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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.

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I would leave the term natural variables to Classical Thermodynamics, where a few state functions (fundamental equations, in the Thermodynamic language), if expressed in terms of their natural variables, give access to the equilibrium properties without the need for thermodynamic integration with the consequent introduction of arbitrary functions.

Partition functions in Statistical Mechanics allow us to evaluate fundamental equations, but even if we did not know anything about fundamental equations, each of them is defined for a system where some thermodynamic variables have been fixed.

Therefore, coming to your specific questions:

  • It turns out that the "derived variables" can be expressed as averages of some observable over the probability distribution of the ensemble. Averages do fluctuate. Partition functions are not averages.
  • State functions that are fundamental expressions if expressed in terms of their natural variables if described in terms of different variables correspond to "derived variables" in some ensemble. This is the core of the so-called "thermodynamic reduction". Therefore, the above-mentioned state functions do fluctuate.
  • The two previous points are entirely general.
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  • $\begingroup$ Thank you very much for your answer! I am not sure I got an aspect of you question, so I am asking explicitly: does it mean that whenever we use the "Thermodynamic square" to compute some variable which is not natural/standard in the ensemble we are using (e.g. we are computing pressure in the canonical ensemble) the result is always the average value of that fluctuating variable? $\endgroup$
    – Andrea
    Aug 29, 2022 at 19:01
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    $\begingroup$ @Andrea The point is that pressure in the canonical ensemble is a partial derivative of the free energy. Therefore, it can be put in the form of the average of a suitable observable (for the pressure that is the average of the viral (apart from the trivial, non average ideal contribution). $\endgroup$ Aug 29, 2022 at 20:22

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