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The Kirkwood closure point process: A solution of the Kirkwood-Salsburg equations for negative activities

Fabio Frommer

TL;DR

The paper proves the existence of the Kirkwood closure process for stable and regular pair potentials, linking the closure to the Kirkwood-Salsburg equations at negative activity and showing the closure is a tempered Gibbs point process with Janossy densities tied to KS solutions. It provides a constructive approach, via finite-volume KS equations and a limiting argument, and establishes GNZ-type identities for the Kirkwood closure, confirming its Gibbsian nature. Moreover, the work extends the closure framework to higher-order (multi-body) closures, giving conditions under which a bounded multi-body KS operator yields a realizable, tempered point process with higher-order correlations. These results strengthen the theoretical underpinning of the Kirkwood closure in statistical mechanics and offer rigorous guarantees for its use in inverse realizability problems and Gibbsian analyses.

Abstract

The Kirkwood superposition is a well-known tool in statistical physics to approximate the $n$-point correlation functions for $n\geq 3$ in terms of the density $ρ$ and the radial distribution function $g$ of the underlying system. However, it is unclear whether these approximations are themselves the correlation functions of some point process. If they are, this process is called the Kirkwood closure process. For the case that $g$ is the negative exponential of some nonnegative and regular pair potential $u$ existence of the the Kirkwood closure process was proved by Ambartzumian and Sukiasian. This result was generalized to the case that $u$ is a locally stable and regular pair potential by Kuna, Lebowitz and Speer, provided that $ρ$ is sufficiently small. In this work, it is shown that it suffices for $u$ to be stable and regular to ensure the existence of the Kirkwood closure process. Furthermore, for locally stable $u$ it is proved that the Kirkwood closure process is Gibbs and that the kernel of the GNZ-equation satisfies a Kirkwood-Salsburg type equation.

The Kirkwood closure point process: A solution of the Kirkwood-Salsburg equations for negative activities

TL;DR

The paper proves the existence of the Kirkwood closure process for stable and regular pair potentials, linking the closure to the Kirkwood-Salsburg equations at negative activity and showing the closure is a tempered Gibbs point process with Janossy densities tied to KS solutions. It provides a constructive approach, via finite-volume KS equations and a limiting argument, and establishes GNZ-type identities for the Kirkwood closure, confirming its Gibbsian nature. Moreover, the work extends the closure framework to higher-order (multi-body) closures, giving conditions under which a bounded multi-body KS operator yields a realizable, tempered point process with higher-order correlations. These results strengthen the theoretical underpinning of the Kirkwood closure in statistical mechanics and offer rigorous guarantees for its use in inverse realizability problems and Gibbsian analyses.

Abstract

The Kirkwood superposition is a well-known tool in statistical physics to approximate the -point correlation functions for in terms of the density and the radial distribution function of the underlying system. However, it is unclear whether these approximations are themselves the correlation functions of some point process. If they are, this process is called the Kirkwood closure process. For the case that is the negative exponential of some nonnegative and regular pair potential existence of the the Kirkwood closure process was proved by Ambartzumian and Sukiasian. This result was generalized to the case that is a locally stable and regular pair potential by Kuna, Lebowitz and Speer, provided that is sufficiently small. In this work, it is shown that it suffices for to be stable and regular to ensure the existence of the Kirkwood closure process. Furthermore, for locally stable it is proved that the Kirkwood closure process is Gibbs and that the kernel of the GNZ-equation satisfies a Kirkwood-Salsburg type equation.

Paper Structure

This paper contains 7 sections, 8 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. The Kirkwood closure process
  4. The Kirkwood-Salsburg operator
  5. Existence of the Kirkwood closure process
  6. The Kirkwood closure process is a Gibbs point process
  7. Extension to higher order closures

Key Result

Theorem A

\newlabelthm:lenard[Lenard Lenard75] Let $(\rho^{(n)})_{n\geq 1}$ be a family of nonnegative symmetric functions that satisfy (eq:Rbound) for some $\xi>0$ such that for all $n\in\mathbb{N}$, all bounded $\Lambda\subset \mathbb{R}^d$ and all $\boldsymbol{{x}}_n\in\Lambda^n$ and Then there exists a point process $\mathsf{P}$ with correlation functions $(\rho^{(n)})_{n\geq 1}$.

Theorems & Definitions (27)

  • Theorem A
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 3.1
  • Remark 3.2
  • Corollary 3.3
  • proof
  • Proposition 3.4
  • ...and 17 more