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Finite-range lattice interaction and conditional energy — Examples

1 · Prerequisites

2 · Summary

These examples distinguish conditional Gibbs specifications from free-boundary marginals and work through the exactly solvable interacting Ising chain. A final regime comparison keeps the sufficient high-temperature uniqueness and low-temperature coexistence hypotheses separate and leaves their intervening parameters unclassified. The examples are exact calculations within stated classical models, not reported experiments or evidence about a particular material.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedjudge pass (gpt-6-luna)Open item page →

Interacting marginals are not free Gibbs laws

Statement refuted

A finite marginal of an interacting Gibbs chain always equals the free Gibbs law obtained by retaining only interaction terms wholly inside the observed sites.

Facts & Assumptions

Given: A three-site zero-field Ising chain with spins $s_{1}, s_{2}, s_{3} \in {- 1, + 1}$ , open-chain Hamiltonian $H = - J (s_{1} s_{2} + s_{2} s_{3})$ , $β > 0$ , and $J \neq 0$ .

[A1]

Put $K = β J$ and define $\sinh K : = (e^{K} - e^{- K}) / 2$ , $\cosh K : = (e^{K} + e^{- K}) / 2$ , and $\tanh K : = \sinh K / \cosh K = (e^{K} - e^{- K}) / (e^{K} + e^{- K})$ . The denominator is positive by positivity of the exponential; since $K \neq 0$ , strict increase gives $e^{K} \neq e^{- K}$ , so $\tanh K \neq 0$ (The real exponential function and the number $e$ by a power series, The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ , The exponential function is strictly increasing).

[A2]

The specification uses conditional laws given the exterior; it does not identify them with unconditional finite marginals (Lattice Gibbs kernels and DLR states).

[F1]

The finite Gibbs weights are proportional to $e^{- β H}$ and have positive finite normalizer (Finite Gibbs relative entropy and free-energy minimum).

Counterexample

1.1givenA1F1algebra

With $K$ , $\sinh K$ , $\cosh K$ , and $\tanh K$ as in [A1], summing over the middle spin gives $Z = 2 (2 \cosh K)^{2}$ and $E (s_{1} s_{3}) = \frac{2 (2 \sinh K)^{2}}{2 (2 \cosh K)^{2}} = \tanh^{2} K > 0$ . Thus the endpoint marginal is correlated.

2.1A2F1step 1.1algebra∎

The only interaction terms are on bonds ${1, 2}$ and ${2, 3}$ , so no term is wholly supported on the retained endpoint set ${1, 3}$ . The claimed retained-term free law therefore has zero energy and assigns independent uniform endpoint spins, with correlation zero. This contradicts step 1.1, refuting the statement.

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Analytic interacting chain and its Gibbs state

Physical scope

One-dimensional classical nearest-neighbor Ising model at finite positive temperature; exact conditional model example, not experimental evidence.

Example

For a one-dimensional zero-field nearest-neighbor Ising chain with coupling energy $J = J_{I} > 0$ joules and temperature $T > 0$ kelvin, let $β = (k_{B} T)^{- 1}$ in $J^{- 1}$ and $K = β J$ , which is dimensionless. The dimensionless pressure per site is $p (K) = \lim_{N \to \infty} N^{- 1} \log Z_{N} = \log (2 \cosh K)$ , where $Z_{N} = \sum_{s \in {- 1, + 1}^{N}} \exp (K \sum_{i = 1}^{N} s_{i} s_{i + 1})$ for periodic chains with $N \geq 2$ and $s_{N + 1} = s_{1}$ , the infinite-volume DLR law is a stationary Markov chain, and for $0 < K < 1 / 2$ this dimensionless pressure has the convergent series from Exact convergent Ising-chain high-temperature series. Under the adopted Gibbs-equilibrium postulate this is a conditional model example, not an experimental observation.

Facts & Assumptions

Given: The finite-range one-dimensional zero-field Ising model, $J > 0$ , $β > 0$ , and the adopted classical equilibrium interpretation.

[F1]

The chain's positive transfer matrix yields a unique stationary Markov DLR law (Positive Ising transfer matrix and stationary Markov law, All-positive-temperature one-dimensional Ising DLR uniqueness).

[F2]

The zero-field chain pressure has the absolutely convergent series for $∣ K ∣ < 1 / 2$ (Exact convergent Ising-chain high-temperature series).

[F3]

Gibbs/DLR states are adopted as the equilibrium model for the chosen Hamiltonian and reference (Interacting classical Gibbs equilibrium adoption).

[F4]

For the periodic Ising chain with finite $β > 0$ and $J, h \in R$ , $Z_{N} = tr (T^{N}) = λ_{+}^{N} + λ_{-}^{N}$ for $T_{s, t} = e^{β J s t + β h (s + t) / 2}$ , with $λ_{+} > ∣ λ_{-} ∣$ and pressure $\log λ_{+}$ (One-dimensional periodic Ising-chain limit).

[A1]

Countable Choice is carried by the chain measure construction (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Verification

1.1A1F1

The transfer-matrix construction in [F1] supplies a stationary Markov probability; the DLR uniqueness theorem gives the finite-volume endpoint limits and identifies that probability as the sole infinite-volume DLR state.

1.2F2F4algebra

Apply [F4] at $h = 0$ with the given finite $J > 0$ and $β > 0$ . Its Hamiltonian is exactly $- J \sum_{i = 1}^{N} s_{i} s_{i + 1}$ , so its counting-measure partition is the $Z_{N}$ in the Example. In the spin order $+ 1, - 1$ , $T = (\begin{array}{cc} e^{K} & e^{- K} \\ e^{- K} & e^{K} \end{array})$ has eigenvalues $λ_{+} = e^{K} + e^{- K} = 2 \cosh K$ and $λ_{-} = e^{K} - e^{- K}$ . Thus $Z_{N} = λ_{+}^{N} (1 + q^{N})$ , where $q = λ_{-} / λ_{+} \in (0, 1)$ . Consequently $0 \leq N^{- 1} \log Z_{N} - \log λ_{+} = N^{- 1} \log (1 + q^{N}) \leq N^{- 1} \log 2 \to 0$ . This proves $p (K) = \log (2 \cosh K)$ for every finite $K > 0$ . When $0 < K < 1 / 2$ , [F2] additionally supplies the series and its absolute and compact-uniform convergence.

2.1F3step 1.1step 1.2∎

The physical reading is conditional on [F3]. It predicts the named Gibbs state for this classical model only; it does not claim a measured pressure, equilibration, or agreement with a particular material.

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Sufficient Ising uniqueness and coexistence regimes

Physical scope

Two-dimensional classical nearest-neighbor ferromagnetic Ising with J>0 and positive temperature: the sufficient uniqueness bound holds at every constant finite field energy h, while the stated coexistence bound requires h=0. Conditional mathematical regimes only; no empirical threshold claim.

Example

For the two-dimensional nearest-neighbor ferromagnetic Ising model, take coupling energy $J = J_{I} > 0$ joules, a constant finite field energy $h \in R$ in joules per dimensionless spin, and temperature $T > 0$ kelvin, set $β = (k_{B} T)^{- 1}$ in $J^{- 1}$ , and write $K = β J$ (dimensionless). The results on this page give a sufficient high-temperature uniqueness regime $4 \tanh K < 1$ at every such $h$ , and a sufficient low-temperature coexistence regime $K \geq 2$ at $h = 0$ . Parameters outside these sufficient hypotheses are not classified by these bounds. These are conditional model predictions, not measured transition temperatures.

Facts & Assumptions

Given: A two-dimensional nearest-neighbor classical Ising model with $J > 0$ , $β > 0$ , a constant finite field energy $h$ , and the adopted Gibbs-equilibrium interpretation; the coexistence branch additionally assumes $h = 0$ .

[A1]

Gibbs/DLR states are adopted as the equilibrium model for the chosen Hamiltonian and reference (Interacting classical Gibbs equilibrium adoption).

[A2]

Countable Choice is retained in the DLR construction (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[F1]

If $4 \tanh K < 1$ , the influence row sum is below one and the DLR state is unique (Conditional Ising equilibrium phase predictions).

[F2]

For the zero-field model $h = 0$ , if $K \geq 2$ , there are distinct stationary DLR states with opposite nonzero magnetizations (Conditional Ising equilibrium phase predictions).

Verification

1.1A1A2F1algebra

For nearest-neighbor interactions in two dimensions, each site has four neighbors, so the sufficient row sum is $4 \tanh K$ . Thus $4 \tanh K < 1$ is a sufficient high-temperature uniqueness condition; [F1] is applicable under the stated Countable Choice hypothesis [A2].

2.1A2F2step 1.1algebra

For $h = 0$ , the low-temperature theorem [F2] applies for $K \geq 2$ under [A2]. At other parameter values outside the high-temperature inequality and this zero-field coexistence regime, neither sufficient conclusion applies in this comparison.

3.1A1A2step 1.1step 2.1∎

The postulate [A1] makes these mathematical DLR conclusions conditional equilibrium-model predictions only. The thresholds are not empirical measurements, and no exact critical point is asserted.

Finite-range lattice interaction and conditional energy — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

5 · Examples, experiments and counterexamples

Interacting marginals are not free Gibbs laws

Statement refuted

Facts & Assumptions

Counterexample

Analytic interacting chain and its Gibbs state

Example

Facts & Assumptions

Verification

Sufficient Ising uniqueness and coexistence regimes

Example

Facts & Assumptions

Verification

Sources