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Physical interacting local phase-space ensemble bridge

Physical scope

Conditional predictions for fixed marked local observables and the named momentum moments in the adopted single-species classical preparation model; no equilibration, experiment, material fit, finite-volume grand limit, or universal shell-width claim.

Statement

Under the hypotheses and adopted preparations below, shell and exact-N canonical expectations of every fixed bounded marked-local observable converge to the same Gaussian-marked positional variational state. The shell also has the kinetic-energy, first-momentum and second-momentum expectation limits stated in Local kinetic energy and first and second momentum expectations. This is a conditional deduction from the model assumptions, not an observation or an equilibration claim.

Facts & Assumptions

Given: AC, the ultrafilter extension principle, DC and HB; an integer dimension $d \geq 1$ ; integer side lengths $L_{n} \to \infty$ , $V_{n} = L_{n}^{d}$ , and exact particle numbers $N_{n} \in N$ with $N_{n} / V_{n} \to ρ > 0$ at an interior attainable density; $T > 0$ kelvin, $k_{B} > 0$ J K $^{- 1}$ , $β_{p h y s} : = 1 / (k_{B} T)$ in J $^{- 1}$ , and $b = E_{0} β_{p h y s} = E_{0} / (k_{B} T)$ ; the superstable regular interaction with periodic boundary condition $b c = p e r$ ; existence of the positional canonical log rate $F_{b^{'}} (ρ)$ for every $b^{'}$ in a neighborhood of $b$ , differentiability at $b$ , a finite supporting $α$ with $P_{L e b} (α, b) = F_{b} (ρ) + α ρ$ , and a singleton positional grand variational minimizer. Adopt the single-species preparations of Classical full-phase interacting ensemble preparations under the exact conditions stated there. By that postulate and Interacting marked phase-space and reference measures, the finite-volume physical Hamiltonian is $E_{0} H_{L_{n}, p e r}$ , and the physical preparation laws match the normalized mathematical canonical and band laws. Here $ℓ_{0}$ is the positive position unit in metres, $E_{0}$ the positive energy unit in joules, $m$ the positive mass in kilograms, $p_{*}$ the positive momentum reference in kg m s $^{- 1}$ , and the local region is fixed in dimensionless $x = q / ℓ_{0}$ coordinates. The numerical intensity $ρ$ corresponds to physical number density $ρ / ℓ_{0}^{d}$ in m $^{- d}$ .

[A1]

The full phase-band, canonical, and grand laws are adopted preparations under Classical full-phase interacting ensemble preparations; no dynamical or empirical premise is added.

[A2]

AC, the ultrafilter extension principle, DC, and HB are the mathematical choice hypotheses carried by the positional convergence suppliers (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, The real dominated-extension principle as an additional hypothesis over ZF).

[F1]

Full interacting shell, canonical, and variational-grand local marked equivalence: shell and exact-N canonical expectations converge for each fixed bounded measurable marked-local observable to the same Gaussian-marked positional variational state; shell convergence also covers |G|≤C(1+N_Δ).

[F2]

Local kinetic energy and first and second momentum expectations: shell expectations of local kinetic energy and the first and second momentum sums have the stated limits.

[F3]

Interacting marked phase-space and reference measures: integrating the finite-volume grand preparation's momentum factors maps full activity $z_{p h}$ to positional activity $z_{p h} κ_{b}$ . Thus the activity matching $e^{α}$ is $e^{α} / κ_{b}$ ; this normalization identity makes no finite-volume grand-convergence claim.

Proof

Given: The hypotheses and adopted preparation above.

1.1A1A2F1F3

A fixed bounded marked-local observable is bounded on its named region, so [F1] applies directly to shell and canonical preparations. Their common limiting law is the independent Gaussian mark lift of the positional grand variational minimizer $P_{*}$ . The finite-volume normalization in [F3] identifies the full-grand activity that matches positional activity $\exp (α)$ as $\exp (α) / κ_{b}$ ; it does not assert convergence of finite-volume grand preparations.

2.1A1F2algebra∎

For the named momentum-moment limits, fix a bounded half-open cube $Δ \subseteq [0, \infty)^{d}$ , or any fixed translate of such a cube, as in [F2]. Apply [F2] on this cube. Restoring units multiplies dimensionless kinetic energy by $E_{0}$ and uses $E_{0} / b = k_{B} T = 1 / β_{p h y s}$ ; the local kinetic-energy limit is $d ρ ∣ Δ ∣ / (2 β_{p h y s})$ joules. The first momentum sum tends to zero, and the momentum-product sum tends to $m ρ ∣ Δ ∣ δ_{i j} / β_{p h y s}$ in SI units. These are conditional model deductions, not measured values or claims of equilibration.

Physical interacting local phase-space ensemble bridge

Statement

Facts & Assumptions

Proof

Depends on

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