# Classical and Quantum Gases on a Semiregular Mesh
## 2 Lattice-Gas Models on a Spherical Mesh
### Finite-Temperature Behavior
This means that for \(\mu=0\) the system has equal probability to be in either of the two phases. Hence, no hysteresis would be observed at the empty-ICO transition, as indeed found. Now consider the ICO-DOD transition at \(\mu^{*}=15/8\). Starting from ICO, the cost to annihilate a particle is \(\Delta H^{*}=15/8\) (since \(\Delta E=0\)); the next step of creating a particle in a dodecahedral site has a minimum cost of \(\Delta H^{*}=31/8\) (since \(\Delta E^{*}\geq 2\)). Thus, for \(\mu^{*}=15/8\) there is a free-energy barrier for the transition to DOD, and the system then remains for long in ICO notwithstanding DOD is more stable. If we instead start from DOD, we should first annihilate a particle (\(\Delta H^{*}=3/8\)) and then create another particle in an icosahedral site (\(\Delta H^{*}\geq 17/8\)). Again, the transition from DOD to ICO is discouraged at low \(T\) (though apparently less so than the opposite transition), meaning that for \(\mu^{*}=15/8\) the system is more likely to be found in DOD than in ICO. As a result, hysteresis will be observed at the ICO-DOD transition. Figure 3: Lattice gas on a PD mesh, for \(\gamma=1/2\) and \(T^{*}=0.1,0.2,\ldots,0.5\). **Top left**: number of particles. **Top right**: reduced energy. **Bottom left**: icosahedral OP. For the lowest temperatures, MC data refer to two distinct sequences of runs where \(\mu^{*}\) is respectively increased or decreased in steps of 0.1 (\(T^{*}=0.1\), blue and cyan dots; \(T^{*}=0.2\), emerald and green; \(T^{*}=0.3\), red and pink; \(T^{*}=0.4\), brown; and \(T^{*}=0.5\), black). For all temperatures but the lowest one, MC data are reported as lines. Hysteresis is evident for \(T^{*}=0.1\) and barely visible already for \(T^{*}=0.2\). Lastly, we consider the DOD-full transition at \(\mu^{*}=5\). If we start from DOD, the first step towards \"full\" is creating a particle in an icosahedral site (\(\Delta E^{*}=5\)) and the probability for this move is one. If we instead start from \"full\", the cost for annihilating a particle in an icosahedral site is zero again, since \(\Delta E^{*}=-5\). Indeed, no hysteresis is detected at the DOD-full transition. In Figure 4 we plot a few response functions for \(\gamma=1/2\). At the lowest temperatures all these quantities exhibit a distinct peak near each \(T=0\) transition point, which is where the energy and particle number are subject to the sharpest variations. In the \"empty\" phase, the reduced isothermal compressibility is close to 1 (the ideal-gas value); upon heating, every asperity in its profile becomes smoothened until \(\rho K_{T}\) becomes a monotonously decreasing function of \(\mu\). As \(T\) grows, both specific heats develop a broad maximum inside the ICO and DOD regions. Admittedly, these are the locations where, in the moderately hot system, the fluctuations of energy and particle number are stronger. Eventually, both maxima gradually deflate, the DOD bump being the last to disappear. Finally notice that inside the \"empty\" region the two specific heats have different behaviors at low temperature: while the covariances involving energy are practically zero, the mean square fluctuation of \(N\) is of order \(N\) (see Equation (2)). Looking at Equations (A6) and (A15), we thus conclude that \(C_{\mu}/k_{\rm B}\approx(\beta\mu)^{2}\gg 1\) and \(C_{N}\approx 0\). As a second example, consider \(\gamma=1\) and \(T^{*}=0.05\) close to \(\mu^{*}=9/2\)--which is where, at \(T=0\), ICO+CUB transforms into ICO+COC. By the same argument put forward before, we would conclude that this transition is accompanied by hysteresis at low temperature. However, when plotting \(N\) as a function of \(\mu\), in a small neighborhood of \(\mu^{*}=9/2\) we see a narrow plateau in the middle between the \(N\) levels in the two phases (top-left panel of Figure 5), and a similar occurrence is found for \(E\) (not shown). This unexpected outcome would suggest the existence of a yet undetected phase near \(\mu^{*}=9/2\), characterized by \(N=22\) and \(E^{*}=33\).