Statistical mechanics, structure of matter Lattice spin systems Example

Lattice gas

Builds on Ising model Grand canonical ensemble

Example

Let $\Lambda$ be a finite box of $\Z^d$ with periodic boundary conditions, so that each site has $q = 2d$ nearest neighbours. Let $\epsilon \in \R$ be the attraction between neighbouring particles, $\mu \in \R$ the chemical potential and $\beta \ge 0$ the inverse temperature. A configuration is $n \colon \Lambda \to \set{0, 1}$ , where $n_x = 1$ means that the site $x$ holds a particle. Its energy is $H(n) = -\epsilon \sum_{\set{x, y}} n_x n_y - \mu \sum_{x \in \Lambda} n_x,$ where the first sum runs over nearest-neighbour pairs. The lattice gas is the probability measure on $\set{0, 1}^\Lambda$ proportional to $e^{-\beta H(n)}$ . It is a grand canonical ensemble of particles that sit on sites, at most one per site.

Remarks

The substitution $n_x = (1 + \sigma_x)/2$ with $\sigma_x \in \set{-1, +1}$ turns it into the Ising model. Indeed $n_x n_y = (1 + \sigma_x + \sigma_y + \sigma_x \sigma_y)/4$ , and each site lies in $q$ nearest-neighbour pairs. So $H = -J \sum \sigma_x \sigma_y - h \sum_x \sigma_x + \text{const}$ with $J = \epsilon/4$ and $h = \mu/2 + q\epsilon/4$ . Thus the liquid-gas transition of the lattice gas corresponds to the magnetization transition of the Ising model, and the particle density corresponds to $(1 + m)/2$ , where $m$ is the magnetization.

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Tag	`W8TEPY844N`
URI	`m2://W8TEPY844N`
Kind	example
Writing	W0 stub
MSC	82B20, 82B26
Instance of	Lattice model
Uses	Ising model, Grand canonical ensemble
Changed	3 November 2026

Lattice gas

Example

Remarks

Reading path

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