Gibbs measure over the cone of vector-valued discrete measures

2025/07/14 by Luca Di Persio, Di Persio, Luca, Yuri Kondratiev +3
Mathematics · #05A40 #46E50 #60G55 #60H40 #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2507.10071

openalex publication_date 2025/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a gas whose each particle is characterised by a pair (x,v_x) with the position x∈ \mathbb R^d and the velocity v_x∈ \mathbb R^d₀= \mathbb R^d∖ \0\. We define Gibbs measures on the cone of vector-valued measures and aim to prove their existence. We introduce the family of probability measures μ_λ on the cone \mathbb K(\mathbb R^d). We define local Hamiltonian and partition functions for a positive, symmetric, bounded and measurable pair potential. Using those above, we define Gibbs's measure as a solution to the Dobrushin-Lanford-Ruelle equation. In particular, we focus on the subset of tempered Gibbs measures. To prove the existence of the Gibbs measure, we show that the subset of tempered Gibbs measures is non-empty and relatively compact.

Citations

• Classical gases with singular densities
• Laplace operators on the cone of Radon measures
• Gibbs states over the cone of discrete measures

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