# Classical and Quantum Gases on a Semiregular Mesh
## 2 Lattice-Gas Models on a Spherical Mesh
### Finite-Temperature Behavior
In fact, when the length of the MC trajectory is increased by a factor of 10 the plateau at \(N^{*}=22\) changes to a more or less smooth crossover between the phases (top-right panel of Figure 5). To clarify things better,  
Figure 4: Lattice gas on a PD mesh, for \(\gamma=1/2\) and \(T^{*}=0.1,0.2,\ldots,0.5\). **Top**: reduced compressibility. **Bottom left**: Constant-\(\mu\) specific heat. **Bottom right**: Constant-\(N\) specific heat. Symbols and notation as in Figure 3.  
we have enumerated the microstates with all icosahedral sites occupied, thus confirming that ICO+CUB (ICO+COC) is the stable phase for \(\mu^{*}<9/2\) (\(\mu^{*}>9/2\)); instead, exactly for \(\mu^{*}=9/2\) we have counted as many as 240 distinct microstates, all with \(N=22\) and \(E^{*}=33\), having the same grand potential as ICO+CUB and ICO+COC. It is the existence of such configurations that makes the transition between ICO+CUB and ICO+COC smoother than expected. We care to stress that this occurrence is rather exceptional; usually, phases are well separated in free energy and compete with each other only in pairs. As anticipated, hysteresis is an annoying problem due to the inadequacy of Metropolis dynamics to overcome free-energy barriers. To solve this issue we have employed the Wang-Landau algorithm [25], which directly computes the density of states and is thus particularly suited for a free-energy landscape with multiple minima. Compared to the original algorithm, the refinement parameter \(\ln f\) was reduced at a slower rate (at regular intervals, \(\ln f\) is divided by \(1.1\) rather than \(2\)), which greatly reduces the (already small) saturation error [26]. We report results for \(\gamma=1/2\) in Figures 6 and 7. In the former figure, \(N\) and \(E^{*}\) are plotted as a function of \(\mu\) across the ICO-DOD transition; we see that Metropolis sampling is indeed adequate for \(T^{*}=0.3\) (notice, in particular, how the Wang-Landau data carefully interpolate the Metropolis data in the low-\(\mu\) region, where a small bump is present in the energy). At lower temperatures, only the Wang-Landau simulation is unaffected by hysteresis. For completeness, for \(T^{*}=0.2\) we plot in Figure 7 the density of states \(g\) as a function of \(N\) and \(U\) (i.e., the sum of the first two terms on the r.h.s. of (1)) and the probability density of the particle number, \(P(N)\). In the ICO region, well before the transition at \(\mu^{*}=15/8\), a second peak builds up in \(P(N)\), which, as \(\mu\) is  
Figure 5: Lattice gas on a PD mesh, for \(\gamma=1\) and \(T^{*}=0.05\): \(N\) vs. \(\mu\) across the transition between ICO+CUB and ICO+COC. We plot data from two sequences of runs, ascending (blue) and descending (cyan). The **top left** and **top right** panels differ for the number of equilibrium cycles performed in each run, which is \(5\times 10^{6}\) and \(5\times 10^{7}\), respectively. **Bottom** panels: OPs for ICO+CUB and ICO+COC, for the simulation with \(5\times 10^{7}\) equilibrium cycles per run. We see a narrow interval of \(\mu^{*}\) values around \(9/2\) where the order is neither ICO+CUB nor ICO+COC. increased, is gradually shifted to larger and larger \(N\) values until becoming centered at \(N=20\). Finally, it is useful to compare MC results with the outcome of a mean-field (MF) theory. The simplest approach is to estimate the grand potential of (1) using the Gibbs-Bogoliubov (GB) inequality with a trial probability density \(\pi[c]\) given as an uncorrelated product of one-site terms:  
\[\pi[c]=\prod_{i=1}^{32}\pi_{i}^{(1)}(c_{i}), \tag{5}\]  
with  
\[\pi_{i}^{(1)}(c)=\left\{ \begin{array}{lcl}\pi(c;\rho_{\rm A})&,&i\in{\rm A} \\ \pi(c;\rho_{\rm B})&,&i\in{\rm B}\\ \pi(c;\rho_{\rm C})&,&i\in{\rm C}\,.\end{array} \right.