# Fluctuations of non-natural state variables in statistical mechanics

Background

In statistical mechanics, every ensemble has its natural variables and thermodynamic potential (e.g. for the canonical ensemble $$T$$, $$V$$, and $$N$$ are the natural variable and $$F = F(T, V, N)$$ is the potential).

When we want to compute some variables which are not "natural" we have to use the thermodynamic relations. For example if we want to compute the temperature in the microcanonical ensemble, we use

$$\frac{1}{T}= \frac{\partial U}{\partial S}\Big|_{V, N}$$

Another example is the pressure computed in the canonical ensemble

$$P= T\frac{\partial F}{\partial V}\Big|_{T, N}$$

however on wikipedia("Free energy, ensemble averages, and exact differentials" paragraph) I am reading that this formula only gives the average pressure of the system.

My questions

• In general, are these "derived variables" always constant (and well defined) when they are not the natural variables of the ensemble or they always fluctuate?
• What about the non-natural thermodynamic potentials, do they always fluctuate? E.g. the internal energy in the canonical ensemble does fluctuate.
• Is there a general rule?

I am assuming we are not in the thermodynamic limit in general.