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I really need to properly evaluate the following integral.

\begin{equation}\label{1} \frac{1}{2}\int dx \, dy \,\frac{\rho(x)\rho(y)}{(x-y)^2} \end{equation}

Where $\rho$ is some density function normalized to 1: $\int dx \rho(x)=1$. However, it is clear that the original integral diverges. We can see this just by evaluating it for a Gaussian or some other standard distribution.

Does anyone know how I should properly regulate this thing? I know I should probably introduce an $\epsilon $ somewhere and expand the integral as a series in $\epsilon$ and then subtract the infinite parts. But I don't have a lot of intuition for how to do this.

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  • $\begingroup$ It is more technical question than conceptual. If you provide more details of the problem, it is easier two answer. Roughly speaking, the upper limit of integration can be cutted with existence of some boundaries and the low limit of integration can be cutted with something like "size of particles". $\endgroup$ Commented Feb 7, 2020 at 10:51
  • $\begingroup$ If not, you can introduce by hand cut-offs on upper/lower (or both) limits of integration and perform integration. It a final answer seems like "cut-off independent part" + "cut-off dependent part", you can assume the calculation is true. $\endgroup$ Commented Feb 7, 2020 at 10:54
  • $\begingroup$ I agree this is a more technical question, but it's more or less the last piece of a question I'm investigating. Essentially I'm trying to take the hydrodynamic limit of a gas with an inverse square potential interaction. So I agree the "size of the particle" cutoff is probably what I need to do, I just have no idea how to implement this. $\endgroup$
    – S Thomas
    Commented Feb 7, 2020 at 10:55
  • $\begingroup$ I believe that my second comment may help. Just try to introduce cut-offs and extract regulator independent part of the answer. $\endgroup$ Commented Feb 7, 2020 at 10:57
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    $\begingroup$ Some physics insight is helpful here. For example, Artem's comment on particle size was useful. $\endgroup$
    – S Thomas
    Commented Feb 7, 2020 at 11:46

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