[Paper Review] Particle systems with weakly attractive interaction

Hanno Gottschalk|ArXiv.org|Sep 14, 2004

Advanced Mathematical Theories and Applications14 references3 citations

TL;DR

This paper introduces a class of classical continuous particle systems with purely attractive, yet thermodynamically stable, interactions via Kac-like potentials derived from concave, linearly bounded energy densities in a static field. Using lattice approximations and FKG correlation inequalities, it constructs infinite-volume Gibbs measures for arbitrary activity and temperature, proving their uniqueness and invariance under Euclidean transformations, even for long-range, integrable interactions and multi-component systems.

ABSTRACT

Systems of classical continuous particles in the grand canonical ensemble interacting through purely attractive, yet stable, interactions are defined. By a lattice approximation, FKG ferromagnetic inequalities are shown to hold for such particle systems. Using these inequalities, a construction of the infinite volume measures is given by a monotonicity and upper bound argument. Invariance under Euclidean transformations is proven for the infinite volume measures. The construction works for arbitrary activity and temperature and for integrable long range interactions. Also, inhomogeneous systems of particles with different "charge" can be treated.

Motivation & Objective

• To define a class of purely attractive, yet stable, particle interactions without a repulsive core, enabling modeling of soft matter and biological systems.
• To overcome limitations of traditional pair-potential models that require a repulsive core for stability, which hinders analysis outside the low-density/high-temperature regime.
• To extend FKG correlation inequalities from discrete spin systems to continuous particle systems, enabling rigorous construction of infinite-volume measures.
• To establish the existence and uniqueness of infinite-volume Gibbs measures for arbitrary activity and temperature, including long-range and inhomogeneous interactions.
• To prove invariance of the resulting measures under Euclidean transformations, ensuring physical consistency in spatially homogeneous systems.

Proposed method

• Define a new class of interaction potentials using concave, linearly bounded energy densities in the static field generated by particle charges, ensuring thermodynamic stability.
• Approximate the continuous particle system via a lattice gas model, enabling application of ferromagnetic FKG inequalities to the discrete approximation.
• Use monotonicity and upper bound arguments based on FKG inequalities to construct the thermodynamic limit of the grand canonical ensemble.
• Apply Minlos' theorem to the characteristic functional of the limiting measure, ensuring weak convergence and existence of the infinite-volume measure.
• Prove invariance under Euclidean transformations by showing that the measure is invariant under translations and rotations via dual action of the transformation on the configuration space.
• Handle long-range interactions by requiring integrability of the interaction kernel, ensuring convergence of energy integrals and stability of the system.

Experimental results

Research questions

• RQ1 Can purely attractive, stable particle interactions be rigorously defined in continuous systems without a repulsive core?
• RQ2 Can FKG correlation inequalities be extended from discrete spin systems to continuous particle systems with weakly attractive interactions?
• RQ3 Is it possible to construct infinite-volume Gibbs measures for such systems at arbitrary activity and temperature, without low-density or high-temperature assumptions?
• RQ4 Do the resulting infinite-volume measures remain invariant under Euclidean transformations, ensuring physical consistency?
• RQ5 Can the framework be extended to multi-component systems with different particle types and charge-like interactions?

Key findings

• A new class of purely attractive, yet stable, interaction potentials is defined via concave, linearly bounded energy densities in a static field, avoiding the need for a repulsive core.
• The FKG inequality is rigorously established for the continuous particle system through lattice approximation, enabling monotonicity-based construction of the thermodynamic limit.
• The infinite-volume Gibbs measure exists and is unique for arbitrary activity and temperature, without requiring low-density or high-temperature conditions.
• The constructed infinite-volume measures are invariant under Euclidean transformations (rotations, translations), ensuring spatial homogeneity and isotropy.
• The method applies to integrable long-range interactions and extends to inhomogeneous systems with multiple particle types.
• The characteristic functional of the limiting measure is shown to be continuous and positive definite, allowing the application of Minlos' theorem to define the measure uniquely.