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The variational principle for a marked Gibbs point process with infinite-range multibody interactions

Benedikt Jahnel, Jonas Köppl, Yannic Steenbeck, Alexander Zass

TL;DR

This work establishes the Gibbs variational principle for a continuum gas with unbounded particle marks and infinite-range multibody interactions in the Asakura–Oosawa depletion framework. By developing a rigorous probabilistic setup based on Palm measures, DLR equations, and tempered boundary conditions, the authors prove existence of infinite-volume Gibbs measures and analyze the energy and entropy densities along with the thermodynamic pressure. The main result demonstrates that the functional $P\mapsto I^z(P)+\beta H(P)$ attains a finite infimum equal to $-\psi(z,\beta)$, and that minimizers are precisely translation-invariant Gibbs measures; conversely, every such Gibbs measure minimizes the functional. This extends the Gibbs variational principle to systems with unbounded radii and infinite-range multibody interactions, providing a foundational framework for continuum statistical mechanics of complex colloid–polymer mixtures.

Abstract

We prove the Gibbs variational principle for the Asakura--Oosawa model in which particles of random size obey a hardcore constraint of non-overlap and are additionally subject to a temperature-dependent area interaction. The particle size is unbounded, leading to infinite-range interactions, and the potential cannot be written as a $k$-body interaction for fixed $k$. As a byproduct, we also prove the existence of infinite-volume Gibbs point processes satisfying the DLR equations. The essential control over the influence of boundary conditions can be established using the geometry of the model and the hard-core constraint.

The variational principle for a marked Gibbs point process with infinite-range multibody interactions

TL;DR

This work establishes the Gibbs variational principle for a continuum gas with unbounded particle marks and infinite-range multibody interactions in the Asakura–Oosawa depletion framework. By developing a rigorous probabilistic setup based on Palm measures, DLR equations, and tempered boundary conditions, the authors prove existence of infinite-volume Gibbs measures and analyze the energy and entropy densities along with the thermodynamic pressure. The main result demonstrates that the functional attains a finite infimum equal to , and that minimizers are precisely translation-invariant Gibbs measures; conversely, every such Gibbs measure minimizes the functional. This extends the Gibbs variational principle to systems with unbounded radii and infinite-range multibody interactions, providing a foundational framework for continuum statistical mechanics of complex colloid–polymer mixtures.

Abstract

We prove the Gibbs variational principle for the Asakura--Oosawa model in which particles of random size obey a hardcore constraint of non-overlap and are additionally subject to a temperature-dependent area interaction. The particle size is unbounded, leading to infinite-range interactions, and the potential cannot be written as a -body interaction for fixed . As a byproduct, we also prove the existence of infinite-volume Gibbs point processes satisfying the DLR equations. The essential control over the influence of boundary conditions can be established using the geometry of the model and the hard-core constraint.
Paper Structure (25 sections, 29 theorems, 162 equations, 1 figure)

This paper contains 25 sections, 29 theorems, 162 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setting and Notation
  3. Point process formalism
  4. Colloidal multi-body interaction
  5. Gibbsian formalism
  6. Specific entropy
  7. Energy density
  8. The pressure
  9. Main result
  10. Strategy of proof
  11. The energy density and its continuity properties
  12. Tempered configurations and measures
  13. Existence of the pressure
  14. The variational principle
  15. Preliminaries
  16. ...and 10 more sections

Key Result

Proposition 2.4

[proposition]prop_specific_entropy_basics For every activity $z > 0$ and $P \in \mathcal{P}_\Theta$, the specific entropy exists. Moreover, the map $P \mapsto I^z(P)$ is lower-semicontinuous and the level sets $\{I^z \leq c\}, c \in \mathbb{R},$ are compact and sequentially compact w.r.t. the $\tau_\mathcal{L}$-topology.

Figures (1)

  • Figure 1: Such a geometric arrangement forces balls with centers outside $\Delta$, which intersect $[-A, A]^d \supset B(0,\epsilon^{-1})$, to contain at least one of the $\Delta_j$.

Theorems & Definitions (57)

  • Remark 2.1: $k$-body interactions via sufficiently small $r$
  • Definition 2.2: Finite-volume Gibbs measures
  • Definition 2.3: Gibbs measures
  • Proposition 2.4: Specific entropy
  • Theorem 3.1: Gibbs variational principle
  • Corollary 3.2: Existence
  • Proposition 4.1
  • Definition 4.2: Temperedness
  • Lemma 4.3: Finite specific entropy implies temperedness
  • Lemma 4.4: Infinite-volume Gibbs measures are tempered
  • ...and 47 more