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A Leibniz rule of distributional pairing and hyperforce sum rule

Takashi Maruyama, Tatsuki Seto, Viktor Zaverkin, Henrik Christiansen

Abstract

We reformulate and generalize the equilibrium hyperforce sum rule, a generalization of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy, by employing the Schwartz space and its dual. We show that the hyperforce sum rule for the Euclidean space and the equilibrium BBGKY hierarchy at arbitrary level are derived through the Leibniz rule of the derivative for the pairing of tempered distributions and Schwartz functions. We also apply the Leibniz rule to obtain the hyperforce sum rule for systems with periodic boundary conditions.

A Leibniz rule of distributional pairing and hyperforce sum rule

Abstract

We reformulate and generalize the equilibrium hyperforce sum rule, a generalization of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy, by employing the Schwartz space and its dual. We show that the hyperforce sum rule for the Euclidean space and the equilibrium BBGKY hierarchy at arbitrary level are derived through the Leibniz rule of the derivative for the pairing of tempered distributions and Schwartz functions. We also apply the Leibniz rule to obtain the hyperforce sum rule for systems with periodic boundary conditions.
Paper Structure (11 sections, 20 theorems, 153 equations, 2 tables)

This paper contains 11 sections, 20 theorems, 153 equations, 2 tables.

Table of Contents

  1. Preliminary and assumption
  2. Bogolyubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy
  3. Hyperforce sum formula
  4. Case: $\mathbb{R}^{d}$
  5. Generalized thermal average and hyperforce distribution
  6. Application
  7. Physical interpretation of distributional hyperforce sum formula
  8. Distributional hyperforce sum rules for periodic boundary systems
  9. Definition and Basic Properties of Tempered Distributions
  10. Multivariate Faà di Bruno's formula
  11. (Partially) extended real numbers

Key Result

Theorem A

Let $N \in \symbb{N}$ and $0 \leq n \leq N-1$. For a tempered distribution $u \in \symscr{S}^{\prime}(\symbb{R}^{d(N-n)\times2})$ and a Schwartz function $\phi \in \symscr{S}(\symbb{R}^{dN\times2})$, we define the localized hyperforce of $u$ and $\phi$ centered at $i$ to be a continuous-function-val Then, the sum of the localized hyperforce coincides with the equilibrium distributional hyperforce

Theorems & Definitions (52)

  • Theorem A: Theorem \ref{['thm:hyperforce_sum_rewrite']} and Corollary \ref{['cor:hyperforcesum']}
  • Theorem B: Lemma \ref{['cor:general_derivation_hyperforce_sum_rule']} and Theorem \ref{['thm:generalized_BBGKY']}
  • Theorem C: Corollary \ref{['cor:derivation_hyperforce_sum_rule']} and Corollary \ref{['cor:generalized_168']}
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Definition 2.3: Generalized thermal average
  • Example 2.4
  • Example 2.5
  • ...and 42 more