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Qualified interacting continuum thermodynamic and positional ensemble limits

Physical scope

Conditional positional-model deductions for the selected classical Gibbs equilibrium postulate. The d=3 SI pressure conversion uses the declared energy and length units. Every positive-temperature branch retains its own exact tail, core, boundary, density, differentiability, and choice hypotheses. Clause 8 is a separate formal dimensionless beta=0 continuation at fixed mathematical activity z, outside the positive-temperature physical premises and without an SI temperature or pressure interpretation; it retains hard exclusions. No momentum-shell result, continuum Gibbs-state existence or uniqueness, minimizer-to-DLR identification, empirical observation, or claim of physical applicability is asserted.

Statement

For clauses 1–7 assume the adopted classical Gibbs-equilibrium postulate and parameter conventions of Interacting classical Gibbs equilibrium adoption and Classical Gibbs quantities and SI reference conventions, with a single positional species and reference exactly matching the mathematical suppliers below. Use the postulate’s thermal-wavelength convention with reference-action identification $a_{0} = h_{r e f}$ and its operationally indistinguishable $1 / N!$ sector factor. Let $d \geq 1$ . Fix a reference length $ℓ_{0} > 0$ in metres and energy scale $E_{0} > 0$ in joules. If the physical inverse temperature is $β_{S I} = (k_{B} T)^{- 1} > 0$ in J $^{- 1}$ and physical activity is $z_{c} > 0$ in m $^{- d}$ as defined in Classical Gibbs quantities and SI reference conventions, the numerical mathematical inputs are $β = E_{0} β_{S I}$ and $z = ℓ_{0}^{d} z_{c}$ ; coordinates and pair energies are expressed in units $ℓ_{0}$ and $E_{0}$ . Thus $k_{B} T = E_{0} / β$ and physical volume is $ℓ_{0}^{d}$ times numerical volume. Clause 8 is a separate formal dimensionless continuation with $β = 0$ and fixed $z > 0$ treated as a mathematical input; it lies outside the positive-temperature physical premises and has no SI temperature or pressure interpretation. Each clause is a separate conditional conclusion; no hypothesis of a stronger clause is implicit in a weaker one.

Stable power-tempered free cubes. Under ${A C}_{ω}$ , for a stable power-tempered pair potential $ϕ$ with the exact constants and tail bound of Stable power-tempered pair-potential model, the free-cube standard positional grand partition sums $Ξ_{L} (z, β)$ have a finite reduced-pressure limit $p (z, β) = \lim_{L \to \infty} L^{- d} \log Ξ_{L} (z, β) .$
Stable regular interior canonical rates and duality. Under full $A C$ , for a stable regular potential with the decreasing integrable tail of Superstable regular pair interactions and configurational ensemble notation, the free-cube exact- $N$ rate $F_{β} (ρ)$ exists for every interior attainable density $ρ > 0$ and every sequence $N_{L} / L^{d} \to ρ$ . The free grand reduced pressure $P (α, β) = \lim_{L \to \infty} L^{- d} \log Ξ_{L} (e^{α}, β)$ satisfies $P (α, β) = \sup_{ρ \geq 0} {F_{β} (ρ) + α ρ}, F_{β} (ρ) = \inf_{α \in R} {P (α, β) - α ρ}$ on the closed effective domain of $F_{β}$ . This does not assert an exact- $N$ limit at an excluded endpoint.
Superstable regular grand variation. Under $A C$ , the ultrafilter extension principle, $D C$ , and $H B$ , for a superstable regular potential and $z, β > 0$ , the normalized unit-Poisson partition rate exists for both free and periodic cubes: $p_{Q} (z, β) = \lim_{L \to \infty} L^{- d} \log E_{Q_{Λ_{L}}} ⁣ [z^{N_{Λ_{L}}} e^{- β H_{L, b}}] = - \min_{P \in P_{s t a t}^{1}} {I (P) + β Φ (P) - ρ (P) \log z}, b \in {f r e e, p e r} .$ Here $Q_{Λ_{L}}$ is unit-intensity Poisson, $P_{s t a t}^{1}$ is the stationary finite-intensity law space, $I$ is specific relative entropy to $Q$ , $ρ$ is intensity, and $Φ$ is positional energy density, all as defined in Superstable regular pair interactions and configurational ensemble notation. The minimizer set is nonempty and compact in the local-tame topology $τ_{L}$ of Local-Tame Topology. The standard $1 / N!$ Lebesgue positional partition has reduced pressure $p_{Q} + 1$ ; in $d = 3$ its physical pressure is $E_{0} (p_{Q} + 1) / (β ℓ_{0}^{3})$ Pa.
Arbitrary free-wall shapes and fixed-particle limits. Under full $A C$ , for the same superstable regular interaction, the free-wall grand reduced pressure along every van Hove sequence, as defined in Van Hove sequence of bounded regions, tends to the free-cube reduced pressure. At every interior attainable $ρ > 0$ , the free-wall exact- $N_{k}$ positional rate along every van Hove sequence with $N_{k} / ∣ Ω_{k} ∣ \to ρ$ tends to $F_{β} (ρ)$ . No stability-only arbitrary-shape upper bound or excluded-endpoint conclusion is asserted.
Periodic and free local positional equivalence. For the same superstable regular interaction and positive $β$ as in clause 3, under the four choice assumptions in that clause, if $ρ > 0$ is interior attainable, $N_{L} / L^{d} \to ρ$ , and $z > 0$ is a finite supporting activity, the periodic exact- $N_{L}$ rate is $F_{β} (ρ)$ and its periodized empirical field concentrates in every open $τ_{L}$ neighborhood of the grand variational minimizer set. The free exact- $N_{L}$ empirical field has the same neighborhood concentration. Here the periodized empirical field $R_{L, ω}$ is the spatial average of translations of the periodic continuation, as defined in Periodized empirical field of a finite configuration, and convergence uses the outer-probability convention explicitly defined in Conditioning grand empirical fields on an exact particle number. These statements concern spatially averaged local-tame observables, not the raw finite-box law at a fixed origin.
A selected shrinking positional energy window. Under clause 5, if $β \mapsto F_{β} (ρ)$ is differentiable at $β > 0$ and the supporting grand variational minimizer is a singleton $P_{*}$ , there exists a specific sequence $δ_{L} ↓ 0$ for which the periodic exact- $N_{L}$ positional shell $∣ H_{L, p e r} / L^{d} + \partial_{β} F_{β} (ρ) ∣ \leq δ_{L}$ is eventually defined and its periodized empirical field converges in probability in $τ_{L}$ to $P_{*}$ . The free and periodic exact- $N_{L}$ laws and the matching periodic grand law have the same limiting empirical field. The width is selected by the proof; no arbitrary shell width or zero-width energy surface is covered.
Uniform tempered exterior boundary. Under full $A C$ , a superstable regular interaction, a genuine finite-valued positive core with divergent radial integral as in Superquadratic cell coercivity from a divergent or hard core, and fixed $z, β > 0$ , the free-wall grand pressure converges uniformly over the exact square-count exterior class $Ω (t)$ defined in Uniform exterior-pressure limit for tempered configurations, for each finite $t$ . Its limit is the free-cube pressure. This class excludes volume-filling external fields.
Retained hard-exclusion zero-parameter branch. Let $d \geq 1$ and let $ϕ$ be superstable and regular in the exact sense of Superstable regular pair interactions and configurational ensemble notation. In the separate formal continuation with fixed dimensionless activity $z > 0$ and $β = 0$ , under $A C$ , the ultrafilter extension principle, $D C$ , and $H B$ , the model that retains the hard-forbidden set $B = {x : ϕ (x) = + \infty}$ has the free and periodic rates and minimizers specified in Zero-inverse-temperature-parameter limit with forbidden pairs retained. This is outside the positive-temperature physical premises. It is the activity-weighted model with forbidden pairs still excluded, not the unrestricted Poisson model unless $∣ B ∣ = 0$ .

All clauses are conditional model deductions. None says that the adopted postulate describes a particular material. The variational minimizers above are not thereby identified as DLR laws.

Facts & Assumptions

Given: For clauses 1–7, the physical postulate and parameter definition in the Statement; for clause 8, the separate formal dimensionless continuation and fixed mathematical activity stated there; the explicit matching of the chosen positional Hamiltonian to each supplier's pair-potential convention; and exactly the clause-specific mathematical hypotheses stated in the Statement.

[A1]

For clauses 1–7, the current draft Interacting classical Gibbs equilibrium adoption adopts the finite-region Gibbs conditional laws for a chosen Hamiltonian and reference when their normalizers are positive and finite, and defines an infinite-volume equilibrium state by the DLR conditional identities. Its particle-sector convention points to the current draft Classical particle counting and physical observables; for this single species use its $1 / N!$ factor and set $a_{0} = h_{r e f}$ as required for the thermal-wavelength convention. These are adopted conventions, not proved or empirically validated claims. Clause 8 does not use this postulate or infer its activity from physical parameters.

[A2]

For clauses 1–7, the current draft Classical Gibbs quantities and SI reference conventions uses SI units, defines $β_{S I} = (k_{B} T)^{- 1}$ in J $^{- 1}$ and continuum activity $z_{c} = e^{β_{S I} μ} λ_{T}^{- d}$ in m $^{- d}$ , with $λ_{T} = h_{r e f} / \sqrt{2 π m_{p} k_{B} T}$ . The reference scales $E_{0}, ℓ_{0}$ convert these physical inputs into the dimensionless mathematical suppliers' convention. These physical definitions do not assign a temperature interpretation to clause 8.

[A3]

In clauses 1–7 the physical model is explicitly restricted to a single-species positional Hamiltonian $H_{N} = \sum_{i < j} ϕ (q_{i} - q_{j})$ with numerical energy in units $E_{0}$ , numerical coordinates in units $ℓ_{0}$ , reference action $a_{0} = h_{r e f}$ , and the exact free, periodic, or exterior boundary and $1 / N!$ reference convention of the named mathematical supplier. Clause 8 instead uses the same dimensionless positional interaction and reference conventions as a formal mathematical continuation, with $z > 0$ held fixed independently of any physical $μ$ or $T$ . These are model-selection hypotheses, not derived identifications of arbitrary material Hamiltonians.

[A4]

The stable power-tempered clause assumes ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )) exactly as in Finite free-cube grand pressure for stable power-tempered pair potentials.

[A5]

The stable regular and superstable regular clauses assume full $A C$ (The Axiom of Choice); Superstable regular pair interactions and configurational ensemble notation is introduced under that assumption.

[A6]

The variational, zero-parameter, periodic empirical-field, and shrinking-shell clauses assume the ultrafilter extension principle (The ultrafilter extension principle (UL/BPI)) exactly where their supplier theorems state it.

[A7]

Those same clauses assume $D C$ (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain) exactly where their supplier theorems state it.

[A8]

Those same clauses assume $H B$ (The real dominated-extension principle as an additional hypothesis over ZF) exactly where their supplier theorems state it. No implication between these additional principles is used.

[F1]

Stable power-tempered pair-potential model defines an even Borel $ϕ : R^{d} \to R \cup {+ \infty}$ , $U_{N} = \sum_{i < j} ϕ (q_{i} - q_{j})$ , stability $U_{N} \geq - B N$ , and the real tail bound $∣ ϕ (x) ∣ \leq C ∥ x ∥_{2}^{- d - ϵ}$ for $∥ x ∥_{2} \geq R$ .

[F2]

Superstable regular pair interactions and configurational ensemble notation defines stable, superstable, and regular potentials; $Z (L, N; β)$ with $1 / N!$ positional normalization; and unit-Poisson partition $Z_{Q} = E_{Q} [z^{N} e^{- β H}]$ . Its variational notation is $Φ = E$ , $I$ specific relative entropy, and $ρ$ stationary particle intensity. The sector-density justification for converting $Z_{Q}$ to the standard positional partition in this item is supplied explicitly by [F20].

[F3]

Poisson Configuration Reference defines the locally finite simple configuration space, its count-generated sigma-algebra, the unit Poisson law $Q$ , and its finite-region restrictions $Q_{Λ}$ .

[F4]

Van Hove sequence of bounded regions defines a bounded measurable sequence by positive finite volumes tending to infinity and vanishing fixed-width two-sided boundary-layer ratio.

[F5]

Local-Tame Topology defines $τ_{L}$ by convergence of expectations of every measurable local function bounded by a constant times $1 + N_{A}$ on a bounded Borel set $A$ .

[F6]

Periodized empirical field of a finite configuration defines $R_{L, ω}$ by averaging translations of the periodic continuation of a finite configuration over its numerical cube.

[F7]

Finite free-cube grand pressure for stable power-tempered pair potentials assumes ${A C}_{ω}$ , stability, the stated power tail, and positive activity and inverse-energy parameter; it proves existence and finiteness of the free-cube reduced-pressure limit.

[F8]

Canonical free-cube limit at every interior attainable density proves the free-cube exact- $N$ rate $F_{β} (ρ)$ for every interior attainable $ρ > 0$ under the stable regular decreasing-integrable-tail assumptions.

[F9]

Canonical–grand-canonical thermodynamic duality identifies $P (α, β) = \lim L^{- d} \log Ξ_{L} (e^{α}, β)$ and proves both displayed convex-duality identities on the closed effective density domain; it does not assert local-observable equivalence.

[F10]

Superstable regular continuum grand pressure and variational formula assumes $A C$ , the ultrafilter extension principle, $D C$ , and $H B$ , along with a superstable regular potential and $z, β > 0$ ; it proves free/periodic normalized-Poisson pressure equality, the variational minimum, and compact nonempty minimizers.

[F11]

Zero-inverse-temperature-parameter limit with forbidden pairs retained assumes the same four choice principles, a superstable regular potential, and retained hard admissibility; it proves the $β = 0$ free/periodic rates and minimizer claims without replacing the constrained law by unrestricted Poisson unless the forbidden set is null.

[F12]

Superquadratic cell coercivity from a divergent or hard core requires either a finite-valued positive core with divergent weighted integral or a stated hard-core radius; it gives the stronger superquadratic cell-occupation coercivity used in the exterior theorem.

[F13]

Uniform exterior-pressure limit for tempered configurations assumes stability, regularity, that genuine positive-core condition, and a fixed finite square-count exterior class $Ω (t)$ ; it proves uniform convergence of the free-wall grand pressure to the free-cube value.

[F14]

Free-wall pressure on arbitrary van Hove shapes under superstability assumes superstability, regularity, and a van Hove sequence; it proves the free-wall grand-pressure limit, not a stability-only arbitrary-shape theorem.

[F15]

Periodic fixed-particle canonical limit assumes the four choice principles, a superstable regular potential, positive $β$ , and interior density; it proves the periodic exact- $N$ rate and concentration of the periodized empirical field.

[F16]

Fixed-particle local ensemble equivalence at interior density gives the corresponding free exact- $N$ empirical-field concentration under a finite supporting activity and the same compactness assumptions.

[F17]

Conditioning grand empirical fields on an exact particle number defines outer probability by infimum over measurable supersets and states the common outer-probability convention for empirical-field convergence.

[F18]

Controlled shrinking energy-density shell and three-ensemble equivalence additionally requires differentiability of $F_{β} (ρ)$ and a singleton minimizer; it constructs one shrinking positional energy-density window and asserts no arbitrary shell-width conclusion.

[F19]

Exact canonical limit on arbitrary van Hove shapes assumes a superstable regular potential, a van Hove sequence, and an interior attainable density; it proves the stated free-wall exact- $N$ rate.

[F20]

Stable finite-range continuum specification normalization and consistency proves that for every bounded Borel region $D$ and $N \geq 0$ , the unit-intensity Poisson $N$ -point sector is the pushforward of $e^{- ∣ D ∣} / N!$ times product Lebesgue measure on ordered simple tuples. In particular this holds for the bounded Borel cube $Λ_{L}$ with $∣ Λ_{L} ∣ = L^{d}$ .

Proof

1.1A1A3A4F1F7

Under clause 1, the assumptions of [F1] and [F7] are exactly the stable power-tempered model, ${A C}_{ω}$ , and positive $z, β$ . Applying [F7] gives the finite limit $p (z, β)$ asserted there. This is a reduced positional partition deduction under the adopted model [A1], not an empirical pressure measurement.

1.2A3A5F2F8F9

Under clause 2, [F2] supplies the stable regular decreasing-integrable-tail convention and full $A C$ [A5]. The density and particle-number hypotheses are exactly those of [F8], so the canonical rate exists and equals $F_{β} (ρ)$ . For each $α$ , [F9] applies to that same rate and free-cube partition, giving both convex-duality identities on its stated closed effective domain. No endpoint canonical limit is inferred from the duality identity.

1.3A3A5A6A7A8F2F3F5F10

Under clause 3, [A5]–[A8] supply the four separately stated choice principles, and [F2] fixes the same superstable regular positional model and Poisson normalization. Thus [F10] gives the free/periodic $p_{Q}$ limit, variational formula, and compact nonempty minimizer set. The definitions [F3] and [F5] identify the reference law and topology used in that theorem.

1.4A3A5F4F8F14F19

For clause 4, full $A C$ is supplied by [A5], and the sequence satisfies [F4]'s explicit van Hove boundary-layer condition, so [F14] gives the free-wall grand-pressure limit. For the exact- $N$ assertion, the same sequence and density hypotheses match [F19], which gives the claimed rate; [F8] identifies that rate as $F_{β} (ρ)$ . Neither implication supplies an excluded endpoint or removes superstability.

1.5A3A5A6A7A8F5F6F8F9F10F15F16F17

In clause 5, the four choice assumptions [A5]–[A8] are exactly those in [F15] and [F16]. The positive finite supporting activity, interior density, superstable regular potential, and exact- $N$ rate meet their hypotheses. By [F6] the observable is the spatially periodized field, by [F5] its topology tests every local observable with at-most-linear count growth, and by [F17] convergence for arbitrary events uses the stated outer probability. Applying [F15] to periodic boundary and [F16] to free boundary gives precisely the two concentration claims.

1.6A3A5A6A7A8F6F10F15F17F18

Clause 6 adds differentiability and a singleton variational minimizer to clause 5. These are exactly the extra hypotheses in [F18]. Its constructed sequence $δ_{L}$ and its proof yield the asserted periodic positional shell, while its conclusion explicitly restricts the width to that selected sequence.

1.7A3A5F2F12F13

Clause 7 has full $A C$ by [A5], the stable regular model [F2], and the genuine finite-valued positive core [F12]. The prescribed square-count class and fixed $z, β$ are the ones in [F13]. Applying that theorem gives uniform convergence to the free-cube pressure; no volume-filling field is included in its hypotheses.

1.8A3A5A6A7A8F2F11

Clause 8 assumes $d \geq 1$ and the superstable regular potential of [F2], exactly the interaction class required by [F11]. It uses the separate formal $β = 0$ continuation with fixed mathematical activity $z > 0$ from [A3] and assumes the four choice principles [A5]–[A8] exactly as in [F11]. Applying [F11] yields its free/periodic zero-parameter conclusions. Its hypothesis and conclusion retain forbidden configurations throughout, so no unrestricted Poisson claim follows when the forbidden set has positive measure; no inference about $β_{S I} > 0$ or an SI pressure is made.

2.1A2A3F2F3F20∎

By [F20], the unit-intensity Poisson $N$ -point sector on the bounded Borel cube $Λ_{L}$ has density $e^{- L^{d}} / N!$ relative to the labelled positional sector. Therefore its normalized partition is $e^{- L^{d}} Ξ_{L}$ , so the logarithmic rates differ by exactly $1$ . With numerical volume $L^{d}$ and physical volume $ℓ_{0}^{d} L^{d}$ , the physical pressure conversion is $k_{B} T p / ℓ_{0}^{d} = E_{0} p / (β ℓ_{0}^{d})$ by [A2]. For $d = 3$ , the units are J/m $^{3}$ = Pa. This conversion changes units only; it does not add momentum variables or an empirical premise.

Remarks

The stationary variational minimizers in clause 3 are not identified with DLR laws by any cited supplier. The finite-range DLR specification defined on this page is a separate local object; no existence or uniqueness of an infinite-volume DLR law is asserted here. No experiment, measured parameter, or claim of equilibration is used.

Qualified interacting continuum thermodynamic and positional ensemble limits

Statement

Facts & Assumptions

Proof

Remarks

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