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I know that there exists a pressure ensemble (NpT): given inverse temperature $\beta$ and pressure $p$, this ensemble has an equilibrium probability distribution \begin{align} \rho(\Gamma) = e^{\beta(H(\Gamma, V) - pV}/Z \end{align}

Even if this ensemble is derived only from information theoretical ponderings (see for example https://bayes.wustl.edu/etj/articles/theory.1.pdf) on an ensemble whose average enery and volume are fixed,we get two lagrangian multipliers $\beta$ and $\beta p$, and we can identify p to be the pressure by the equation

\begin{align} p = - \langle\frac{\partial H}{\partial V} \rangle \end{align} (Which only holds in equilibrium).

This ensemble is usually referred to as an ensemble with fixed pressure p (see for example this answer: https://physics.stackexchange.com/a/446947/102243). However, I don't understand what we mean by "fixed" in this case. At least "fixed" is not used in the same way we that we use it when we talk about Energy or Volume. A fixed Energy ensemble (microcanonical ensemble) is an ensemble where energy is constant for every allowed microstate. Similar, a fixed volume ensemble (canonical ensemble) doesn't have a varying volume for any possible microstate of the system.

As an analogue I now ask: What is the fixed quantity, fixed in the sense that it doesn't change for any of the microstates, for the NPT ensemble?

We know that $p = - \langle\frac{\partial H}{\partial V} \rangle$, so p is some average of some function of te microstates (some "Observable"),but it isn't a given that this expectation value is sharp, in eiter of the ensembles.

So - is $\frac{\partial H}{\partial V}$ a sharp quantity in the NPT ensemble (I doubt it)? If it isn't, in what sense is pressure "fluctuating" more in the Volume ensemble, than it is in the pressure ensemble? Is there even any ensemble with sharp $\frac{\partial H}{\partial V}$?

One answer made a distinction between $\frac{\partial H}{\partial V} and p, and stated that the former would fluctuate in most ensembles. Is there however a way one can describe an ensemble with fixed V with a "let loose" parameter p?

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Your doubts could also apply to the temperature in the canonical ensemble: on the one hand, it is a fixed parameter characterizing the thermostat. On the other hand, we can obtain it from the average kinetic energy of the system. The key concept is that the thermostat parameter is a constant non-fluctuating quantity, while the system's kinetic energy is fluctuating.

The situation for the fixed pressure ensemble is the same. In addition to the thermostat (fixed temperature), there is a barostat, i.e., a surrounding characterized by a fixed pressure value. In addition to such external parameter, it is always possible to measure the system pressure as the average of the fluctuating $-\frac{\partial H}{\partial V}$. It should coincide with the external pressure at equilibrium, although it is not a fixed quantity.

Notice that $-\frac{\partial H}{\partial V}$ is a fluctuating observable both in the canonical and isothermal-isobaric ensemble. However, its fluctuations do not need and are not the same in the two ensembles.

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  • $\begingroup$ Yes, that's what I thought as well (fixed pressure simply meaning that there is a barostat given by the parameter p). You are right, I could have asked the same for the temperature, but I didn't, because (unlike pressure) I am not in general able to express it as the mean of some microscopic observable. To formulate my question a bit looser: Is there any sense in which pressure fluctuates more in the canonical ensemble, compared to the isothermal-isobaric ensemble? $\endgroup$ Jan 6, 2022 at 10:32
  • $\begingroup$ @Quantumwhisp as far as temperature, it is not difficult: $T_{system}=\frac{2}{3k_B}\frac{1}{(N-1)}\langle K \rangle$. Why pressure should fluctuate less in the isothermal-isobaric? A handwaving argument is that the variation of volume may absorb part of the effect of the fluctuations due to the varying interparticle distances. A more sound statement would require a detailed comparison of the fluctuations in the two ensembles. $\endgroup$ Jan 6, 2022 at 11:24
  • $\begingroup$ for the temperature, the formula you have given only holds for the canonical ensemble, no? Since $\frac{\partial H}{\partial V}$ fluctuates in both ensembles, and since you made the distinction between p and $\frac{\partial H}{\partial V}$, is there any way in which one could talk about a "fluctuating p" ? (I'm asking, knowing that p is a parameter here). I deliberately try to carry on the fluctuating vs fixed properties of V, N, E (and so on) to p, $\beta$, $\mu$, this is basically where I'm coming from. $\endgroup$ Jan 6, 2022 at 18:25
  • $\begingroup$ @Quantumwhisp the barostat $p$ is fixed. The configurational value of $-\frac{\partial H}{\partial{V}}$ is configuration dependent and therefore, fluctuating. $\endgroup$ Jan 6, 2022 at 23:45
  • $\begingroup$ I gave it a thought, and what I'm looking for to be fluctuating in the fixed V ensemble is exactly the barostat. Fixed V mean I arranged a system in which the wall that borders the volume always stays were it is, which means that any Force stemming from the system ( $\frac{ - \partial H}{\partial V}$ will be cancelled out completely by an equal opposite $p$. $\endgroup$ Jan 13, 2022 at 11:13
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e.g. a gas in a cylinder with a movable piston, where the force on the outside of the piston is held constant. In this situation the volume and temperature of the gas can fluctuate together but the pressure will stay constant. If the pressure were to start to increase, the piston would move out.

Another way is to have a pressure sensor and implement a servo control.

Having said all that, pressure is not a conserved quantity like energy and volume so it is harder to keep it constant. There can still be fluctuations where the pressure goes up at one point and down at another.

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