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).