We are all familiar with the typical example of statistical physics; $N$ particle of gas in a box of average volume $\overline{V}$ and average energy $\overline{E}$.
The equation of state of of the canonical ensemble of this system is $TdS=d\overline{E}+pd\overline{V}$, where $T$ and $p$ are Lagrange multipliers and are constant in the body at thermodynamic equilibrium.
The quantities $T$ and $p$ do not have thermal fluctuations, but the quantities $\overline{V}$ and $\overline{E}$ do. Explicitly, in the case of $\overline{V}$, the fluctuations are $\langle \Delta \overline{V} \rangle ^2=\frac{\partial^2 \ln{Z}}{\partial (\beta p)^2}$, and in the case of $\overline{E}$, the fluctuations are $\langle \Delta \overline{E} \rangle ^2=\frac{\partial^2 \ln{Z}}{\partial \beta^2}$.
Questions:
What is an average volume, in this case. I don't get it. A box is a box, why would the volume vary. If the volume is an average $\overline{V}$, then it means that at some time, the volume of the gas inside the box occupy a volume slightly higher than the box, and some times slightly less than the box. As this makes no sense, $V$, at most, should be an upper bound, not an average.
In the case of a box with a fixed volume $V$, would it not make more sense to construct the ensemble with an average pressure $\overline{p}$ and an average energy $\overline{E}$. In this case the equation of state would be $TdS=d\overline{E}+Vd\overline{p}$, with an uniform volume as a Lagrange multiplier, and with fluctuations over the pressure --- essentially reversing the definition of the first case? But this case is never presented in introductory statistical physics; all authors seem to prefer the case where the fluctuations are on the volume. What am I missing here?
Generally, do Lagrange multipliers experience fluctuations as the priors do? Specifically, does $T$ and $p$ in $TdS=d\overline{E}+pd\overline{V}$ and $V$ in $TdS=d\overline{E}+Vd\overline{p}$, the Lagrange multipliers, experience fluctuations? I believe that they do not, but I would like a confirmation.