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Suppose I have two position vectors $\mathbf{r}_1$ and $\mathbf{r}_2$.

The magnitude of both of these vectors are chosen independently from the same probability distribution of known mean and variance. The direction (again for both vectors) is chosen from an isotropic distribution (i.e. uniform in both angular directions).

The quantity I would like to calculate is, $$ \left\langle \frac{1}{|\mathbf{r}_1 - \mathbf{r}_2|} \right\rangle $$

There have been various similar posts (see, here and here) which calculate the average distance. However, I am interested in particular case of a non-uniform distribution of points in the radial direction.

Any help or links to literature would be appreciated! Thanks in advance.

MarcosMFlores
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  • In general the mean and variance do not determine a distribution and are therefore insufficient information to calculate a general expectation value (such as the quantity you desire). The expectation value you have written is unlikely to even be finite. Take for instance $R_1,R_2$ independently from $N(0,1)$ (standard normal). Then $R_1 - R_2 \sim N(0, 2)$ and $E[1/|R_1 - R_2|] \propto \int_{-\infty}^\infty e^{-r^2/4}/r dr$, and this last integral diverges. – snar Feb 16 '21 at 22:56

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