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.

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.