The Fourier transformed density $f(\mathbf{k},t)$ plays a central in dynamic light scattering. The classic text is by <A HREF="https://www.eng.uc.edu/~beaucag/Classes/Properties/Books/Bruce%20J.%20Berne,%20Robert%20Pecora%20-%20Dynamic%20Light%20Scattering_%20With%20Applications%20to%20Chemistry,%20Biology,%20and%20Physics-John%20Wiley%20&%20Sons,%20Inc.%20(2000).pdf">Berne and Pecora</A> (B&P). The correlator
$$F(\mathbf{k},t)=\langle f(-\mathbf{k},0)f(\mathbf{k},t)\rangle$$
contains information on the Brownian motion of particles suspended in a fluid (see equation 5.4.2 in B&P). For non-interacting particles it decays as
$$F(\mathbf{k},t)=e^{-k^2 Dt},\;\;t>0,$$
with $D$ the diffusion constant. In a fluid the particles may interact by the perturbation of the velocity field (hydrodynamic interaction), resulting in a non-exponential decay.