## Appendix C Ensemble Representation of Thermodynamic Conditions
}}=\mathcal{S}\), at the grand-potential minimum:  
\[\min_{\hat{\text{D}}}\mathbf{\Omega}\mathbf{\Omega}\text{=}\mathbf{\Omega}[\text{D}(\mu, T;\mathbf{v})]\equiv\mathbf{\Omega}(\mu,T;\mathbf{v}). \tag{108}\]  
The externally imposed parameters (\(\mu\), \(T\)) then determine the optimum probabilities of the ensemble stationary states, eigenstates of the Hamiltonians \(\{\hat{\text{H}}(N_{i},\mathbf{v})\}\), in the (mixed) equilibrium state of the grand ensemble,  
\[P_{j}^{\,i}(\mu,T;\mathbf{v})=\Xi^{-1}\exp[\beta(\mu N_{i}-E_{j}^{\,i})], \tag{109}\]and the associated density operator  
\[\hat{\mathsf{D}}(\mu,T;v)=\sum_{i}\sum_{j}|\psi_{j}{}^{i}\rangle P_{j}^{i}(\mu,T; v)\langle\psi_{j}{}^{i}|\equiv\hat{\mathsf{D}}_{\text{eq}}. \tag{107}\]  
Here, \(\Xi\) stands for the grand-ensemble partition function and \(\beta=(k_{\text{B}}T)^{-1}\). The equilibrium probabilities of Equation (106) represent eigenvalues of the grand-canonical statistical operator acting in [PERSON]'s space:  
\[\hat{\mathsf{d}}(\mu,T;v)=\exp\{\beta[\mu\hat{\mathsf{N}}-\hat{\mathsf{H}}(v)] \}/\text{tr}\{\beta[  
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