ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Smooth free energy for a finite two-state model

Physical scope

A constructed two-state canonical model at finite positive temperature; the formulas are model predictions, not measured material properties or a thermodynamic-limit result.

Example

Let $X = {0, 1}$ have energies $E (0) = 0$ J and $E (1) = ϵ$ J with $ϵ > 0$ , both weights $g_{i} = 1$ , and $β = (k_{B} T)^{- 1} > 0$ in $J^{- 1}$ . This illustrates the finite-state canonical-sum clause of Finite-state thermodynamic-limit contract and Countable canonical model; it supplies no thermodynamic-limit sequence. Its partition sum and free energy are

Z (β) = 1 + e^{- β ϵ}, F (β) = - β^{- 1} \log (1 + e^{- β ϵ}) .

Facts & Assumptions

Given: The two-state set, energies, weights, and fixed $ϵ > 0$ in the Example; $T > 0$ and $k_{B} > 0$ use the SI convention of SI quantities and unit conventions.

[F1]

Countable canonical model: For a finite state set with finite energies and positive weights, the partition sum is the finite sum $\sum_{i} g_{i} e^{- β E_{i}}$ ; its admissible parameter set requires $0 < Z < \infty$ .

[F2]

The exponential function is smooth and $(\exp)^{'} = \exp$ and The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: $(e^{u})^{'} = e^{u} u^{'}$ for differentiable $u$ , and $(\log y)^{'} = y^{'} / y$ when $y > 0$ .

[F3]

The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ and Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ : derivatives of compositions and finite sums/products have their usual formulas.

[F4]

Continuity and derivatives of positive-base real powers: for every real $a$ , $x \mapsto x^{a}$ is continuous and differentiable on $(0, \infty)$ , with derivative $a x^{a - 1}$ . Higher regularity of the reciprocal is deduced locally below.

[F5]

A real-analytic function on an open subset of $R$ is locally represented by a convergent real power series: real analyticity means a convergent local real power series. The exponential series converges absolutely for every real argument and The sum of a real power series is real analytic throughout the open interval determined by its radius make the exponential real analytic. [The power series for log(1+x) on (-1,1], including the Abel endpoint](/item/thm-log-one-plus-x-power-series) supplies $\log (1 + x) = \sum_{n \geq 1} (- 1)^{n + 1} x^{n} / n$ for $∣ x ∣ < 1$ , and Real-analytic functions are closed under sums, products and compositions, and under quotients where the denominator is nonzero supplies the indicated algebra, nonzero quotients and local compositions. Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm gives $\log (a b) = \log a + \log b$ .

Verification

1.1givenF1F2F3algebra

The finite sum gives $Z = 1 + e^{- β ϵ} > 1$ , so $F$ is finite for every $β > 0$ . Differentiating with [F2] yields

F^{'} (β) = \frac{\log Z (β)}{β^{2}} + \frac{ϵ e^{- β ϵ}}{β Z (β)},

whose terms are continuous on $β > 0$ .

2.1F2F3F4F5step 1.1algebra∎

Each derivative of $e^{- β ϵ}$ is $(- ϵ)^{m} e^{- β ϵ}$ by induction on $m$ , so $Z$ is $C^{\infty}$ . Since $Z > 0$ , repeated application of the logarithm derivative formula and quotient/product rules makes every derivative of $\log Z$ continuous on $β > 0$ . The same induction gives smoothness of $β^{- 1}$ there. Thus $F = - β^{- 1} \log Z$ is $C^{\infty}$ on $β > 0$ . To establish the stronger analytic conclusion, fix $β_{0} > 0$ and $Z_{0} = Z (β_{0}) > 0$ . The exponential series and [F5] make $Z$ real analytic. On a sufficiently small neighborhood of $β_{0}$ , $h (β) : = (Z (β) - Z_{0}) / Z_{0}$ has $∣ h ∣ < 1$ , and $\log Z (β) = \log Z_{0} + \log (1 + h (β))$ by the logarithm law. The convergent log series and local analytic composition make this real analytic. The reciprocal $β^{- 1}$ is analytic on $β > 0$ by the nonzero-quotient clause, so $F$ is real analytic there by [F5]. Since $T \mapsto (k_{B} T)^{- 1}$ is likewise an analytic nonzero quotient on $T > 0$ , the same holds in the temperature variable. Thus no nonanalytic free-energy point occurs at positive finite temperature; this says nothing about a sequence limit.

Smooth free energy for a finite two-state model

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources