# Classical and Quantum Gases on a Semiregular Mesh
## 2 Lattice-Gas Models on a Spherical Mesh
### Finite-Temperature Behavior  
For \(T>0\) the study of model (1) cannot be fully analytical since the number of microstates, 232, is huge. Thus, we have resorted to grand-canonical Monte Carlo (MC) simulation for the computation of thermal averages. We employ the standard Metropolis algorithm with single-site moves: at each step of the Markov chain, a move is attempted by changing the state of a randomly selected site \(j\) from \(c_{j}\) to \(1-c_{j}\). Calling \(\Delta H\) the virtual change in \(H\), the move is accepted according to a probability given by \(\min\{1,\exp(-\beta\Delta H)\}\) with \(\beta=(k_{\text{B}}T)^{-1}\), as usual. Once equilibrium has been reached, statistical averages are computed over no less than five million MC cycles (one cycle consisting of 32 trial moves). We monitor a number of equilibrium properties as a function of \(\mu\): the number of particles (\(N=\left\langle\sum_{i}c_{i}\right\rangle\)) and the energy (\(E=\left\langle H\right\rangle+\mu N\)), together with their self- and cross-correlations; the reduced isothermal compressibility,  
\[\rho k_{\text{B}}TK_{T}=\frac{\left\langle\delta N^{2}\right\rangle}{N}\;\; \text{with}\;\;\rho=N/32\;\;\text{and}\;\delta N=\sum_{i}c_{i}-N\,; \tag{2}\]  
and two specific heats, namely  
\[C_{\mu}=\frac{T}{N}\frac{\partial S}{\partial T}\bigg{|}_{V,\mu}\;\;\text{and }\;\;C_{N}=\frac{T}{N}\frac{\partial S}{\partial T}\bigg{|}_{V,N} \tag{3}\]  
(\(S\) being the entropy), expressed in terms of grand-canonical averages through Equations (A6) and (A15) of Appendix A. In addition, we also determine a number of order parameters (OPs, see Appendix B for a definition), in order to establish the nature of the system \"phase\" and the crossover behavior at its boundaries. For the sake of illustration, take \(\gamma=1/2\). For \(T=0\) the succession of phases is  
\[\text{empty}\;\stackrel{{ 0}}{{\longrightarrow}}\;\;\text{ICO}\; \stackrel{{ 1.875}}{{\longrightarrow}}\;\;\text{DOD}\;\stackrel{{ 5}}{{\longrightarrow}}\;\;\text{full}\,. \tag{4}\]  
We explore the phase behavior of this model at five temperatures, \(T^{*}=0.1,0.2,\ldots 0.5\), and in the \(\mu^{*}\) range from \(-2\) to \(7\). To account for the possibility of hysteresis, we carry out our simulation runs in sequence: for a given value of \(\mu^{*}\), the run is started from the last configuration generated in the previous run at a slightly larger or smaller \(\mu^{*}\). Our results are collected in Figures 3 and 4. In the former figure, we plot \(N,E,O_{\text{ICO}}\), and \(O_{\text{DOD}}\) as a function of \(\mu^{*}\). As \(T\) progressively grows, icosahedral and dodecahedral orders become increasingly weakened, and the crossover region between distinct \"phases\" gets wider and wider. An interesting behavior occurs for \(T^{*}=0.1\), where we observe a large hysteresis loop near the ICO-DOD \"transition\". In other words, despite the absence of sharp transitions for \(T>0\), at sufficiently low temperature the order can be so robust that we observe hysteresis--the most characteristic feature of a first-order transition. This is the clue to insufficient sampling of the equilibrium distribution, which can be cured by either substantially increasing the length of the MC trajectory or changing the MC algorithm (see more below). Curiously enough, hysteresis is only found for one of the three transitions present at \(T=0\). To clarify this point, it is instructive to compute the acceptance probability of the Metropolis move(s) that initially drive the system out of one phase into another. The crucial quantity to look at is the ensuing variation in \(H\), i.e., \(\Delta H=\Delta E-\mu\Delta N\), with \(\mu\) given by the transition-point value. Starting from the empty mesh at \(\mu=0\), the repeated addition of particles in icosahedral sites occurs with probability 1, since at each step \(\Delta E=0\) and \(\Delta N=1\). If we start from ICO, the repeated removal of particles again occurs with probability 1, since at each step \(\Delta E=0\) and \(\Delta N=-1\).