Thermodynamic potential and partition function

Question

I am a bit confused by the relation between thermodynamic potential and partition functions.

From my understanding, we can generate all thermodynamical quantities by taking partial derivatives to the thermodynamic potential (or $\ln(Z)$ where $Z$ is the partition function). Thus, I was expecting them to be proportional to each other.

For canonical ensembles, the partition function is $Z=\sum e^{-\beta E}$ and the corresponding thermodynamic potential is the Helmholtz free energy $F=-k_BT\ln(Z)$.

Similarly, for the grand canonical ensembles, the partition function is $Z=\sum e^{-\beta E-\alpha N}$ and the corresponding thermodynamic potential is the grand potential $J=-k_BT\ln(Z)$.

However, for microcanonical ensemble, it does not seem to work anymore. The partition function $Z=\sum1=\Omega$ so it will be the entropy which is proportional to $\ln(Z)$. However, the thermodynamic potential is the internal energy $U$ in this case. It seems a bit wired to me.

I am wondering whether there is some deep reason behind this? Thanks!

Answer 1

The thermodynamic potential for microcanonical ensemble should be $S$ instead of $U$, for two reasons: first, by definition microcanonical ensemble describes an isolated system whose energy is conserved. So in this sense, $U$ is treated like an external condition (like the temperature of a canonical ensemble). Second, thermodynamic potentials satisfy extremal principles: that is, the thermodynamic equilibrium state is the minimum/maximum of the potential. The entropy $S$ does exactly that for isolated systems.

Answer 2

Strictly speaking, thermodynamic potential should be a name reserved for the internal energy as a function of the extensive variables ($S,V,N$, for simple systems) and its Legendre transforms (Helmholtz free energy, enthalpy, Gibbs free energy, to stick on the most well known). Thermodynamics allows for other fundamental equations (i.e. those functions of the state which encode the whole thermodynamic information about a system), originating from the entropy as a function of its extensive variables. Legendre transforms are collectively known as Massieu's functions (see Callen's textbook on thermodynamics).

Even if it is not widely appreciated, the natural set of fundamental equations for statistical mechanics is not the set of thermodynamic potentials, but the set of Massieu functions. In therm of them, each fundamental equation of the different ensembles displays the same relation with the appropriate partition function.

Indeed, starting with the entropy $S(U,V,N)$, we have the following Massieu functions relevant for the canonical and grand-canonical ensemble: $$ \begin{align} S_1\left(\frac{1}{T},V,N\right) &= S-\frac{U}{T}=-\frac{F}{T}\\ S_2\left(\frac{1}{T},V,\frac{\mu}{T}\right) &= S-\frac{U}{T}+\frac{\mu}{T}N=\frac{PV}{T}=-\frac{J}{T}. \end{align} $$ It is immediate to see that in terms of these functions, the relation between partition function and thermodynamics is uniform: $$ \begin{align} S&= k_B \ln \Omega\\ S_1&= k_B \ln Z\\ S_2&= k_B \ln \Xi \end{align} $$ where $\Xi$ is the grand-canonical partition function.