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Superstable regular continuum grand pressure and variational formula

Statement

Assume AC, the ultrafilter extension principle, DC, and HB. Let $ϕ$ be superstable and regular as in Superstable regular pair interactions and configurational ensemble notation, and fix $z, β > 0$ . Use the stationary finite-intensity law space $P_{s t a t}^{1}$ , the specific relative entropy $I (P)$ , the intensity $ρ (P)$ , and the positional energy density $Φ (P) = E (P)$ defined in Superstable regular pair interactions and configurational ensemble notation and Local-Tame Topology. The existence and lower semicontinuity of $Φ$ under these hypotheses are established below. For free and periodic cubes define $Z_{L, z, β, b c} = E_{Q_{Λ_{L}}} ⁣ [z^{N_{Λ_{L}}} e^{- β H_{L, b c}}], b c \in {f r e e, p e r} .$ Then the limit $p_{Q} (z, β) = \lim_{L \to \infty} L^{- d} \log Z_{L, z, β, b c}$ exists, is independent of $b c$ , and $p_{Q} (z, β) = - \min_{P \in P_{s t a t}^{1}} {I (P) + β Φ (P) - ρ (P) \log z} .$ The minimizer set is nonempty and compact in the local-tame topology $τ_{L}$ . This theorem concerns positional configurations only. If $Ξ_{L, z, β}$ is instead normalized against Lebesgue measure with $1 / N!$ , then $Ξ_{L, z, β} = e^{L^{d}} Z_{L, z, β, f r e e}$ , so its logarithmic rate is $p_{Q} + 1$ .

Facts & Assumptions

Given: AC, the ultrafilter extension principle, DC, HB, a superstable regular pair potential, $z, β > 0$ , and the unit-intensity Poisson reference.

[A1]

The choice assumptions are carried explicitly because Compactness of Finite-Specific-Entropy Levels in the Local-Tame Topology uses AC, the ultrafilter extension principle, DC, and HB for compactness and sequential compactness.

[F1]

Superstable regular pair interactions and configurational ensemble notation: superstable means a stable part plus a nonnegative repulsive core, regularity supplies the decreasing integrable lower and upper tail, and $Z_{Q}$ uses normalized unit-intensity Poisson measure.

[F2]

Cell-quadratic lower bound from superstability: there are $A > 0, B \geq 0$ such that $H_{ϕ} (ω) \geq A \sum_{i} N_{C_{i}} (ω)^{2} - B N (ω)$ .

[F3]

Compactness of Finite-Specific-Entropy Levels in the Local-Tame Topology: for every finite $c$ , the laws with $I (P) \leq c$ are compact and sequentially compact in $τ_{L}$ under the assumptions in [A1].

[F4]

Spatial Mean-Information Theorem: if a stationary law is ergodic under all translations and has finite $I (P) = γ$ , then $N^{- d} \log (d P_{Λ_{N}} / d Q_{Λ_{N}}) \to γ$ in $L^{1} (P)$ for integer side lengths $N \to \infty$ . The real-side extension used below is proved in step 1.3.

[F5]

Local-Tame Topology: $τ_{L}$ is convergence of expectations of every measurable local observable bounded by a constant times $1 + N_{A}$ on some bounded set $A$ .

[F6]

Free-cube grand-pressure limit for decreasing integrable tails: for the regular tail class here, the free-cube grand partition is finite and its free pressure limit exists.

[F7]

Monotone convergence for the integral: nonnegative cell-count sums increase to their extended expectation.

[F8]

Specific Relative Entropy Density gives the real-side cube entropy limit; Relative Entropy Variational, Chain, and Conditioning Rules and Relative Entropy of Probability Measures: finite-region entropy has the conditional chain rule, is nonnegative, and its variational formula applies to bounded measurable tests.

[L1]

Georgii's Theorem 1 (1994, Section 3) states that for a superstable lower-regular pair potential, $Φ (P) = \lim L^{- d} E_{P} H_{L, f r e e}$ exists in $R \cup {+ \infty}$ for every stationary finite-intensity law and is lower semicontinuous in the local topology. Its hypotheses match [F1]–[F2]; the full 1994 proof was read. This supplies the energy-density existence interface, not a certification stamp.

[L2]

Georgii's Theorem 2 (1994, Section 3, equations (2.20)–(2.22)) states that for a superstable regular potential and $z, β > 0$ , the free and periodic normalized-Poisson partition functions have the same limiting rate $p (z, β) = - \min_{P} [I (P) + β Φ (P) - ρ (P) \log z]$ , and the associated rate functional has compact local-tame sublevel sets. Its proof is in Sections 4–5; the complete source proof was read. The local Poisson/entropy inputs used there are supplied by [F2]–[F8].

Proof

Given: The assumptions and suppliers in Facts & Assumptions.

1.1F1F2L1algebra

Georgii's Theorem 1 [L1] gives existence of $E (P)$ and lower semicontinuity for every stationary finite-intensity law; by the Definition, this is $Φ (P)$ . The source uses the same maximum norm $∣ x ∣_{\infty}$ in (2.3)–(2.7). Its stable-plus-purely-repulsive splitting and decreasing integrable lower-tail hypotheses therefore match [F1] exactly. Thus $Φ$ has exactly the domain and convention in the Statement.

1.2A1F1F2F3F7F8algebra

The cell estimate [F2] and $β > 0$ imply $J = I + β Φ - ρ \log z$ is lower semicontinuous and coercive on bounded- $J$ sets: the Poisson entropy variational inequality with bounded truncations of $a N_{C_{0}}$ gives $I (P) \geq a ρ (P) - (e^{a} - 1)$ ; the pointwise square inequality gives $E_{P} N_{C_{0}}^{2} \geq ρ (P)^{2}$ . Hence bounded $J$ bounds $ρ$ , $I$ , and $Φ$ from above. A finite candidate exists: take independent Bernoulli occupancy of widely separated small cubes, at most one absolutely continuously distributed point per cube, and a uniform random lattice phase. Each tile has finite relative entropy to Poisson, and the summable regular tail bounds its absolute intertile energy per volume. Thus $J$ is proper. By [F3] and the choice assumptions [A1], its bounded sublevels are compact in $τ_{L}$ .

1.3A1F4F8algebra

To extend [F4] to real side lengths, fix its stationary ergodic $P$ with $I (P) = γ < \infty$ , put $N = ⌊ L ⌋$ , and write $h_{L} = d P_{Λ_{L}} / d Q_{Λ_{L}}$ and $D_{L} = D (P_{Λ_{L}} ∥ Q_{Λ_{L}})$ . Every such entropy is finite by Specific Relative Entropy Density. The chain rule [F8] factors $h_{L} = h_{N} k$ $P$ -almost surely, where $k$ is the conditional density of the remaining region relative to its independent Poisson reference. Against the reference probability $P_{Λ_{N}} \otimes Q_{Λ_{L} ∖ Λ_{N}}$ , $k$ is a density of mass one. Thus $E_{P} (- \log k)^{+} \leq 1 / e$ , and $E_{P} \log k = D_{L} - D_{N}$ , so $∥ \log h_{L} - \log h_{N} ∥_{1} \leq D_{L} - D_{N} + 2 / e$ . The specific-entropy supplier gives $D_{L} / L^{d} \to γ$ and $D_{N} / N^{d} \to γ$ ; since $N / L \to 1$ , the right side divided by $L^{d}$ tends to zero. Together with [F4] this proves $L^{- d} \log h_{L} \to γ$ in $L^{1} (P)$ .

2.1A1F1F3F4F5F8L1L2step 1.1step 1.3

For the finite-volume Gibbs tilt $G_{L, z, β, b c}$ , direct integration gives $L^{- d} \log Z_{L, z, β, b c} = - L^{- d} D (G_{L, z, β, b c} ∥ Q_{Λ_{L}}) - β L^{- d} E_{G} H_{L, b c} + L^{- d} \log z E_{G} N .$ Georgii's Theorem 2 [L2], applied with the present superstable regular interaction and $F = 0$ , supplies the matching upper and lower empirical-field bounds and the free/periodic pressure limit. Its upper proof uses this entropy identity, block stationarization, and the superstable cell cutoff; its lower proof conditions finite blocks on bounded finite-energy events, inserts empty corridors, and applies the spatial likelihood limit. The inspected source hypotheses match [F1]–[F8]: the Poisson reference is independent unit intensity, $τ_{L}$ is the local-tame topology [F5], entropy compactness is [F3], and the energy-density interface is [L1]. The theorem therefore yields $p_{Q} (z, β) = - \min_{P} {I (P) + β Φ (P) - ρ (P) \log z}$ for both boundary conditions, with compact variational sublevels.

2.2A1F3L2step 1.2

The minimum is attained: a minimizing sequence lies in a finite sublevel by the sparse finite candidate in step 1.2; compactness [F3] gives a convergent subsequence and lower semicontinuity from step 1.1 gives a minimizer. The set of minimizers is the closed minimum level within that compact sublevel, hence nonempty and compact. This agrees with the compact-sublevel conclusion in [L2].

3.1F1F6L2algebra∎

The unit-intensity Poisson law on $Λ_{L}$ has density $e^{- L^{d}} / N!$ relative to Lebesgue measure on the $N$ -point sector. Therefore the physical partition $Ξ_{L, z, β} = e^{L^{d}} Z_{L, z, β, f r e e}$ ; its log rate is $p_{Q} + 1$ . This is only a positional normalization conversion.

Superstable regular continuum grand pressure and variational formula

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources