Hamiltonian particle system and its frequency domain

Question

I am interested in the following question. So let suppose we have finite number of point particles on plane $\mathbb{R}^2$. We can assume that every $j$ point is represented by Dirac delta function $\delta(\vec{r} - \vec{r}_j)$.

Fourier transformation maps $\delta(\vec{r} - \vec{r}_j) \rightarrow e^{-2 \pi i (\vec{k} \cdot \vec{r}_j) }$

Being applied to the whole system it leads to the following frequency value $f (\vec{k}) = \sum_j e^{-2 \pi i (\vec{k} \cdot \vec{r}_j)}$

The points are moving according to Hamiltonian dynamics which means frequency function changes over time. And it satisfies the following differential equation:

$ \frac{\partial f}{\partial t} = -2 \pi i \sum_j (\vec{p}_j \cdot \vec{k}_j) e^{-2 \pi i (\vec{k} \cdot \vec{r}_j)} $

Is there anything known in this field? Like how to evaluate asymptotics of frequency domain $t \rightarrow \infty$? I would really appreciate some references as paper or book.

Answer 1

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.}