Isobaric-isothermal ensemble

Figure: A system kept at constant pressure and temperature in the heat bath. The Volume $V$ and the energy $E$ fluctuate.

In the microcanonical ensemble the system has fixed energy $E$, volume $V$ and number of particles $N$. These are all quantities private to the system that are not exchanged with any reservoir. In the canonical ensemble we allow the energy to be exchanged with a heat bath, and the quantities that remain private to the system are the volume $V$ and the number of particles $N$. The temperature $T$ is also fixed, but it is not private, as it is determined by the heat bath. In the gran canonical ensemble we removed also the constraint on the number of particles, that can now be exchanged with the reservoir. The corresponding variable that is kept fixed is the chemical potential $\mu$, which again is determined by the reservoir, and the only variable that is still private to the system is its volume $V$. If we continue our generalisation and allow also the volume to change, in response to a pressure difference between the system and the reservoir, we lose the last variable private to the system, meaning that there is nothing we can use to identify the system and tell it apart from the reservoir, as all variables $T$, $\mu$ and $P$ are set by the reservoir. Note that these are all intensive variables, and so there is nothing to tell how big the system is, nothing to confine it. We therefore need to have at least one extensive variable still in play. Our last useful choice then would be to allow energy exchanges and volume changes but to keep the number of particles constant. This is the isothermal-isobaric ensemble, characterised by constant $N,P,T$. Here $N$ is private to the system, and $P$ and $T$ are set by the reservoir by the condition of equilibrium. Following on the discussions of the previous sections it will be no surprise to find that the isothermal-isobaric probability distribution for the microstates is:

$$p_{V,r} =\frac{1}{\Delta} \exp \{ -\beta(PV+E_{V,r})\}$$ (3.54)

and the partition function is

$$\Delta(N,P,T) = \sum_V \sum_r \exp \{-\beta (PV+ E_{V,r})\}= \sum_V \exp\{-\beta P V\} Z(N,V,T),$$ (3.55)

which is a function of $P,T$ and $N$. Note that here the energies now depend on the volume, $E_{V,r}$, and we have assumed that also the volume can only change by discontinuous amounts. By repeating the analysis of the previous sections we also find that the average volume is given by

$$\bar{V} = -\frac{1}{\beta}\frac{\partial \ln \Delta}{\partial P}.$$ (3.56)

This volume is proportional to the number of particles for a system held at constant pressure and therefore it is an extensive quantity, meaning that also $\ln \Delta$ is an extensive quantity. The average quadratic fluctuation of the volume around its average value, $(\delta V)^2$ can be obtained by taking the second derivative of $\ln \Delta$ w.r.t. $P$, and as such that is also extensive, meaning once again that $\delta V/ V \propto N^{-1/2}$, and so for a macroscopic system held at constant pressure the volume is also effectively constant.

As for the Boltzmann and the Gibbs probabilities, if the energy levels $E_{V,r}$ have degeneracy $\Omega(E_{V,r})$, and $S(E_{V,r}) = k_{\rm B}\ln \Omega(E_{V,r})$, the probability for the system to have energy $E_{V,r}$ is:

$$p(V,E_{V,r}) =\frac{1}{\Delta} \exp \{ -\beta[PV+ E_{V,r} - TS(E_{V,r})]\}.$$ (3.57)