The Fourier transformed density $f(\mathbf{k},t)$ plays a central in dynamic light scattering. The classic text is by Berne and Pecora (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.
Personal note: I started out my scientific life calculating how the decay is modified by hydrodynamic interactions.