Why do Green functions show up in treating dielectric media microscopically?

I'm reading the treatment of Thomas-Fermi screening in Ashcroft and Mermin (ch. 17). They write some strange equations which I've never seen before, anywhere.

First of all, they write the usual local relation $$\mathbf{D} = \epsilon \mathbf{E}$$ as a non-local relation

$$\mathbf{D}(\mathbf{r}) = \int \epsilon(\mathbf{r},\mathbf{r'})\mathbf{E}(\mathbf{r}')d\mathbf{r}'$$

And then proceed to write the same type of equation linking external potential $$\phi^{ext}$$ and potential $$\phi$$, and induced charge density $$\rho^{ind}$$ and potential $$\phi$$:

$$\phi^{ext}(\mathbf{r}) = \int \epsilon(\mathbf{r},\mathbf{r'})\phi(\mathbf{r'})d\mathbf{r}'$$

$$\rho^{ind}(\mathbf{r}) = \int \chi(\mathbf{r},\mathbf{r'})\phi(\mathbf{r'})d\mathbf{r}'$$

I've seen enough Green's functions to recognise them here, and also enough Green's functions to intuitively see it makes a tiny little bit of sense to use them. After all, the electric displacement field is influenced by only the local electric field only when averaging over long length scales. And that is exactly what Ashcroft and Mermin do not do in treating Thomas-Fermi screening.

Still, I've never seen this language before in this context and I'm a little stumped as to where it comes from. Two questions:

1. Ashcroft and Mermin decide to map $$\mathbf{D}\to \phi^{ext}$$ and $$\mathbf{E}\to \phi$$ in all their equations. Why is this allowed?
2. What is the theoretical background that validates using these integrals here? Is it an ad hoc assumption they make, which just happens to work out? Or do I just have a strangely shaped hole in my knowledge of electrostatics?
• Found myself asking the same question recently. Did you find an answer elsewhere, by any chance? Oct 4, 2021 at 12:29