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Local kinetic energy and first and second momentum expectations

Statement

Assume AC and the hypotheses of Spatial random-occupation conditional mark entropy bound. Fix a bounded half-open cube $Δ \subseteq [0, \infty)^{d}$ ; then $Δ \subseteq Λ_{L_{n}}$ for all sufficiently large $n$ . By torus translation invariance this placement gives the same limits as any fixed translate of the cube. Let $N_{Δ} (q) = # (q \cap Δ)$ . For $b > 0$ and $K (p) = ∣ p ∣^{2} / (2 m E_{0})$ , the full shell law satisfies $E_{M_{n}^{p h}} \sum_{x \in q \cap Δ} K (p_{x}) \to d ρ ∣ Δ ∣ / (2 b), E_{M_{n}^{p h}} \sum_{x \in q \cap Δ} p_{x, i} \to 0,$ $E_{M_{n}^{p h}} \sum_{x \in q \cap Δ} p_{x, i} p_{x, j} \to (m E_{0} ρ ∣ Δ ∣ / b) δ_{i j} .$ The corresponding canonical and marked variational-grand laws have the same limits by their independent Gaussian marking. Momentum is in kg m s^-1, E_0K is in joules, and b=E_0/(k_BT) is dimensionless. These conclusions concern only the named moments, not every unbounded mark observable.

Facts & Assumptions

Given: A fixed cube, the model and Gaussian kernel, the shell and canonical laws, and the local conditional KL bound.

[A1]

AC is used in the relative-entropy variational inequality (The Axiom of Choice).

[F1]

Spatial random-occupation conditional mark entropy bound: expected conditional KL for the marks attached to points in $Δ$ , relative to $ν_{b}^{\otimes N_{Δ}}$ , tends to zero.

[F2]

Gaussian mark normalization, moments, and exponential controls: for g=K,p_i,p_ip_j the centered log moment-generating function c_g(t) is finite near zero and O(t²); the respective means are d/(2b), 0, and (mE_0/b)δ_ij.

[F3]

Total-energy concentration and controlled canonical windows and Interacting marked phase-space and reference measures: the shell is supported on $∣ H / V_{n} - e_{b} ∣ \leq δ_{n}$ , stability bounds the positional energy below, and the positional shell and canonical laws are torus-translation invariant.

[F4]

Relative Entropy Variational, Chain, and Conditioning Rules: for every bounded measurable h, D(P||Q)≥P(h)-log Q(e^h).

[F5]

Full phase shell marginal satisfies the positional variational bound: the exact-N positional canonical expectations of every local-tame test, including $N_{Δ}$ , converge to $P_{*}$ , and $N_{Δ}$ is local-tame.

[F6]

Existence of the independent Gaussian-marked positional lift: $P_{*}$ has a probability lift whose marks, conditional on positions, are independent with law $ν_{b}$ .

Proof

Given: The stated assumptions and supplier definitions.

1.1A1F1F2F3F4construct

Fix one named $g$ and write $S_{Δ} = \sum_{x \in q \cap Δ} (g (p_{x}) - E_{ν} g)$ . For positive $t$ in reciprocal units to $g$ , the unbounded test $t S_{Δ}$ is dimensionless; define $h_{M} : = \max (- M, \min (M, t S_{Δ}))$ and first use this bounded truncation in [F4]. The bounds $∣ h_{M} ∣ \leq t ∣ S_{Δ} ∣$ and $e^{h_{M}} \leq 1 + e^{t S_{Δ}}$ , together with shell integrability from [F3] and the Gaussian exponential moment [F2], permit dominated convergence as $M \to \infty$ . Since $N_{Δ} \leq N_{n}$ and $c_{g} (t)$ is finite near zero, the reference exponential moment is finite at each $n$ . Thus the variational inequality extends to $t S_{Δ}$ and gives $t E_{b_{n, Δ} (. ∣ q)} S_{Δ} \leq D (b_{n, Δ} (. ∣ q) ∣ ∣ ν_{b}^{\otimes N_{Δ}}) + N_{Δ} c_{g} (t)$ for sufficiently small t>0. Testing with -tS_Δ gives the analogous inequality for -E S_Δ and c_g(-t). The integrals are finite: the shell bounds total kinetic energy at each n, and the Gaussian exponential moments exist by [F2]. Averaging over B_n yields $∣ E_{M_{n}^{p h}} S_{Δ} ∣ \leq {E_{B_{n}} D (b_{n, Δ} ∣ ∣ ν_{b}^{\otimes N_{Δ}}) + E_{B_{n}} N_{Δ} \max (c_{g} (t), c_{g} (- t))} / t .$

2.1A1F1F2F3F5F6step 1.1algebra∎

For any fixed exact- $N_{n}$ configuration on the torus, averaging $N_{Δ}$ over all torus translations gives $N_{n} ∣ Δ ∣ / V_{n}$ : each of its $N_{n}$ points is translated into $Δ$ for a set of translation parameters of Lebesgue measure $∣ Δ ∣$ . Translation invariance of $B_{n}$ and $C_{n}^{p o s}$ therefore gives $E_{B_{n}} N_{Δ} = E_{C_{n}^{p o s}} N_{Δ} = N_{n} ∣ Δ ∣ / V_{n} \to ρ ∣ Δ ∣$ . For fixed $t > 0$ , the first numerator term in step 1.1 vanishes by [F1]. The limsup is at most $ρ ∣ Δ ∣ \max (c_{g} (t), c_{g} (- t)) / t$ ; this tends to zero as $t ↓ 0$ because $c_{g} (\pm t) = O (t^{2})$ by [F2]. Hence $E_{M_{n}^{p h}} S_{Δ} \to 0$ . Adding $E_{B_{n}} N_{Δ} E_{ν_{b}} g$ and substituting the three means in [F2] proves the shell formulas. By [F5], $E_{P_{*}} N_{Δ} = \lim_{n} E_{C_{n}^{p o s}} N_{Δ} = ρ ∣ Δ ∣$ . Conditional independent marking [F6] gives the same three limits for the variational-grand marked law; the exact canonical law has them by independent marking as well.

Local kinetic energy and first and second momentum expectations

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources