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From the first law of thermodynamics relation and using Legendre Transform of the Internal Energy, we can possibly define 8 Thermodynamic potentials out of which we know 5. I could not find any names of the other 3 and any use thereof. Also, for the microcanonical ensemble, we devise $S=S(E,V,N)$ and precisely $E,V,N$ are the parameters held fixed in the formulation of the ensemble. This property seems to be true for all the 5 known potentials (for instance, in the grand potential, we derive $\phi=\phi(T,V,\mu)$ which are the quantities held fixed in the grand canonical ensemble). So for the other 3 "not known" potentials, can we cook up other ensembles and if there are any, what are there uses? In the image, I have put $K_i$ for ease of referencing while answering.

List of potentials and related ensembles with the natural variables

Qmechanic
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quirkyquark
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    If they have names then those names are not widely known so you don't need the names if your aim is purely to learn physics. – Andrew Steane Jul 14 '21 at 16:40
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    $_3$ defines a very weird potential that is always null and its equations are equivalent to Gibbs–Duhem equation. Also, the ensemble for $_3$ is very restrictive. It would be equivalent to a system where the flux of particles, temperature and pressure are fixed by the environment. But apparently it can be of use when various species are involved, see this question https://physics.stackexchange.com/questions/26873/mu-p-t-pseudo-ensemble-why-is-it-not-a-proper-thermodynamic-ensemble?rq=1 – Mauricio Jul 14 '21 at 18:32
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    Note that this table can get infinitely more complicated if you add magnetic fields or different species $\mu_1 N_1,\mu_2 N_2,\cdots$ – Mauricio Jul 14 '21 at 18:37
  • @Mauricio Actually while making the Legendre transform wrt $\mu N$, I meant $\sum_i \mu_i N_i$. But what you say is also on point. – quirkyquark Jul 15 '21 at 16:39
  • I have updated my answer as a consequence of some additional information about the IUPAP/IUPAC convention about the names of a couple of Massieu functions. – GiorgioP-DoomsdayClockIsAt-90 Jul 18 '21 at 16:39

1 Answers1

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The correspondence between ensembles and relevant thermodynamic potentials requires a partial correction.

Let me start with a general remark about names. Within thermodynamics, the generic name for a function of state embodying the full information about the system is fundamental equation. Energy as a function of $S,V,N$, is a fundamental equation. Moreover, energy and its Legendre transforms are collectively known as thermodynamic potentials.

However, there are fundamental equations that are not energy or any of its Legendre transforms. This is the case of the entropy as a function of energy $U$, $V$, and $N$ and its Legendre transforms, sometimes called Massieu functions (see for instance H.B.Callen's textbook Thermodynamics and an introduction to thermostatistics). There is no uniform conventional symbol (or specific names) for the Massieu functions, although the chemistry community (IUPAC) and IUPAP chose symbols and name for a couple of them. Sometimes they have been indicated as $S_1$, $S_2$, and so on. However, on the one hand, it is worth recalling that also thermodynamic potentials lack a completely uniform notation. Internal energy is sometimes indicated as $E$, sometimes as $U$. The Helmholtz free energy may be indicated as $F$, $A$, or $\Phi$, to cite the most frequent notations. This status of things requires checking the meaning of symbols carefully in a paper or a book. On the other hand, the Massieu functions can be written in terms of thermodynamic potentials, although as functions of different variables.

Thermodynamic potentials are better suited for discussing experimental data. Entropy-based fundamental equations (the Massieu functions) are the natural environment for statistical mechanics. This should not be a surprise since measurements of energy transfers are typically accessible in the experiments. At the same time, statistical mechanics was built around the concept of counting the number of states.

Therefore we have two tables, one more relevant for statistical mechanics and another more relevant for thermodynamics. I show in the following the most important entries of both. Notice that the natural variables of the usual statistical mechanics ensembles are those pertaining to entropy and Massieu functions. No usual ensemble in statistical mechanics has entropy among its natural variables.

Entropy and Massieu functions

Name/symbol of the fundamental equation natural variables ensemble
entropy/$S$ $U,V,N$ microcanonical
Massieu function$^{(*)}$/$J^{(*)}=S_1=S-\frac{U}{T}=-\frac{F}{T}$ (?, after division by $k_B$, it is the Helmholtz free-energy in units of $k_BT$, changed of sign) $\frac{1}{T},V,N$ canonical
Planck function$^{(*)}$/$Y^{(*)}==S_3=S-\frac{PV}{T}-\frac{U}{T}=-\frac{G}{T}$ (?,after division by $k_B$, it is the Gibbs free-energy in units of $k_BT$, changed of sign) $\frac{1}{T},\frac{P}{T},N$ isothermal-isobaric
?/$S_4=S-\frac{U}{T}+\frac{\mu N}{T}=\frac{PV}{T}$ (?,after division by $k_B$, it is the grand potential, in units of $k_BT$, changed of sign) $\frac{1}{T},V,\frac{\mu}{T}$ isothermal-isochoric

$^{(*)}$: according to the IUPAC document Quantities, Units and Symbols in Physical Chemistry (section 2.1) and to the IUPAP-SUNAMCO Red book (section 4.4).

Other Massieu functions and the corresponding ensembles can be conceived, and occasionally have been used in computer simulations.

Thermodynamic potentials

Name/symbol of the fundamental equation natural variables
internal energy/$U$ $S,V,N$
Helmholtz free energy/$F=U-TS$ $T,V,N$
enthaply/$H=U+PV$ $S,P,N$
Gibbs free energy/$G=U-TS+PV=\mu N$ $T,P,N$

A final comment is about the special cases of the Massieu function obtained as Legendre transform of entropy with respect to all its natural variables: $$ \zeta=S-\frac{U}{T}-\frac{PV}{T}+\frac{\mu N}{T} $$ and the thermodynamic potential $$ Z=U-TS+PV-\mu N. $$ As a consequence of the homogeneity of degree one of $U$ and $S$, as functions of their own variables, it should be $\zeta = Z=0$. For this reason, somebody called $Z$ the "zero"-thermodynamic potential. However, in real conditions these fundamental equations are not useless. Their theoretical vanishing is just a consequence of the exact homogeneity predicted at the thermodynamic limit. For finite-size systems, the "zero"-potential does not vanish exactly, due to sub-dominant contributions, like the surface terms. Therefore, they can be used as a magnifying glass for surface thermodynamics.

Crisco
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GiorgioP-DoomsdayClockIsAt-90
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    The last point about the zero-potential being zero only in the thermodynamic limit is a nice point, something that I feel people often forget. – quirkyquark Jul 15 '21 at 16:37
  • If you are in the zero-ensemble of the zero-potential, we have indeed $Z=0$. Only in the case where you are in another ensemble this potential is not zero. Similar to how the grand canonical potential $\Omega\to-PV$ in the thermodynamic limit only, unless you are already in the grandcanonical ensemble then $\Omega=-PV$. – Mauricio Jul 15 '21 at 16:51
  • @Mauricio For finite systems, $Z\neq 0$. The Euler relation holds only if the internal energy is a homogeneous function of degree 1 of its natural variables. This would imply an exact scaling of $U$ like the volume. However, it is well known that apart from the trivial case of the ideal gas, finite-size systems introduce surface terms contributions making non-zero the zero-potential. Notice that this is a thermodynamic argument. It can be translated into Statistical mechanics, but in that case, one should start from the expression of the probability density in the extended phase space. – GiorgioP-DoomsdayClockIsAt-90 Jul 15 '21 at 20:56
  • @GiorgioP What do you mean by extending the probability density to the extended phase space? Also I don't get what the surface terms (lets call them $\gamma S$) have to do here. You can always create a new $Z'=Z-\gamma S$, and remove that contribution and create a new zero potential. – Mauricio Jul 17 '21 at 18:06
  • @Mauricio I mean that in the vase of a fixed pressure ensemble, the probability distribution will depend on pressure, in addition to coordinates and momenta. Similarly for a fixed chemical potential ensemble, there will be a dependence on chemical potential. About your $Z'$, its introduction is formally feasible, but there is no proof that it should be exactly zero for finite-size systems. You should remember that sub-dominant finite-size corrections remain even in bulk systems treated with period boundary conditions (i.e. without an explicit boundary surface). I would expect them even with Z' – GiorgioP-DoomsdayClockIsAt-90 Jul 17 '21 at 18:43
  • @GiorgioP I agree, but just to be clear, would you agree that in the grand canonical ensemble $\Omega=-PV$ no matter if it is a macroscopical or a finite system? I was trying to make an argument from the statistical side more than on the thermodynamical side. – Mauricio Jul 17 '21 at 18:58
  • @Mauricio I agree, but for the identification of $\Omega = \log \Xi$, where $\Xi $ is the grand canonical partition function, with the thermodynamic $PV$, there is a subtle point, again connected to finite-size effects and the to the need of thermodynamic limit to get the usual thermodynamic behavior. The main issue is about the properties of such $P$. However, it is neither simple nor appropriate to discuss it in the space of the comments. I will consider a self-answered question to explain all details. Maybe I'll have time tomorrow. – GiorgioP-DoomsdayClockIsAt-90 Jul 17 '21 at 19:49