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Measurable canonical normalization and identities

Statement

Let $(X, Σ, μ)$ be a measure space, let $E : X \to R$ be measurable, and let $J \subset R$ be open. Suppose $Z (β) : = \int e^{- β E} d μ$ satisfies $0 < Z (β) < \infty$ on $J$ , and for every compact interval $K \subset J$ there is $G_{K} \in L^{1} (μ)$ with $(1 + ∣ E ∣^{2}) e^{- β E} \leq G_{K}$ pointwise for every $β \in K$ . Define $p_{β} = e^{- β E} / Z (β)$ , $⟨ Y ⟩_{β} : = \int Y p_{β} d μ$ for integrable $Y$ , and ${Var}_{β} (E) : = \int (E - ⟨ E ⟩_{β})^{2} p_{β} d μ$ . Then $Z \in C^{2} (J)$ , $Z^{'} = - \int E e^{- β E} d μ$ , $Z^{''} = \int E^{2} e^{- β E} d μ$ , and $- \partial_{β} \log Z = ⟨ E ⟩_{β}$ , $\partial_{β}^{2} \log Z = {Var}_{β} (E)$ . The reference-relative entropy $- \int p_{β} \log p_{β} d μ$ is finite and equals $\log Z + β ⟨ E ⟩_{β}$ .

Facts & Assumptions

Given: As stated in the item.

[A1]

$(X, Σ, μ)$ , $E$ , $J$ , positivity/finiteness of $Z$ , and the local integrable envelope are the hypotheses of this statement.

[F1]

Measure space and integral defines the reference measure integral, integrability, and signed/complex conventions; inclusion of simple-minorant sets gives monotonicity of the nonnegative integral. Monotone convergence for the measure integral supplies additivity and nonnegative homogeneity of that integral and linearity on real integrable functions.

[F2]

Dominated convergence for the measure integral: a measurable pointwise-limit representative of measurable functions dominated by an integrable envelope is integrable, and the limit passes through the integral. On a noncomplete measure space the named representative must be measurable; the canonical measurable representative always exists. The limits used below are measurable by [F7].

[F3]

The real exponential function and the number $e$ by a power series defines the exponential series, and The exponential function is smooth and $(\exp)^{'} = \exp$ supplies smoothness and $\exp^{'} = \exp$ . Positivity is supplied by The natural logarithm as the inverse of the exponential function, which identifies $\exp : R \to (0, \infty)$ as the inverse bijection of $\log$ .

[F4]

The natural logarithm as the inverse of the exponential function, Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm, and The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: logarithms convert positive products to sums, are continuous on positive reals, and differentiate as $1 / x$ .

[F5]

The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ : scalar difference quotients of the exponential are bounded by intermediate derivatives.

[F6]

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, finite sums, products and quotients obey their stated rules; quotient differentiation requires a nonzero denominator. In particular, parameter differentiation of $e^{- β E (x)}$ uses composition with the exponential.

[F7]

Arithmetic and lattice operations preserve measurability whenever they are defined and Composition with a Borel measurable outer map preserves measurability: finite sums/products and composition with continuous scalar functions are measurable.

Proof

Given: A measure space, a measurable real energy $E$ , a positive finite partition function, and the compact-interval envelope.

1.1A1F1F2F3F5F6F7

For each fixed parameter, measurable energy/observable coordinates and [F7] make every finite linear combination, exponential weight, weighted first/second moment and parameter difference quotient measurable; their displayed pointwise derivative limits are the same measurable compositions. For $β$ in the interior of a compact interval $K \subset J$ , the difference quotient of $e^{- β E (x)}$ tends pointwise to $- E (x) e^{- β E (x)}$ . The mean-value theorem [F5] bounds its absolute value, for small $h$ , by $∣ E ∣ e^{- θ E}$ for an intermediate $θ \in K$ ; this is at most $G_{K}$ because $∣ E ∣ \leq 1 + ∣ E ∣^{2}$ . Apply dominated convergence [F2] along any sequence of nonzero increments tending to zero; by [F1] the integral of the difference quotient is the difference quotient of $Z$ . Every such sequence has the same limit, so $Z^{'} = - \int E e^{- β E} d μ$ . Applying the same argument to $- E e^{- β E}$ , dominated by $G_{K}$ , gives $Z^{''} = \int E^{2} e^{- β E} d μ$ ; for each of the derivative kernels $- E e^{- γ E}$ and $E^{2} e^{- γ E}$ , pointwise continuity in $γ$ and the same envelope let dominated convergence prove continuity of both derivatives.

2.1A1F1F3F4F6step 1.1

The density $p_{β}$ is nonnegative and integrates to one. The envelope makes its first and second energy moments finite. Differentiating $\log Z$ using [F4] gives $- Z^{'} / Z = \int E p_{β} d μ$ and $Z^{''} / Z - (Z^{'} / Z)^{2} = \int E^{2} p_{β} d μ - (\int E p_{β} d μ)^{2}$ . The latter is the variance by expanding its defining square.

3.1A1F1F3F4step 2.1∎

Since $\log p_{β} = - β E - \log Z$ almost everywhere and $\int ∣ E ∣ p_{β} d μ < \infty$ , the bound $p_{β} ∣ \log p_{β} ∣ \leq ∣ β ∣ ∣ E ∣ p_{β} + ∣ \log Z ∣ p_{β}$ makes the entropy integrand absolutely integrable. Substitution and normalization yield $- \int p_{β} \log p_{β} d μ = β ⟨ E ⟩_{β} + \log Z$ . This entropy depends on the chosen reference measure $μ$ .

Measurable canonical normalization and identities

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources