Isobaric-Isothermal ensemble
Note that allowing other constraints (like volume fluctuation) would add new lagrange multipliers appearing in the exponential of the expression of $p_i$.
The isobaric-isothermal ensemble, where both $U$ and $V$ are constrained to a mean value. It is easy to show.
\[\boxed{p_i = \frac{1}{\Delta} e^{-\beta E_i + P V_i}}\]Where $\Delta$ is the Isobaric-Isothermal partition function
\[\boxed{\Delta=\sum_i e^{-\beta E_i + P V_i}}\]We will not cover in detail this ensemble here, as all proof are very similar to the ones done for previous ensembles.
Overview of the ensembles
All ensembles have
\[p_i = \frac{\text{Weight}}{\text{Partition function}}\]| Ensemble | Fixed Variables | Probability Weight ($p_i \propto \dots$) | Partition Function | Thermodynamic Potential |
|---|---|---|---|---|
| Microcanonical | $N, V, E$ | $1$ | $N$ or $\Omega$ | Energy $U$ or entropy $S = \ln \Omega$ |
| Canonical | $N, V, T$ | $e^{-\beta E_i}$ | $Z = \sum_i e^{-\beta E_i}$ | Free Energy: $A = - T \ln Z$ |
| Grand Canonical | $\mu, V, T$ | $e^{-\beta(E_i - \mu N_i)}$ | $\Xi = \sum_i e^{-\beta(E_i - \mu N_i)}$ | Grand Potential: $\Phi_G = - T \ln \Xi$ |
| Isobaric-Isothermal | $N, P, T$ | $e^{-\beta(E_i + PV_i)}$ | $\Delta = \sum_i e^{-\beta(E_i + PV_i)}$ | Gibbs Free Energy: $G = - T \ln \Delta$ |
Bla
| Ensemble | Definition | Differential Form ($d\Phi$) |
|---|---|---|
| Microcanonical | $S = \ln \Omega$ | $\text{d}S = \frac{1}{T}dU + \frac{P}{T}\text{d}V - \frac{\mu}{T}\text{d}\mathcal{N}$ |
| Canonical | $A = U - TS$ | $\text{d}A = -S\text{d}T - P\text{d}V + \mu \text{d}\mathcal{N}$ |
| Grand Canonical | $\Phi_G= A - \mu N$ | $\text{d}\Phi_G = -S\text{d}T - P\text{d}V - N\text{d}\mu$ |
| Isobaric-Isothermal | $G = A + PV$ | $\text{d}G = -S\text{d}T + V\text{d}P + \mu \text{d}\mathcal{N}$ |