Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Bulk log-partition pressure for a bounded finite-range lattice model

Physical scope

Finite-alphabet, bounded, translation-invariant finite-range interactions on free cubic lattices with a declared cell volume; the pressure convention is specific to this lattice model and implies no continuum-particle or long-range pressure law.

Statement

Use the finite-alphabet, bounded, translation-invariant finite-range lattice model of Bounded finite-range lattice model, with physical interaction energies in joules, inverse temperature $β = (k_{B} T)^{- 1} > 0$ , and partition sum $Z_{L} (β)$ . Suppose each site represents a fixed cell of volume $v_{0} > 0$ in $m^{3}$ . Define the reduced log-partition density $p_{L} = L^{- d} \log Z_{L}$ and the lattice pressure convention $P_{L} = (k_{B} T / v_{0}) p_{L}$ in pascals. Then $p_{L}$ and $P_{L}$ have finite limits as $L \to \infty$ . If a second boundary-energy model on the same configuration sets differs pointwise by at most $C_{b} L^{d - 1}$ joules for fixed $C_{b}$ , it has the same limit.

The pressure convention in this statement is assigned to this lattice model. Equating it with a derivative-based macroscopic pressure or a measured mechanical pressure requires a separate model identification.

Facts & Assumptions

Given: A finite alphabet, finite-range bounded lattice interactions, positive $T$ , the canonical partition sum, and a fixed physical cell volume.

[F1]

Bounded finite-range lattice model: for integer $d, L \geq 1$ , finite alphabet $A$ , bounded finite-range translation-invariant interactions, and $β > 0$ , $H_{L}$ and $Z_{L} = \sum_{s} e^{- β H_{L} (s)}$ are defined with free boundary.

[F2]

Finite-range lattice log-partition limit (current batch 11 draft): the sequence $p_{L} = L^{- d} \log Z_{L}$ has a finite limit for every fixed $β > 0$ , and a boundary-energy perturbation bounded by $C_{b} L^{d - 1}$ has the same limit. Its full current proof was inspected; its draft status remains conditional.

[F3]

SI quantities and unit conventions: $k_{B}$ has units $J K^{- 1}$ , $T$ is in kelvins, $v_{0}$ is in $m^{3}$ , and $k_{B} T / v_{0}$ is in pascals.

[F4]

Canonical sampling assumption for finite and normalized statistical models: the positive finite lattice partition sum defines the adopted finite-volume canonical law.

[A1]

$v_{0}$ is fixed as $L$ varies, and the physical interaction parameters are held fixed in the sequence of cubic boxes.

Proof

Given: The bounded finite-range lattice data, $β > 0$ , a fixed cell volume, and the stated boundary perturbation bound.

1.1F1F2F4A1

The finite lattice data in [F1] and $β > 0$ match the model hypotheses of [F2]. Thus $p_{L} = L^{- d} \log Z_{L}$ converges to a finite $p_{\infty}$ ; the same supplier gives the identical limit for the stated boundary perturbation. By [F4], these finite partition sums give the adopted lattice canonical law.

2.1F3step 1.1algebra∎

Since $v_{0}$ and $T$ are fixed along the volume sequence, multiplying by the constant $k_{B} T / v_{0}$ preserves convergence: $P_{L} = (k_{B} T / v_{0}) p_{L} \to (k_{B} T / v_{0}) p_{\infty}$ . The factor has units of pressure by [F3]. This proves only the declared lattice pressure convention.

Bulk log-partition pressure for a bounded finite-range lattice model

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources