Macroscopic quantities (like isothermal compressibility) from fluctuations and the partition function

Here's a formal derivation, ignoring any subtlety about different ensembles, etc. For brevity, I'm just going to write the partition function begin{align} Z(Y) = int dX ,e^{-XY} end{align} Then begin{align} langle X rangle(Y) = frac{1}{Z(Y)}int dX,X e^{-XY}. end{align} Differentiating with respect to $Y$, begin{align} -partial_Y langle X rangle &= -partial_Yleft(frac{1}{Z}right)int dX ,X e^{-XY} - frac{1}{Z}partial_Yleft(int dX X ,e^{-XY}right) &= -frac{1}{Z^2}left(int dX,X e^{-XY}right)^2 + frac{1}{Z}left(int dX X^2 ,e^{-XY}right) &= langle X^2rangle - langle X rangle^2. end{align}

Regarding the second part, the grand canonical ensemble is definitely helpful here since you can consider an arbitrary subvolume V of an extensive thermodynamic system, and then examine the fluctuations in total particle number N in that subvolume. The variance in N will be proportional to V. Such a proportionality means it is in fact fluctuations in density. The value $langle delta N^2 rangle / V$ is then independent of the subvolume size, thus it's some kind of intensive quantity and indeed it turns out to be directly related to isothermal compressibility.

Now clearly, this trick only works because we allow N to vary, otherwise variance in N would be zero. But what if we want to use this trick to calculate the density fluctuations in another quantity? It's less obvious but we still have to pay attention about what ensemble we are using. To illustrate why, a couple of examples:

Let's look at trying to calculate fluctuations in energy, and consider the case where each particle carries a huge potential energy offset. Every time a particle jumps in and out of our subvolume, its total energy varies a great deal. If we used the canonical ensemble, this would have no impact on our energy variance. In the grand canonical ensemble it could dominate our energy variance. Both energy variances are nonzero and proportional to V, but have very different values, and the canonical one is wrong since it doesn't reflect the fact that particles can move in and out of the subvolume.

Alternatively let's say we are looking at fluctuations in magnetization in some gas of paramagnetic atoms, in a nonzero magnetic field. Each atom in our subvolume has some random (but biased) magnetic moment. Thus our fluctuations in total magnetization in a subvolume will be influenced by that per-atom variance, but will also have a contribution from the bias multiplied by the randomness in number of atoms that happen to find themselves in our subvolume. The canonical ensemble would only pick up the first term.

(On the other hand, if you know that you have placed exactly 85823135 atoms in a box and you want to calculate the variance in magnetization for the full box, then it would be a mistake to use the grand canonical ensemble in this calculation and you must use the canonical ensemble for N=85823135.)

For further reading, I suggest Landau & Lifshitz Volume 5 (Statistical Physics) section 112.