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Periodic fixed-particle canonical limit

Statement

Assume AC, the ultrafilter extension principle, DC, and HB, and the superstable regular interaction of Superstable regular continuum grand pressure and variational formula, $β > 0$ , and an interior attainable density $ρ > 0$ . For integers $N_{L}$ with $N_{L} / L^{d} \to ρ$ , let $Z_{p e r} (L, N_{L}; β)$ be the $1 / N_{L}!$ -normalized positional partition on the periodized cube, with the image convention of Superstable regular pair interactions and configurational ensemble notation. Then $L^{- d} \log Z_{p e r} (L, N_{L}; β) ⟶ F_{β} (ρ),$ the same rate as the free cube. Let $C_{L, N_{L}}^{p e r}$ be the periodic exact- $N_{L}$ canonical law and $R_{L, ω}$ its periodized empirical field from Periodized empirical field of a finite configuration. For any finite supporting activity $z > 0$ of $F_{β}$ at $ρ$ , these empirical fields concentrate in every open $τ_{L}$ neighborhood of the minimizer set of $I + β Φ - ρ \log z$ . Writing $M_{z, β}$ for this minimizer set, the concentration means $(C_{L, N_{L}}^{p e r})^{*} {ω : R_{L, ω} \notin U} \to 0$ for every open $τ_{L}$ neighborhood $U$ of it. Here outer probability is as defined in Conditioning grand empirical fields on an exact particle number, and equals ordinary probability for measurable events. A singleton minimizer gives convergence in probability in this outer-probability sense in $τ_{L}$ .

Facts & Assumptions

Given: AC, the ultrafilter extension principle, DC, HB, and a superstable regular even pair potential, positive inverse temperature, and an interior canonical density.

[F1]

Superstable regular pair interactions and configurational ensemble notation: stability and regularity give the finite canonical integrals and the stated periodic-image convention.

[F2]

Diagonal gap for a decreasing integrable interaction tail: separated-cell tails are summable and admit gaps with vanishing interaction per volume.

[F3]

Cell-quadratic lower bound from superstability: $H \geq A \sum_{i} N_{C_{i}}^{2} - B N$ , with $A > 0$ .

[F4]

Canonical free-cube limit at every interior attainable density: the free exact- $N$ cube rate is $F_{β} (ρ)$ at each interior density.

[F5]

Canonical–grand-canonical thermodynamic duality: the canonical rate and grand pressure are convex duals; a supporting activity realizes equality.

[F6]

Superstable regular continuum grand pressure and variational formula: free and periodic normalized-Poisson grand pressures have the same limit $p_{Q} = - \min_{P} {I (P) + β Φ (P) - ρ (P) \log z}$ , with a nonempty compact minimizer set; the $1 / N!$ -normalized Lebesgue grand pressure is $p_{Q} + 1$ .

[F7]

Conditioning grand empirical fields on an exact particle number: for either boundary condition, if the exact- $N$ rate is $F_{β} (ρ)$ , its periodized empirical fields concentrate near the grand variational minimizer set. The four named choice hypotheses are required, and convergence uses outer probabilities as defined in that supplier.

[F8]

Holder's inequality for integrals, including the endpoint cases: Hölder’s inequality makes the log of a finite positive Laplace integral convex in its scalar parameter.

[F9]

Periodized empirical field of a finite configuration and Stationarity and intensity of a periodized empirical field: a finite configuration determines a stationary empirical field with intensity equal to its particle number divided by volume.

[L1]

Georgii, Lemma 4.3, states for an empty corridor of width $k$ that the periodic/free energy discrepancy is bounded by $c_{d} δ_{k} T_{L}$ , where $δ_{k} \to 0$ is the lattice tail and $T_{L} = \sum_{i} N_{C_{i}}^{2}$ . The complete lemma and proof were read; this is used only under its superstable regular hypotheses.

Proof

Given: The assumptions and suppliers in Facts & Assumptions.

1.1F4F5F6algebra

For the upper bound, $e^{α N_{L}} Z_{p e r} (L, N_{L}; β) \leq Ξ_{p e r} (L, e^{α}, β)$ for every real $α$ . Divide logarithms by $L^{d}$ , use the free/periodic grand limit [F6], and then take the infimum over $α$ . The canonical-grand duality [F5] gives $lim sup L^{- d} \log Z_{p e r} \leq F_{β} (ρ)$ .

1.2F1F2F3F4L1algebra

Fix corridor width $k$ larger than the tail cutoff. Restrict the $N_{L}$ particles to the inner cube of side $L - k$ , leaving the corridor empty. The exact- $N_{L}$ density is still $ρ$ . The diagonal-gap supplier [F2] gives the vanishing regular-tail cutoff used here. By [L1], the periodized energy differs from the free inner-cube energy by at most $c_{d} δ_{k} T_{L - k}$ . The cell bound [F3] implies $T_{L - k} \leq A^{- 1} (H_{L - k} + B N_{L})$ . Consequently $Z_{p e r} (L, N_{L}; β) \geq e^{- β c_{d} δ_{k} B N_{L} / A} Z (L - k, N_{L}; β (1 + c_{d} δ_{k} / A)) .$ The canonical theorem [F4] applies at the perturbed inverse temperature, giving the corresponding lower rate.

2.1F4F8step 1.2algebra

At fixed interior $ρ$ , $F_{β} (ρ)$ is finite and convex as a function of $β > 0$ , by Hölder’s inequality [F8], since each finite-volume log partition is convex and the pointwise limit is finite. A finite convex function is continuous on the open interval: on a compact subinterval, its secant slopes are bounded by the slopes to fixed exterior endpoints. Let $L \to \infty$ in step 1.2 and then $k \to \infty$ , so $δ_{k} \to 0$ ; continuity yields $lim inf \geq F_{β} (ρ)$ .

3.1F5F6F7F9step 2.1algebra∎

Step 2.1 established the periodic exact- $N_{L}$ rate $F_{β} (ρ)$ , and [F5] supplies a finite supporting activity. The grand pressure identity in [F6] and the periodized empirical field in [F9] satisfy the hypotheses of [F7] with periodic boundary condition. Hence the outer probability outside each open neighborhood of the minimizer set tends to zero; if the set is a singleton, this is convergence in probability in the stated sense.

Periodic fixed-particle canonical limit

Statement

Facts & Assumptions

Proof

Depends on

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