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Conditional mean-field particle distribution limit

Physical scope

The exact adopted smooth mean-field model/compact iid preparation; justified propagation of chaos in this regime, no Boltzmann-Grad interpretation.

Statement

Adopt the finite-particle model and compact-support iid preparation of Selected mean-field particle model and iid preparation, including its stated full Axiom of Choice for constructing the infinite iid sequence. There are two separate conditional consequences within that model.

First, if the initial $N$ -particle law is a normalized symmetric smooth density with the rapid momentum tails specified in Exact normalized BBGKY hierarchy, its evolved marginals satisfy the exact normalized BBGKY hierarchy for every $1 \leq k \leq N$ . This is a finite- $N$ identity and retains all correlations.

Second, for the compact iid preparation with common law $μ_{0}$ , for every fixed $k$ and finite $T$ , the first $k$ particle law converges to $μ_{s}^{\otimes k}$ uniformly for $∣ s ∣ \leq T$ in bounded-Lipschitz tests. For each bounded Lipschitz $Ψ : Z^{k} \to R$ and $N \geq k$ , its error is at most

$2 ∥ Ψ ∥_{\infty} \frac{k (k - 1)}{2 N} + k Lip (Ψ) e^{(1 + 2 A) T} E W_{1} (μ_{N} (0), μ_{0}),$

where $A = ∥ D k ∥_{\infty}$ , and the expected initial error tends to zero by the compact-support estimate of Compact iid empirical convergence and propagation of chaos. This is a deduction from the adopted model and preparation assumptions; it is not a measured observation, a real-gas approximation, or a Boltzmann-Grad limit.

Facts & Assumptions

Given: The adopted particle model and iid preparation, full Axiom of Choice for the infinite product construction, countable choice for the completed product argument in the exact hierarchy, and the regularity branch stated above.

[A1]

Selected mean-field particle model and iid preparation: adopt the specified smooth periodic $N^{- 1}$ Hamiltonian dynamics and compact-support iid preparation; its Ionescu--Tulcea construction is under full Axiom of Choice.

[F1]

Exact normalized BBGKY hierarchy: under the normalized symmetric smooth-density and rapid-tail hypotheses, the finite- $N$ marginals satisfy the displayed exact hierarchy; its completed-product Tonelli/Fubini input assumes The Axiom of Countable Choice ( ${A C}_{ω}$ ).

[F2]

Compact iid empirical convergence and propagation of chaos: for compactly supported iid initial data, expected empirical $W_{1}$ tends to zero and the fixed- $k$ test error is bounded by the collision term plus $k Lip (Ψ) e^{(1 + 2 A) T} E W_{1} (μ_{N} (0), μ_{0})$ .

[A2]

The Axiom of Choice: retain the full Choice assumption used by the countable iid construction; no weaker choice principle is substituted.

[A3]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): assume countable choice for the exact completed-product hierarchy argument.

Proof

1.1A1A3F1given

Under the first stated regularity branch, the model's smooth complete finite- $N$ flow transports the normalized symmetric density. Apply [F1] with its exact smoothness, symmetry, rapid-tail and countable-choice assumptions [A3]. It yields the normalized hierarchy for every $k$ , with coefficient $(N - k) / N$ and no factorization premise; this conclusion is entirely finite- $N$ .

1.2A1A2A3F1F2given

Under the iid branch, [A1] supplies the adopted common law and the countably many iid coordinates under full Choice [A2]. Countable choice [A3] is separately the assumption carried by the completed-product argument in [F1]; neither choice use is inferred from the physical model. The compact iid theorem [F2] applies to these same first $N$ coordinates and their finite- $N$ flow. Its finite-k estimate is precisely the inequality in the Statement.

2.1F2givenalgebra∎

For fixed $k$ and $T$ , the first term in the [F2] bound tends to zero as $N \to \infty$ , and its second term tends to zero because the initial empirical expected Wasserstein distance tends to zero for compactly supported iid data. The constants are uniform for $∣ s ∣ \leq T$ . Therefore the fixed- $k$ laws converge in every bounded-Lipschitz test on each finite time interval. This is a conditional model deduction with the stated preparation and scaling; no experimental sampling or alternate collision limit enters the argument.

Remarks

The exact hierarchy and the iid limit are distinct branches: the former requires a smooth rapidly decreasing $N$ -body density, while the latter requires compactly supported iid preparation. The statement does not claim that either premise is empirically realized.

Conditional mean-field particle distribution limit

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

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