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Statistical mechanics, structure of matter Definition

Lattice model

Builds on GraphProbability measureIsing modelHard spheres

Definition

A (classical) lattice model consists of the following data.

  1. A countable set LL of sites, usually the vertex set of a graph such as ℤd\Z^d with nearest-neighbour edges.
  2. A finite set SS of single-site states (spin values); a configuration is a map σ ⁣:L→S\sigma \colon L \to S.
  3. An interaction: for each finite set A⊆LA \subseteq L, a function ΦA ⁣:SA→ℝ\Phi_A \colon S^A \to \R, such that ΦA=0\Phi_A = 0 unless the diameter of AA is at most some fixed range RR.

For a finite set Λ⊆L\Lambda \subseteq L, the energy of σ∈SΛ\sigma \in S^\Lambda (with free boundary condition) is HΛ(σ)=∑A⊆ΛΦA(σ∣A)H_\Lambda(\sigma) = \sum_{A \subseteq \Lambda} \Phi_A(\sigma|_A). At inverse temperature β≥0\beta \ge 0, the finite-volume Gibbs distribution is the probability measure on SΛS^\Lambda given by μΛ,β(σ)=e−βHΛ(σ)ZΛ,β,ZΛ,β=∑τ∈SΛe−βHΛ(τ).\mu_{\Lambda, \beta}(\sigma) = \frac{e^{-\beta H_\Lambda(\sigma)}}{Z_{\Lambda, \beta}}, \qquad Z_{\Lambda, \beta} = \sum_{\tau \in S^\Lambda} e^{-\beta H_\Lambda(\tau)}.

Remarks

Example: the Ising model on ℤd\Z^d has S={−1,+1}S = \set{-1, +1}, Φ{u,v}(σ)=−Jσ(u)σ(v)\Phi_{\set{u,v}}(\sigma) = -J \sigma(u)\sigma(v) for neighbours u,vu, v, Φ{v}(σ)=−hσ(v)\Phi_{\set{v}}(\sigma) = -h \sigma(v), and ΦA=0\Phi_A = 0 otherwise. All three clauses hold, with range R=1R = 1.

Non-example: a fluid of hard spheres has particle positions in ℝ3\R^3 rather than variables attached to the sites of a lattice, so clause 1 fails; it is a continuum model.

Variants allow an infinite or continuous spin space SS with a reference measure, as for the XY model. Other variants allow interactions of infinite range that decay fast enough, or variables on edges instead of sites. Fixing the spins outside Λ\Lambda to a configuration η\eta adds the terms with AA meeting both Λ\Lambda and its complement, and gives the Gibbs distribution with boundary condition η\eta.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Canonical ensemble, concept, no evidence levelCanonical ensembleconceptGraph, concept, no evidence levelGraphconceptIsing model, example, no evidence levelIsing modelexampleMayer f-function, concept, no evidence levelMayer f-functionconceptProbability measure, concept, no evidence levelProbability measureconceptHard spheres, example, no evidence levelHard spheresexampleLattice model, definition, no evidence levelLattice modeldefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Canonical ensemble concept
  2. Graph concept
  3. Ising model example
  4. Mayer f-function concept
  5. Probability measure concept
  6. Hard spheres example
  7. Lattice model definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Lattice_model_(physics)
About this page
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DefinesLattice model
UsesGraph, Probability measure, Ising model, Hard spheres
Changed3 November 2026

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