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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.

Analytic interacting chain and its Gibbs state

Example

Facts & Assumptions

Verification

Depends on

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