= Lattice gas
A <lattice gas> describes <particle> positions by occupations of fixed cells. In the single-occupancy model, $n_\alpha\in\{0,1\}$ and $N=\sum_\alpha n_\alpha$. Its <grand canonical partition function> sums configurations with weight $\exp[-\beta(U-\widetilde\mu N)]$. Discretizing a classical position <integral> into cells of <volume> $b$ gives $\widetilde\mu=\mu+k_BT\ln(b/\lambda^3)$, with <thermal de Broglie wavelength> $\lambda$.
For <energy> $U=-4J\sum_{\langle\alpha\alpha'\rangle}n_\alpha n_{\alpha'}$ on a regular lattice of coordination $q$, the substitution $n_\alpha=(1+\sigma_\alpha)/2$ maps the model to the <Ising model> with coupling $J$ and effective field $h=qJ+\widetilde\mu/2$. The mean Ising spin is a density imbalance, $\langle\sigma\rangle=2bn-1$. For $J>0$ below the critical <temperature>, the field-driven Ising transition becomes gas-liquid <phase coexistence>.
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