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Exact normalized BBGKY hierarchy

Statement

Fix the smooth periodic finite-N model and complete flow $Φ_{t}^{N}$ of Complete finite-N flow and permutation-invariant Liouville law. Let $F_{N} (t) = F_{N}^{0} \circ Φ_{- t}^{N}$ be the transported normalized symmetric $C^{\infty}$ probability density relative to canonical phase volume, with all phase derivatives rapidly decreasing in momentum uniformly in position on each finite time slab. Here $z_{i} = (q_{i}, p_{i})$ , and rapid decrease means that for every position multi-index $α$ , momentum multi-index $β$ , and integer $r \geq 0$ , $(1 + ∣ p ∣)^{r} ∣ \partial_{q}^{α} \partial_{p}^{β} F_{N} (t, q, p) ∣$ is uniformly bounded on that slab. For $1 \leq k \leq N$ , define the normalized marginal

$f_{k} (t, z_{1}, \dots, z_{k}) : = \int F_{N} (t, z_{1}, \dots, z_{N}) d z_{k + 1} \dots d z_{N} .$

Then, for $1 \leq k \leq N$ ,

$\partial_{t} f_{k} + \sum_{i = 1}^{k} \frac{p_{i}}{m} \cdot \nabla_{q_{i}} f_{k} + \frac{1}{N} \sum_{1 \leq i < j \leq k} K (q_{i} - q_{j}) \cdot (\nabla_{p_{i}} - \nabla_{p_{j}}) f_{k} + \frac{N - k}{N} \sum_{i = 1}^{k} \nabla_{p_{i}} \cdot \int K (q_{i} - q_{k + 1}) f_{k + 1} (t, z_{1}, \dots, z_{k + 1}) d z_{k + 1} = 0.$

When $k = N$ , the last term is zero and no $f_{N + 1}$ is used. This identity is exact at finite $N$ and makes no factorization or molecular-chaos assumption.

Facts & Assumptions

Given: The smooth complete finite-N Hamiltonian flow and its transported symmetric rapidly decreasing density from Complete finite-N flow and permutation-invariant Liouville law, the canonical phase volume, and the countable-choice condition below.

[F1]

Complete finite-N flow and permutation-invariant Liouville law: for the smooth even periodic potential, the complete flow is volume-preserving and permutation-equivariant; a symmetric smooth density with all phase derivatives rapidly decreasing in momentum is transported to a symmetric classical Liouville solution with the same regularity.

[F2]

Tonelli and Fubini for the completed product, with only almost-everywhere section measurability: assuming countable choice, integrable functions on completed products of sigma-finite measure spaces have measurable integrable sections almost everywhere and equal iterated integrals.

[F3]

Smooth periodic finite-N and scaled mean-field phase data: $K = - \nabla W$ for a smooth even periodic potential $W$ , so $K$ is odd; the canonical Hamilton equations are ${\dot{q}}_{i} = p_{i} / m$ and ${\dot{p}}_{i} = N^{- 1} \sum_{j \neq i} K (q_{i} - q_{j})$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): assume the Axiom of Countable Choice, as required by [F2].

Proof

1.1A1F1F3algebra

Use the Hamilton equations and odd force $K$ from [F3], with the classical Liouville solution from [F1] under the countable-choice density measure assumption [A1]. Substituting these equations into $\partial_{t} F_{N} + \sum_{i} ({\dot{q}}_{i} \cdot \nabla_{q_{i}} F_{N} + {\dot{p}}_{i} \cdot \nabla_{p_{i}} F_{N}) = 0$ and grouping the two terms for each unordered pair $i < j$ yields $\partial_{t} F_{N} + \sum_{i = 1}^{N} \frac{p_{i}}{m} \cdot \nabla_{q_{i}} F_{N} + \frac{1}{N} \sum_{1 \leq i < j \leq N} K (q_{i} - q_{j}) \cdot (\nabla_{p_{i}} - \nabla_{p_{j}}) F_{N} = 0.$ The sign in the second paired derivative uses $K (q_{j} - q_{i}) = - K (q_{i} - q_{j})$ .

2.1A1F1F2step 1.1

Fix $k < N$ and integrate step 1.1 over $z_{k + 1}, \dots, z_{N}$ . The integrations and time differentiation are legitimate by the all-phase-derivative rapid tails (the time derivative from step 1.1 adds at most one factor of momentum), compact periodic position variables, and [A1, F2]. A position derivative in an integrated variable has zero integral by periodicity; a momentum derivative in an integrated variable has zero boundary term by the stated decay. Thus one-body terms for $i \leq k$ and interaction pairs entirely inside ${1, \dots, k}$ give the first three terms of the target equation. Pairs with both labels greater than $k$ integrate to zero by the same momentum integration-by-parts argument.

3.1F1F2step 2.1algebra

For a cross pair $i \leq k < j$ , the term containing $\nabla_{p_{j}} F_{N}$ integrates to zero because $K (q_{i} - q_{j})$ is independent of $p_{j}$ and the $p_{j}$ -tail vanishes at infinity. The term containing $\nabla_{p_{i}} F_{N}$ remains. By permutation symmetry of $F_{N}$ , for fixed $i \leq k$ every label $j > k$ has the same integrated contribution, equal to $\nabla_{p_{i}} \cdot \int K (q_{i} - q_{k + 1}) f_{k + 1} (t, z_{1}, \dots, z_{k}, z_{k + 1}) d z_{k + 1} .$ There are exactly $N - k$ such labels, each carrying coefficient $1 / N$ . This gives the factor $(N - k) / N$ and all terms of the displayed hierarchy.

4.1step 2.1step 3.1algebra∎

If $k = N$ , there are no cross pairs, so the same integration argument leaves only the one-body transport and internal-pair terms; the formula's cross coefficient is $(N - N) / N = 0$ . Thus the statement holds for every $1 \leq k \leq N$ , with no factorization assumption at any step.

Depends on

Dependency tree · 2 levels

Sources

François Golse, The Mean-Field Limit for the Dynamics of Large Particle Systems, Journées Équations aux dérivées partielles short course (2003)