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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:

  1. 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.

  2. 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?

  3. 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.

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First, we have to address a misconception in the premise of your question.

When you say "The quantities $T$ and $p$ do not have thermal fluctuations, but the quantities $\overline{E}$ and $\overline{V}$ do," you are choosing to talk about the isothermal-isobaric ensemble, not the canonical ensemble.

There are four general archetypes of thermodynamic ensemble that are usually talked about:

  • Microcanonical ensemble: $N$, $V$, $E$ fixed, while $\mu$, $p$, $T$ fluctuate (models thermally-isolated closed system)

  • Canonical ensemble: $N$, $V$, $T$ fixed, while $\mu$, $p$, $E$ fluctuate (models system in thermal contact with fixed-temperature bath)

  • Isothermal-isobaric ensemble: $N$, $p$, $T$ fixed, while $\mu$, $V$, $E$ fluctuate (models system in thermal contact with fixed-temperature bath and with a movable wall)

  • Grand canonical ensemble: $\mu$, $V$, $T$ fixed, while $N$, $p$, $E$ fluctuate (models open system in contact with fixed-temperature bath and exchanging with a particle reservoir)

You have chosen to talk about the isothermal-isobaric ensemble. Keeping that in mind, addressing your questions:

1) The volume of the box fluctuates because that's part of the definition of the isothermal-isobaric ensemble. In intuitive terms, keeping the pressure in the box constant is accomplished using a movable wall that reacts to any internal energy fluctuations by moving outward and inward.

2) The case where the box has fixed volume, temperature, and particle number is the canonical ensemble. It's a different object that is constructed to model a different physical situation.

3) The quantities that are held constant in the ensemble definition are held constant. (I apologize for the tautology.) Which state variables are held constant depends on which ensemble you're talking about.

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  1. Suppose we fix $T$, $P$ and $N$. Imagine the gas to be inside a very elastic balloon in thermal and mechanical equilibrium with the surroundings (say, the atmosphere). Neither the energy nor the volume of this system are fixed, therefore they can fluctuate around their average values $\bar E$ and $\bar V$.
  2. In ensembles we fix one variable from each pair of extensive/intensive from the set $(U,T), (p, V), (\mu_i,n_i)$. If we fix an extensive variable, the corresponding intensive variable appears as a Lagrange multiplier. If we fix an intensive property, the corresponding extensive variable fluctuates. The extensive property is never obtained as a Lagrange multiplier.

  3. Lagrange multipliers do not fluctuate, and that's a consequence of the way we construct the ensembles. To construct an ensemble at fixed intensive property ($p$, for example), we fix the mean of the corresponding extensive variable ($\bar V$) and collect microstates with all possible volumes as long as their mean is $\bar V$. This in turn fixes $p$ (through a Lagrange multiplier) and defines it as a constant for all microstates in this ensemble.

On Ensembles and Fluctuations

Fluctuations are associated with extensive properties. In the usual derivation of the canonical ensemble we consider a large number of systems with fixed total energy $E_\text{tot}$ and assign different microstates to each system under the condition that the total energy is always $E_\text{tot}$. Each assignment of microstates represents a distribution of energies, and a distribution represents "fluctuations," i.e., not all systems have the same energy. In this picture fluctuations are associated only with extensive properties because only extensive properties can be partitioned such that their total is fixed.

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