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I'm reading about classical non-ideal gases from this source. On page 5, they say that the non-ideal partition function is a product of the ideal partition function and the non-ideal partition function, which makes perfect sense.

They write the interaction part of the non-ideal partition function in equation (3.1.4) as

$$ \frac{1}{V^N} \int \mathrm{e}^{-\left(\sum_{i<j} U(q_i, p_i)\right)/kT}\, \mathrm{d}^{3N}q $$

with no explanation as to where the factor $ 1/V^N$ comes from. I can't seem to figure it out in any sort of detail. Any ideas?

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2 Answers 2

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It is just a convention choice, not every textbook uses this. It's been a while, but I think the choice to have that factor is due to the fact that the positions are uniformly distributed over the volume $V$, so we choose to perform the integral with this factor to represent this probability. So,

$$Q = \int \prod_i \left(\frac{d^3q_i}{V}\right) U(q_1, ... q_n)$$

It doesn't change the physics, since $V$ is a constant in these integrals anyway.

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It's just a normalisation factor.

The total volume for one particle would be $V = \int \mathrm{d}^3q$, so that for $N$ particles is $(\int \mathrm{d}^3q)^N = V^N.$

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