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Conditioning grand empirical fields on an exact particle number

Statement

Assume AC, the ultrafilter extension principle, DC, and HB, and let $ϕ$ be superstable and regular as in Superstable regular pair interactions and configurational ensemble notation. Let $Λ_{L} = [0, L)^{d}$ , $β, z > 0$ , and $b \in {f r e e, p e r}$ . On the simple labelled sector $S_{L, N} = {q \in Λ_{L}^{N} : q_{i} \neq q_{j} (i \neq j)}$ use $H_{L, f r e e} (q) = \sum_{i < j} ϕ (q_{i} - q_{j})$ and the periodic energy $H_{L, p e r}$ from Superstable regular pair interactions and configurational ensemble notation. For $N \geq 1$ define $Z_{b} (L, N; β) = \frac{1}{N!} \int_{S_{L, N}} e^{- β H_{L, b} (q)} d q_{1} \dots d q_{N}, Z_{b} (L, 0; β) = 1,$ with $e^{- β (+ \infty)} = 0$ , and put $Ξ_{b} (L, z, β) = \sum_{N \geq 0} z^{N} Z_{b} (L, N; β)$ . The unit-intensity Poisson reference $Q_{Λ_{L}}$ has sector density $e^{- L^{d}} / N!$ relative to labelled Lebesgue measure, so its normalized grand partition is $Z_{Q, b} (L; z, β) = E_{Q_{Λ_{L}}} [z^{N_{Λ_{L}}} e^{- β H_{L, b}}] = e^{- L^{d}} Ξ_{b} (L, z, β) .$ The exact- $N$ canonical law $C_{L, N}^{b}$ has density $e^{- β H_{L, b}} / (N! Z_{b} (L, N; β))$ on $S_{L, N}$ , and the grand-canonical law $G_{L, z, β}^{b}$ has density $z^{N_{Λ_{L}}} e^{- β H_{L, b}} / Z_{Q, b} (L; z, β)$ relative to $Q_{Λ_{L}}$ . Let $ρ > 0$ be an interior attainable density, choose $α = \log z$ supporting the free canonical rate $F_{β}$ at $ρ$ , and take $N_{L} / L^{d} \to ρ$ . Assume $L^{- d} \log Z_{b} (L, N_{L}; β) ⟶ F_{β} (ρ) .$ Let $R_{L, ω}$ be the periodized empirical field of Periodized empirical field of a finite configuration, and let $M_{z, β} : = {argmin}_{P \in P_{s t a t}^{1}} {I (P) + β Φ (P) - ρ (P) \log z} .$ For any finite-volume law $C$ and arbitrary event $E$ , write $C^{*} (E) = \inf {C (A) : E \subseteq A, A measurable}$ for its outer probability. Here measurability uses the configuration sigma-algebra; on law space the evaluation sigma-algebra is generated by $P \mapsto P (B)$ for measurable configuration events $B$ . Convergence in probability in $τ_{L}$ means that the outer probability of leaving each open neighborhood tends to zero; for measurable events this is ordinary probability. Then for every open $τ_{L}$ neighborhood $U$ of $M_{z, β}$ there is $c_{U} > 0$ such that ${lim sup}_{L \to \infty} L^{- d} \log (C_{L, N_{L}}^{b})^{*} {ω : R_{L, ω} \notin U} \leq - c_{U} .$ If $M_{z, β} = {P_{*}}$ , then $R_{L, ω} \to P_{*}$ in probability in $τ_{L}$ under $C_{L, N_{L}}^{b}$ . This is a statement about spatially averaged local observables through the empirical field, not about the raw finite-box law at a fixed origin.

Facts & Assumptions

Given: AC, the ultrafilter extension principle, DC, HB, a superstable regular interaction, $β, z > 0$ , an interior density $ρ$ , exact particle numbers $N_{L} / L^{d} \to ρ$ , the displayed canonical rate limit, and the same finite-volume boundary condition in the canonical and grand-canonical laws.

[A1]

The Axiom of Choice, The ultrafilter extension principle (UL/BPI), The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, and The real dominated-extension principle as an additional hypothesis over ZF are assumed, carrying the explicit compactness hypotheses of [F3].

[F1]

Superstable regular pair interactions and configurational ensemble notation: the periodic image energy is absolutely defined under the stated tail convention, and the unit-intensity Poisson reference has sector density $e^{- L^{d}} / N!$ ; together with the Statement's local finite-volume definitions this gives $Z_{Q, b} = e^{- L^{d}} Ξ_{b}$ for either boundary condition.

[F2]

Canonical–grand-canonical thermodynamic duality: a supporting $α = \log z$ gives physical pressure $P (α, β) = F_{β} (ρ) + α ρ$ .

[F3]

Superstable regular continuum grand pressure and variational formula: the normalized grand pressure is $p_{Q} (z, β) = P (α, β) - 1 = - \min_{P} {I (P) + β Φ (P) - ρ (P) \log z}$ , and its minimizer set is nonempty and compact in $τ_{L}$ .

[F4]

Local-Tame Topology, Periodized empirical field of a finite configuration and Stationarity and intensity of a periodized empirical field: $R_{L, ω}$ is the stationary empirical field used by the grand empirical-field large-deviation principle.

[L1]

Georgii's Theorem 2 (1994, Sections 3–5) gives, for superstable regular interactions, the large-deviation upper and lower bounds for evaluation-measurable sets in $τ_{L}$ for these periodized empirical fields under both free and periodic grand-canonical Gibbs laws, with rate $I (P) + β Φ (P) - ρ (P) \log z + p_{Q} (z, β)$ . The evaluation sigma-algebra is distinguished from the topology on printed p.172; Theorem 2 requires measurable functionals, and its specialization to sets on pp.175–176 requires measurable sets. The complete Sections 3–5 proof was read; the chosen $b$ must be one of the two stated boundary conditions.

Proof

Given: The hypotheses and suppliers in Facts & Assumptions.

1.1givenF1F2F3algebra

The canonical law is the grand law conditioned on $N = N_{L}$ , since the grand weight on the $N_{L}$ sector differs from the canonical density only by the constant activity factor $z^{N_{L}}$ . Its conditioning probability is $G_{L, z, β}^{b} (N = N_{L}) = \frac{z^{N_{L}} e^{- L^{d}} Z_{b} (L, N_{L}; β)}{Z_{Q, b} (L; z, β)} .$ Divide the logarithm by $L^{d}$ . The numerator terms tend to $ρ \log z + F_{β} (ρ) - 1$ by the assumed canonical rate and [F1]; the denominator tends to $p_{Q} (z, β)$ . By [F2]–[F3], $p_{Q} (z, β) = F_{β} (ρ) + ρ \log z - 1$ , so $L^{- d} \log G_{L, z, β}^{b} (N = N_{L}) ⟶ 0.$ Thus the exact-number conditioning event has subexponential, positive grand probability for all sufficiently large $L$ .

1.2A1F3F4L1algebra

Fix an open neighborhood $U$ of $M_{z, β}$ . Each $P$ in this compact set has a basic neighborhood contained in $U$ , described by finitely many strict inequalities in tame expectations from [F4]. These expectations are evaluation-measurable, so each such basic neighborhood is measurable. A finite subcover of $M_{z, β}$ gives a measurable open $V$ with $M_{z, β} \subseteq V \subseteq U$ . Set $F = V^{c}$ , which is closed and measurable. The good rate $J$ in [L1] has zero set exactly $M_{z, β}$ . Its infimum on $F$ is positive: otherwise the nonempty nested closed sets $F \cap {J \leq 1 / n}$ in the compact sublevel ${J \leq 1}$ would have a common point, a zero outside $M_{z, β}$ . If $F$ is empty the estimate is immediate. The measurable-set upper bound in [L1] gives some $c > 0$ with $lim sup L^{- d} \log G_{L, z, β}^{b} {R_{L, ω} \in F} \leq - c$ . Since ${R \notin U} \subseteq {R \in F}$ , this also bounds its outer probability.

2.1step 1.1step 1.2algebra∎

The measurable event ${R \in F}$ contains ${R \notin U}$ , so conditioning on $N = N_{L}$ yields $(C_{L, N_{L}}^{b})^{*} {R_{L, ω} \notin U} \leq \frac{G_{L, z, β}^{b} {R_{L, ω} \in F}}{G_{L, z, β}^{b} (N = N_{L})} .$ The numerator decays exponentially by step 1.2 and the denominator has zero exponential rate by step 1.1, so the normalized logarithm of the left side has limsup at most $- c$ . Taking $c_{U} = c$ proves the stated exponential bound, and the ratio tends to zero. This proves concentration on every neighborhood of $M_{z, β}$ . If the minimizer is unique, these neighborhood bounds are exactly convergence in probability in $τ_{L}$ .

Conditioning grand empirical fields on an exact particle number

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

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