de Gennes narrowing

 In colloidal solutions, a widely-used relation connects the scale-dependent collective diffusion constant and the structure factor:

$$\begin{equation} D_c(q) =\frac{D_0}{S(q)} \label{eq:dGn} \end{equation}$$ and is generally known as de Gennes narrowing since its use by de Gennes in the context of quasi-elastic neutron scattering from liquids [1].

Intuitively, equation $\eqref{eq:dGn}$ makes sense: if $S(q)$ is high for a certain value of $q$ then fluctuations with that particular $q$ are frequent, meaning that their energetic cost is low and that they will decay slowly. However, I have not yet found in the literature a simple yet rigorous derivation. This is what I will attempt below.

Consider a system characterized by a conserved scalar parameter $\phi(\mathbf{r})$ (for instance, the local particle concentration in a suspension $\phi(\mathbf{r}) = \rho(\mathbf{r}) - \rho_0$, with $\rho_0$ the equilibrium value). We are interested in the excess free energy due to inhomogeneities of this field: $\mathcal{F} - \mathcal{F}_0 = \mathcal{F}[\phi(\mathbf{r})]$ or, in terms of its Fourier components: $\mathcal{F} - \mathcal{F}_0 = \mathcal{F}[\phi(\mathbf{q})]$.

For an isotropic system in the absence of applied fields, all fluctuations $\phi(\mathbf{q})$ with $\left |\mathbf{q} \right | > 0$ will eventually decay to zero. To fix the ideas, we will consider an overdamped relaxation (model B in the Hohenberg-Halperin classification [2], but of course far from criticality).

Let us write Fick's laws, with $\mathbf{j}$ and $\mu$ the current and chemical potential associated to $\phi$ (this is similar to the presentation in [2], Eqs. (2.2)-(2.9)):

$$\begin{equation} \label{eq:defs} \frac{\partial \phi(\mathbf{r},t)}{\partial t} = - \nabla \mathbf{j} \, ; \quad \mathbf{j} = - \lambda \nabla \mu \, ; \quad \mu = \frac{\delta \mathcal{F}}{\delta \phi(\mathbf{r},t)} \, \Rightarrow \, \frac{\partial \phi(\mathbf{r},t)}{\partial t} = \lambda \nabla ^2 \frac{\delta \mathcal{F}}{\delta \phi(\mathbf{r},t)}\end{equation}$$ where $\delta$ denotes the functional derivative. The second relation in $\eqref{eq:defs}$ serves as a definition for the transport coefficient $\lambda$.
Introducing the Fourier components yields: $$\begin{equation} \label{eq:relax} \frac{\partial \phi(\mathbf{r},t)}{\partial t}= \lambda \nabla ^2 \frac{\delta \mathcal{F}}{\delta \phi(\mathbf{r},t)}  \Rightarrow \frac{\partial \phi(\mathbf{q},t)}{\partial t} = - \lambda q^2 \frac{\text{d} \mathcal{F}}{\text{d}\phi(\mathbf{q},t)} \end{equation}$$ If the different Fourier modes are uncoupled, we can write the equipartition relation:
$$\begin{equation} \label{eq:equi} \mathcal{F} = \mathcal{F}_0 + \sum_{\mathbf{q}}  \frac{A(\mathbf{q})}{2} |\phi(\mathbf{q})| ^2 \Rightarrow \frac{\text{d} \mathcal{F}}{\text{d}\phi(\mathbf{q},t) } = A(\mathbf{q}) \phi(\mathbf{q}) \end{equation}$$ Let us introduce the time-dependent structure factor $$\begin{equation} \label{eq:Sqt} S(\mathbf{q},t) = \left \langle \phi(\mathbf{q},t) \phi(\mathbf{-q},0) \right \rangle \end{equation}$$ with $\left \langle \cdot \right \rangle$ the ensemble average.

Plugging $\eqref{eq:equi}$ into $\eqref{eq:relax}$, multiplying by $\phi(-\mathbf{q},0)$ and averaging yields the evolution of mode $\mathbf{q}$: $$\begin{equation} \label{eq:evol} \frac{\partial}{\partial t} \left \langle \phi(\mathbf{q},t) \phi(-\mathbf{q},0) \right \rangle =- \lambda q^2 A(\mathbf{q}) \left \langle \phi(\mathbf{q},t) \phi(-\mathbf{q},0) \right \rangle \Rightarrow \frac{S(\mathbf{q},t)}{S(\mathbf{q},0)} = \exp \left \lbrace - D_c(q) q^2 t \right \rbrace  \end{equation}$$ where we invoked the isotropy of the system. The collective diffusion coefficient is given by: $$\begin{equation} \label{eq:Dc} D_c(q) = \lambda A(q)  \end{equation}$$ On the other hand,equipartition also implies: $$\begin{equation} S(\mathbf{q}) = S(\mathbf{q},0) = \left \langle \phi(\mathbf{q},0) \phi(\mathbf{-q},0) \right \rangle = \frac{k_B T}{A(\mathbf{q})} . \label{eq:Sq} \end{equation}$$ From $\eqref{eq:Dc}$ and $\eqref{eq:Sq}$ we finally obtain:
$$\begin{equation} \label{eq:final} D_c(q) = \frac{\lambda k_B T}{S(q)} \end{equation}$$
[1] P. G. de Gennes, Liquid dynamics and inelastic scattering of neutrons, Physica A 25, 825-839 (1959).
[2] P. C. Hohenberg and B. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys. 49, 435-479 (1977).