Grand canonical ensemble

The grand canonical ensemble treats the system as a collection of microstates.

Unlike the canonical ensemble, the grand canonical ensemble allows the local compositional fluctuation.

For a binary system \(A_{x}B_{1-x}\) or a pseudobinary system \(A_{x}B_{1-x}C\) , the grand canonical partition function, Z, is calculated as follows:

\[Z = \sum_{\sigma}\exp\left({ -\frac{\Delta E ^{total} _\sigma - \Delta \mu N_A }{k_B T} }\right)\]

, where \(N_A\) is the number of atom A in \(A_{x}B_{1-x}C\), and

\[\Delta \mu = \mu _A - \mu _B\]

\(\sigma\) indicates a configuration (microstate). \(\mu\) is a chemical potential of a species of atom. Total mixing energy \(\Delta E ^{total}\) is the sum of freely relaxed energy and the strain energy induced by compositional fluctuation.

\[\Delta E ^{total} _\sigma =\Delta E _\sigma + E^{strain}\]

The probability of a configuration is calculated as:

\[P _\sigma = \exp\left( -\frac{\Delta E ^{total} _\sigma - \Delta \mu N_A }{k_B T} \right) / Z\]

Then the entropy of this system is

\[S= \sum _\sigma P _\sigma \ln P _\sigma\]

The ensemble average of a property of interest, Y is given as:

\[\bar Y = \sum_{\sigma} Y _{\sigma} P _{\sigma}\]

When the Y is \(\Delta E\), the average energy is calculated. With the entropy \(S\), the free energy \(F\) at a finite temperature T can be calculated as:

\[ \begin{align}\begin{aligned}S= -k_B \sum _\sigma P _\sigma \ln P _\sigma\\\Delta F= \Delta E - T \Delta S\end{aligned}\end{align} \]

\(\Delta \mu\) is determined by following equation to satisfy the self-consistency for composition.

\[x = \sum_{\sigma} x _\sigma P _\sigma\]

A detailed explaination is in this paper: J. Phys. D: Appl. Phys. 54, 045104 (2021)