When we examine real polymer chains we have to consider the interactions between single monomers. Therefore we consider a Lennard-Jones-like potential for bringing two monomers together and calculate the probability of the distance r between the monomers with a Boltzmann-distribution.
Then we define the so-called "Mayer f-function" as the difference between the Boltzmann factor and the case of no interaction or infinite distance:
$$f(r) = exp[-U(r)/k_BT]-1$$
The derivation can be visualized with the following graphs:
Now we define the excluded volume v as the negative of the integral of the Mayer f-function over all space:
$$v = -\int f(r)dr = \int 1-exp[-U(r)/k_BT]dr$$
This excluded volume is now interpreted and used as the volume occupied by one monomer, which can therefore not be occupied by anything else (Excluded volume - Wikipedia).
How can I derive this interpretation from the mathematical calculation? I have my difficulties with getting a clear picture of the physical meaning of the excluded volume by just looking at the above derivation.