# Classical and Quantum Gases on a Semiregular Mesh
## 2 Lattice-Gas Models on a Spherical Mesh
### Finite-Temperature Behavior
\tag{6}\]  
In the previous equation,  
\[\pi(c;\rho_{\rm A})=\rho_{\rm A}\delta_{c,1}+(1-\rho_{\rm A})\delta_{c,0}, \tag{7}\]  
with \(0\leq\rho_{\rm A}\leq 1\), and similarly for B and C. The rationale behind Equation (6) is that the average occupancy takes a possibly different value in each set of equivalent nodes, being \(\rho_{\rm A}\) for the icosahedral set, \(\rho_{\rm B}\) for the cubic set, and \(\rho_{\rm C}\) for the co-cubic set. An upper bound to the exact grand potential \(\Omega\) is the GB grand potential \(\Omega^{*}\),  
\[\Omega^{*}=\left\langle H\right\rangle+k_{\rm B}T\langle\ln\pi\rangle\,, \tag{8}\]  
Figure 6: MC results for \(\gamma=1/2\) and \(T^{*}=0.1,0.2\), and \(0.3\). The colored points and lines are the same MC data shown in Figure 3 (\(T^{*}=0.1\), blue and cyan dots; \(T^{*}=0.2\), emerald and green squares; \(T^{*}=0.3\), red and pink). The superimposed black lines are the outcome of a Wang-Landau simulation. **(Left)**: average number of particles. **(Right)**: average energy. Figure 7: Wang-Landau simulation for \(\gamma=1/2\) and \(T^{*}=0.2\). **(Left)**: logarithm of the density of states \(g\), defined in terms of \(N\) and \(U\) (see text). The four points where \(g=1\) correspond to the \(T=0\) phases: “empty”, perfect ICO, perfect DOD, and “full”. **(Right)**: probability function of the particle number \(N\), plotted as a function of the reduced chemical potential. where \(\langle O[c]\rangle=\sum_{\{c\}}\pi[c]O[c]\). Then, it is a simple matter to show that  
\[\Omega^{*} = 24V_{1}\rho_{\rm A}\rho_{\rm B}+36V_{1}\rho_{\rm A}\rho_{\rm C}+24 \gamma V_{1}\rho_{\rm B}\rho_{\rm C}+6\gamma V_{1}\rho_{\rm C}^{2}-12\mu\rho_{\rm A }-8\mu\rho_{\rm B}-12\mu\rho_{\rm C} \tag{9}\] \[+12k_{\rm B}T[\rho_{\rm A}\ln\rho_{\rm A}+(1-\rho_{\rm A})\ln(1- \rho_{\rm A})]+8k_{\rm B}T[\rho_{\rm B}\ln\rho_{\rm B}+(1-\rho_{\rm B})\ln(1- \rho_{\rm B})]\] \[+12k_{\rm B}T[\rho_{\rm C}\ln\rho_{\rm C}+(1-\rho_{\rm C})\ln(1- \rho_{\rm C})]\,.\]  
Observe that the value of \(\Omega^{*}\) in the putative ground states listed in Section 2.1 exactly reproduces their respective grand potentials. The stationary values of (9) fulfill the coupled equations  
\[\rho_{\rm A} = \frac{1}{e^{\beta(2V_{1}\rho_{\rm B}+3V_{1}\rho_{\rm C}-\mu)}+1}\,\ \ \rho_{B}=\frac{1}{e^{\beta(3V_{1}\rho_{\rm A}+3\gamma V_{1}\rho_{\rm C}-\mu)}+1 }\,\ \ {\rm and}\] \[\rho_{\rm C} = \frac{1}{e^{\beta(3V_{1}\rho_{\rm A}+2V_{1}\rho_{\rm B}+\gamma V_{ 1}\rho_{\rm C}-\mu)}+1}\,. \tag{10}\]  
These equations are solved numerically, seeking the solution that provides the absolute minimum \(\Omega^{*}\) for the given \(T\) and \(\mu\). To have a flavor of how MF theory works, we draw in Figure 8 the theoretical phase diagram on the \(\mu\)-\(T\) plane for \(\gamma=2/3\), corresponding to a lattice gas where the icosahedral phase covers a \(\mu\) range as wide as that of the dodecahedral phase, see Figure 2. Owing to a symmetry property of Equation (10), the phase diagram is symmetric around \(\mu=5/2\) (see also Section 3.2). At \(T=0\) the phase boundaries in Figure 8 are exact. As \(T\) grows, the theory predicts a gradual weakening of ICO and DOD orders, as witnessed by the decrease of \(|\rho_{\rm A}-\rho_{\rm B}|\) on heating, eventually resulting in an abrupt (first-order) transition to either \"empty\" (\(\rho_{\rm A}=\rho_{\rm B}=\rho_{\rm C}<0.5\)) or \"full\" (\(\rho_{\rm A}=\rho_{\rm B}=\rho_{\rm C}>0.5\)), according to whether \(\mu<5/2\) or \(\mu>5/2\). Clearly, this singularity is an artifact of MF theory, since no sharp transition is present in our system for \(T>0\). Another unphysical prediction of the theory concerns the behavior of the model in a narrow strip of temperatures and chemical potentials near \(\mu^{*}=0\) and \(\mu^{*}=5\). In Figure 8 we have denoted ICO\({}^{\prime}\) a phase where \(0.5>\rho_{\rm A}>\rho_{\rm B}=\rho_{\rm C}\) and DOD\({}^{\prime}\) a phase where \(0.5<\rho_{\rm A}<\rho_{\rm B}=\rho_{\rm C}\).