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 
$f (\vec{k}) = \sum e^{-2 \pi I (\vec{k} \cdot \vec{r}_j)$

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 (\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.