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DLR Equations for the Superstable Bose Gas at any Temperature and Activity

Guillaume Bellot, David Dereudre, Mylène Maïda

TL;DR

The paper establishes a rigorous thermodynamic limit for the grand canonical Bose gas with superstable, finite-range interactions within the Feynman–Kac path-space framework, for all inverse temperatures $β>0$ and chemical potentials $μ∈ℝ$. It introduces and analyzes three equivalent representations—FK path space, rooted loops, and marked points—then proves the existence of an infinite-volume, stationary FK measure $\mathbb{P}_{\infty}^{(FK)}$ via an entropic method, and shows that under finite-range interactions this limit satisfies DLR equations describing local conditional Gibbs structure. The work further proves thermodynamic-limit results for the rl and mp models, establishes their equivalence with FK in finite volume, and extends locality concepts to the infinite-volume setting, paving the way to study interlacements and Bose–Einstein condensation phenomena within a Gibbsian infinite-volume framework. These contributions provide a solid probabilistic foundation for analyzing infinite cycles and possible interlacement formation in interacting Bose gases. Overall, the paper offers a unified, rigorous route from FK representations to DLR-described infinite-volume Gibbs states, with implications for understanding BEC and related percolation phenomena in quantum gases.

Abstract

We construct a thermodynamic limit for the grand canonical Bose gas in dimension $d\geqslant1$ (in its Feynman-Kac representation) with superstable interaction at any inverse temperature $β>0$ and any chemical potential $μ\in\mathbb{R}$. Our infinite volume model is naturally a distribution over configurations of finite loops and possibly interlacements. We prove the limiting process to solve a new class of DLR equations involving random permutations and Brownian paths.

DLR Equations for the Superstable Bose Gas at any Temperature and Activity

TL;DR

The paper establishes a rigorous thermodynamic limit for the grand canonical Bose gas with superstable, finite-range interactions within the Feynman–Kac path-space framework, for all inverse temperatures and chemical potentials . It introduces and analyzes three equivalent representations—FK path space, rooted loops, and marked points—then proves the existence of an infinite-volume, stationary FK measure via an entropic method, and shows that under finite-range interactions this limit satisfies DLR equations describing local conditional Gibbs structure. The work further proves thermodynamic-limit results for the rl and mp models, establishes their equivalence with FK in finite volume, and extends locality concepts to the infinite-volume setting, paving the way to study interlacements and Bose–Einstein condensation phenomena within a Gibbsian infinite-volume framework. These contributions provide a solid probabilistic foundation for analyzing infinite cycles and possible interlacement formation in interacting Bose gases. Overall, the paper offers a unified, rigorous route from FK representations to DLR-described infinite-volume Gibbs states, with implications for understanding BEC and related percolation phenomena in quantum gases.

Abstract

We construct a thermodynamic limit for the grand canonical Bose gas in dimension (in its Feynman-Kac representation) with superstable interaction at any inverse temperature and any chemical potential . Our infinite volume model is naturally a distribution over configurations of finite loops and possibly interlacements. We prove the limiting process to solve a new class of DLR equations involving random permutations and Brownian paths.

Paper Structure

This paper contains 20 sections, 30 theorems, 310 equations, 7 figures.

Table of Contents

  1. Introduction and Results
  2. Interaction and Assumptions
  3. Feynman-Kac State Space
  4. Locality and Tameness
  5. Thermodynamic Limit
  6. DLR Equations
  7. Equivalent Models
  8. Rooted Loops (rl)
  9. Marked Points (mp)
  10. Proof of Equivalences
  11. Proofs
  12. Entropic Method
  13. Entropic Bounds
  14. Permutation in Infinite Volume
  15. Thermodynamic Limit
  16. ...and 5 more sections

Key Result

Proposition 1.3.5

Let $f$, $\Delta\subset\mathbb{R}^d$ a compact and $a>0$ be such that for any $\gamma,\gamma'\in \textnormal{ConfPerm}^{(\textnormal{FK})}$, where $\textnormal{cyc}\mleftright\br{\gamma,\Delta,n}$ denotes the set of cycles of $\sigma^{(\textnormal{FK})}\mleftright\br{\gamma,\cdot}$ of length at most $n$ whose trajectories intersect $\Delta$. Then $f$ is $\cap^n$-Lipschitz.

Figures (7)

  • Figure 1: Illustration of various projection maps
  • Figure 2: Example of (FK) configuration
  • Figure 3: Exterior and interior configurations relatively to $\Delta$
  • Figure 4: Resampling in the DLR equations
  • Figure 5: Use of the $p$ and $u$ marks in the (mp) encoding
  • ...and 2 more figures

Theorems & Definitions (110)

  • Definition 1.1.1
  • Remark 1.1.2
  • Definition 1.1.3
  • Definition 1.2.1
  • Definition 1.2.2
  • Definition 1.2.3
  • Definition 1.3.1
  • Definition 1.3.2
  • Definition 1.3.3
  • Remark 1.3.4
  • ...and 100 more