel{\u20d7}{k},0)\u3009;,$(7)

where the asterisk (*) indicates complex conjugate. Substitution of eq. (1) into eq. (5) yields the time evolution equation for $I(\stackrel{\u20d7}{k},\tau )$

$\frac{{d}^{2}I}{d{\tau}^{2}}+{A}_{1}(\stackrel{\u20d7}{k})\frac{dI}{d\tau}+{A}_{2}(\stackrel{\u20d7}{k})I=0,$(8)where ${A}_{1}(\stackrel{\u20d7}{k})\equiv (1+{l}_{1}{l}_{2}{k}^{2})/{\tau}_{\stackrel{\u20d7}{J}}$ and ${A}_{2}(\stackrel{\u20d7}{k})\equiv D{k}^{2}/{\tau}_{\stackrel{\u20d7}{J}}$ are two free parameters which are combinations of phenomenological coefficients ${\tau}_{\stackrel{\u20d7}{J}}$, *l*_{1}, *l*_{2}. For given $I(\stackrel{\u20d7}{k},0)$ with vanishing time derivative ${I}^{{}^{\prime}}(\stackrel{\u20d7}{k},0)=0$ its solution reads
$\frac{I(\overrightarrow{k},\tau )}{I(\overrightarrow{k},0)}=\mathrm{exp}\left(-\frac{1}{2}{A}_{1}\tau \right)\left(\mathrm{cos}R\tau +\frac{{A}_{1}}{2R}\mathrm{sin}R\tau \right)$(9)

with $R(k)=[{A}_{2}(k)-{A}_{1}^{2}(k)/4{]}^{1/2}>0$, which imposes the condition ${A}_{2}(k)>{A}_{1}^{2}(k)/4$. It must be emphasized once again that the inclusion of spatial inhomogeneities through the non-conserved variable $\overleftrightarrow{\mathrm{\Im}}$ 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 ${\tau}_{\overleftrightarrow{\mathrm{\Im}}}\to 0$, ${l}_{1}{l}_{2}\to 0$, in eq. (9) which yields the classical exponential decay of $I(\stackrel{\u20d7}{k},\tau )$

$I(\overrightarrow{k},\tau )=I(\overrightarrow{k},0)\mathrm{exp}\left(-\frac{1}{2}{A}_{1}\tau \right).$(10)Furthermore, $I(\stackrel{\u20d7}{k},\tau )$must satisfy certain restrictions, the so-called sum rules [7], which yield a relation between the static structure factor $S(\stackrel{\u20d7}{k})$ and ${A}_{2}(\stackrel{\u20d7}{k})$, namely (see eq. (3.11) in [12])
$S(\stackrel{\u20d7}{k})=\frac{{k}_{B}T}{{A}_{2}(\stackrel{\u20d7}{k})}{k}^{2}=\frac{{k}_{B}T}{D}{\tau}_{\stackrel{\u20d7}{J}}.$(11)

Since $S(\stackrel{\u20d7}{k})$ 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 ${\tau}_{\stackrel{\u20d7}{J}}$. This would leave us with one free parameter ${A}_{1}(\stackrel{\u20d7}{k}),$ or equivalently *l*_{1}*l*_{2}. 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 *A*_{1} or *A*_{2} are known for the binary mixture, but since the condition ${A}_{2}(k)>{A}_{1}^{2}(k)/4$ 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 ${{J}}_{t}\equiv \mathrm{\partial}{J}(\stackrel{\u20d7}{k},t)/\mathrm{\partial}t$, ${{J}}_{tt}\equiv {\mathrm{\partial}}^{2}{J}(\stackrel{\u20d7}{k},t)/\mathrm{\partial}{t}^{2}$. 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 ${J}(\stackrel{\u20d7}{k},0)$, ${{J}}_{t}(\stackrel{\u20d7}{k},0)=0$ its solution becomes
$\frac{{J}\left(\stackrel{\u20d7}{k},t\right)}{{J}\left(\stackrel{\u20d7}{k},0\right)}={e}^{-t/2}\left(cos\text{\hspace{0.17em}}Mt+\frac{1}{2M}sin\text{\hspace{0.17em}}Mt\right),$(14)

where $M={\left(a-1/4\right)}^{1/2}$. By taking the Laplace transform of ${J}\left(\stackrel{\u20d7}{k},t\right)$ with respect to time and setting $s=i\omega $ gives the structure factor

$S\left(\stackrel{\u20d7}{k},\omega \right)=S\left(\stackrel{\u20d7}{k}\right)\frac{{A}_{1}\left(\stackrel{\u20d7}{k}\right){A}_{2}\left(\stackrel{\u20d7}{k}\right)}{\left({\omega}^{2}+{B}_{1}\right)\left({\omega}^{2}+{B}_{2}\right)},$(15)where we have defined $S\left(\stackrel{\u20d7}{k}\right)={J}\left(\stackrel{\u20d7}{k},0\right)$ 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)
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