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Full uniform phase shell has vanishing specific canonical KL

Statement

Assume AC and the hypotheses and notation of Total-energy concentration and controlled canonical windows. Let $R_{n} = R_{L_{n}, N_{n}}$ be the finite phase reference of Interacting marked phase-space and reference measures, let $A_{n}$ be the constructed band, and set $M_{n}^{p h} (E) = R_{n} (E \cap A_{n}) / R_{n} (A_{n})$ . Let $C_{n}^{p h}$ be the full fixed- $N_{n}$ canonical probability. Then $D_{n} = D (M_{n}^{p h} ∥ C_{n}^{p h})$ satisfies $D_{n} / V_{n} \to 0$ . This is a mathematical statement about a normalized measure restriction; it does not assert $D_{n} \to 0$ or global total-variation convergence.

Facts & Assumptions

Given: AC, the finite-reference phase measure, the canonical law, stability, and the window hypotheses.

[A1]

AC is used by the conditional entropy chain rule (The Axiom of Choice).

[F1]

Total-energy concentration and controlled canonical windows: $C_{n}^{p h} (A_{n}) \to 1$ , $δ_{n} \to 0$ , and $A_{n}$ is a finite positive-reference-measure energy band $∣ H / V_{n} - e_{b} ∣ \leq δ_{n}$ at fixed $N_{n}$ .

[F2]

Interacting marked phase-space and reference measures: relative to phase reference $R_{n}$ , $C_{n}^{p h}$ has density $e^{- b H} / Z_{n}$ ; by the definition in the Statement, $M_{n}^{p h} = R_{n} (\cdot \cap A_{n}) / R_{n} (A_{n})$ .

[F3]

Relative Entropy of Probability Measures and Conditional entropy chain and disjoint-block bounds: relative entropy is nonnegative and the exact directional convention is $D (P ∥ Q)$ .

Proof

Given: The hypotheses and supplier definitions above.

1.1F1F2algebra

On A_n, the density of M_n^ph relative to C_n^ph is Z_n e^{bH}/R_n(A_n). Therefore $D_{n} = - \log R_{n} (A_{n}) + \log Z_{n} + b E_{M_{n}^{p h}} H .$ The lower band edge gives $R_{n} (A_{n}) = Z_{n} E_{C_{n}^{p h}} [1_{A_{n}} e^{b H}] \geq Z_{n} C_{n}^{p h} (A_{n}) e^{b V_{n} (e_{b} - δ_{n})} .$

2.1A1F1F3step 1.1algebra∎

The upper band edge gives $E_{M_{n}^{p h}} H \leq V_{n} (e_{b} + δ_{n})$ . Substituting the step 1.1 lower bound for $R_{n} (A_{n})$ yields $D_{n} \leq - \log C_{n}^{p h} (A_{n}) + 2 b V_{n} δ_{n} .$ Relative entropy is nonnegative by [F3]; since $C_{n}^{p h} (A_{n}) \to 1$ and $δ_{n} \to 0$ by [F1], this proves $0 \leq D_{n} / V_{n} \to 0$ .

Full uniform phase shell has vanishing specific canonical KL

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources