Integration bounds for the canonical ensemble's partition function

Question

As far as I understand, when calculating the partition function we are summing over all of phase space. So, for ideal gas for example we have

$$H =\sum\limits_i\frac{p^2_i}{2m}$$

and so

$$Z=\int\limits_{-\infty}^\infty e^{\sum\limits_i \beta p^2_i/2m}d^{3N}qd^{3N}p = \int\limits_{-\infty}^\infty d^{3N}q \cdot \int\limits_{-\infty}^\infty e^{\sum\limits_i \beta p^2_i/2m}d^{3N}p$$

at this point the textbook makes the claim that $$ \int\limits_{-\infty}^\infty d^{3N}q = V^N$$

I would understand this claim if the integrand had some indicator function that zeroes all states outside our "box", but I don't see any, so how does that make sense? If this is some "approximation" we do to compensate for the approximated hamiltonian (no interactions between particles), is there a general rule for when I can just ignore a region in phase space?

Answer 1

On the page linked in a comment, there is a clear indication that one is working with $N$ particles in a volume $V$.

Statistical Mechanics of $N$-particle systems is always done by confining the $N$ particles in a finite volume V because we can only work with a finite number of particles (to avoid divergencies), and a finite number of particles in an infinite volume would imply a vanishing particle density.

However, such a confinement can be obtained in different ways. The most common method used in modern textbooks is confining the positions inside a finite volume $V$ with a one-body potential equal to zero inside the volume and equal to $ +\infty$ outside. In such a case, the integration of the positions could be formally extended to the whole space, But all points outside the volume will give a vanishing contribution and then the integration could be limited to the volume $V^N$. An alternative possibility, exploited in the past, would be to add a smooth confining potential, like a harmonic potential, such that the average number density in the central region is close to the number density of the state one is interested in. In such a case, integration on the position variables is performed on the whole space. In both cases, thermodynamics is recovered only at the thermodynamic limit. I.e., in the limit where volume and number of particles both diverge by keeping their ratio constant.