Cross-posted from physics SE. I am studying statistical mechanics, and I am studying ideas surrounding potential of mean force and n-body density functions.
In a derivation, they mention that $$-\left\langle \frac{\partial U}{\partial r_1} \right\rangle=k_BT\frac{d}{dr_1}\ln g(|r_1,r_2|)$$ and then they say, "since integrating over the average force yields work, we get", $$w(r)= -k_BT\ln g(r)$$
But I feel like I don't understand how the integration works here. I understand $\langle F\rangle = -k_BT\frac{d}{dr_1}\ln g(r_1,r_2)$ But now what I am doing is I am dragging particle 2 from $\infty$ to some distance $r$ from particle 1. So, the integration should look like: $$w(r) = \int_{\infty} ^{r_1+r} \int_{0}^R \langle F \rangle \cdot r dr_1dr_2 = -\int_{\infty}^{r_1+r} \int_{0}^{\infty} k_BT\frac{d}{dr_1}\ln g(|r_1-r_2|) dr_1 dr_2$$
I am not sure how to safely reduce these integrals into the nice statement $$w(r) = -k_BT\ln g(r)$$ How does one go about reducing such an integral? Or if there is an error in the way I have set this up, I would appreciate knowing that too.
I would appreciate any advice you have for me.