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arxiv: 2604.14318 · v1 · submitted 2026-04-15 · 🧮 math.PR

The free energy of the interacting Bose gas: a variational description with loops and interlacements

Wolfgang K\"onig This is my paper

Pith reviewed 2026-05-10 11:58 UTC · model grok-4.3

classification 🧮 math.PR
keywords interacting Bose gasfree energyBrownian loop souprandom interlacementsvariational formulaspecific relative entropylarge deviationsthermodynamic limit
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The pith

The limiting free energy of the interacting Bose gas equals the infimum of interaction energy plus specific relative entropy over stationary point processes of loops and interlacements at fixed density.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper derives a variational description for the free energy of the interacting Bose gas in the thermodynamic limit using a path-integral approach based on the Brownian loop soup. It incorporates random interlacements to handle long loops, which are presumed to form the condensate according to Feynman's suggestion, allowing the formula to apply for any temperature and density. The expression minimizes the sum of the interaction energy and a new specific relative entropy density taken with respect to the Brownian loop soup together with a Markov kernel for path shreds. The derivation and the existence of the entropy term both rely on large-deviation principles. A sympathetic reader would care because the result supplies explicit control over particle numbers in short and long loops and removes the small-density restriction of earlier work.

Core claim

The paper establishes that the limiting free energy equals the infimum, taken over all stationary point processes consisting of loops and interlacements with prescribed particle density, of the sum of the interaction energy and the specific relative entropy density with respect to the reference Brownian loop soup combined with an independent Markov kernel that describes collections of path shreds inside large boxes.

What carries the argument

The variational problem over stationary point processes of loops and interlacements, whose entropy term is the specific relative entropy density relative to the Brownian loop soup and the Markov kernel for path shreds.

If this is right

  • The variational formula holds for every positive temperature and every positive particle density.
  • In dimensions three and higher the long loops are represented by the Brownian interlacement Poisson point process with beta-spacing.
  • The formula yields separate control on the particle numbers carried by short loops and by long interlacements.
  • Existence of the specific relative entropy density is proved via large-deviation theory for the same class of processes.
  • The approach recovers and extends the small-density results of earlier work that lacked a description of long loops.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical minimization of the functional over discretized or parametrized point processes could yield practical approximations to the free energy in regimes where analytic solutions are unavailable.
  • Techniques developed for random interlacements, such as those from percolation theory, might be imported to study the onset of condensation within this variational setting.
  • Verification against the exactly solvable ideal Bose gas could provide a direct test of whether the entropy term correctly captures the non-interacting limit.

Load-bearing premise

The specific relative entropy density with respect to the Brownian loop soup together with the Markov kernel for path shreds is well-defined and finite for the relevant processes.

What would settle it

An independent computation of the free energy, for instance by direct Monte Carlo sampling of the path measure in a large finite box followed by extrapolation to the thermodynamic limit, that fails to equal the value of the variational infimum at the same temperature and density.

Figures

Figures reproduced from arXiv: 2604.14318 by Wolfgang K\"onig.

Figure 1
Figure 1. Figure 1: Illustration (taken from [CJK23]) of a realisation of the Bose gas with 14 particles (grey and black bullets), organised in three Brownian bridges, attached to three Poisson points (black bullets). The red cycle has six particles, the blue and green ones each four. 1.4. Purpose of this paper. The main purpose of the present paper is to identify, in the ther￾modynamic limit, the limiting free energy with di… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration (taken from [CJK23]) of two realisations of a Bose gas: left with low density with only short loops, and right with higher density with one additional long loop (red). The number of all the other particles is denoted by N (ℓ,¬R) Λ (ω) = X x∈ζ∩Λ ℓ(fx)1l{fx is not R-crossing}, (1.15) i.e., the number of particles in loops whose particles are entirely contained in one of the subboxes z + W. Note … view at source ↗
read the original abstract

We consider the interacting Bose gas in the thermodynamic limit in a large box in $\R^d$ at positive temperature $1/\beta\in(0,\infty)$ with particle density $\sim\rho\in(0,\infty)$. We follow a path-integral approach and adopt from \cite {ACK10} a description of the free energy in terms of the {\it Brownian loop soup}, a Poisson point process consisting of Brownian bridges, also called loops or cycles. It is the objective of this paper to derive, for any values of $\beta$ and $\rho$, a formula for the limiting free energy with explicit control on the particle numbers in the short and in the long loops. The latter are presumed to play the role of the condensate, according to Feynman's \cite{F53} famous, vague suggestion, and they turn into {\it random interlacements} (bi-infinite, locally finite random processes in $\R^d$) in our formula. In \cite{ACK10} there was no concept that could describe the long loops; only small $\rho$ could be handled successfully. In the present paper we represent the limiting free energy in terms of a variational formula, ranging over the set of all stationary point processes with loops and with interlacements, having each a given particle density, and minimizing the sum of the interaction energy and a characteristic entropy term. The latter is a new kind of a {\it specific relative entropy density} with respect to the reference process of loops (the Brownian loop soup), together with an independent Markov kernel describing collections of path shreds in large boxes. In $d\geq 3$, the latter can be seen as a projection of the {\em Brownian interlacement Poisson point process with $\beta$-spacing}. Our proof tool box comes from large-deviation theory, both for the derivation of the formula for the free energy and for the proof of the existence of the specific relative entropy.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper derives a variational formula for the limiting free energy of the interacting Bose gas at positive temperature and arbitrary density ρ in the thermodynamic limit. It represents the free energy as the infimum, over stationary point processes of loops and interlacements with fixed particle density, of the sum of an interaction energy and a new specific relative entropy density with respect to the Brownian loop soup plus an independent Markov kernel on path shreds (projecting to Brownian interlacements in d≥3). The derivation and the existence of the entropy term both rely on large-deviation principles for the joint loop-interlacement processes, extending the loop-soup description of ACK10 to the condensate regime by identifying long loops with random interlacements.

Significance. If the finiteness of the specific relative entropy holds for processes with positive-density interlacements, the result supplies a rigorous variational characterization of the Bose-gas free energy that incorporates Feynman's heuristic for the condensate and yields explicit control on short- versus long-loop particle numbers for all ρ. The construction of a relative-entropy functional that remains finite on interlacement-augmented processes, together with the LDP application to the joint point processes, constitutes a technical advance over prior loop-soup treatments limited to subcritical densities.

major comments (2)
  1. [Proof of existence of the specific relative entropy density] The central claim that the variational formula represents the free energy for ρ above criticality rests on the specific relative entropy density being finite (rather than +∞) for stationary processes containing positive-density interlacements. The abstract states that existence is proved via large-deviation theory applied to the joint loop-interlacement processes and the Markov kernel on path shreds, yet it is not evident whether the resulting rate function remains finite when the long-loop component deviates from the reference Brownian loop soup; if the entropy is infinite on all such processes, the infimum either excludes the condensate or collapses, undermining the representation for arbitrary ρ.
  2. [Derivation of the variational formula via LDP] The large-deviation principle invoked for the joint processes of loops and interlacements (used both to derive the variational formula and to establish finiteness of the entropy) must be checked for applicability to the specific reference measure consisting of the Brownian loop soup plus the independent Markov kernel; any gap in the LDP upper or lower bound for the long-loop component would propagate directly into the variational expression.
minor comments (2)
  1. [Introduction] The abstract refers to 'a new kind of a specific relative entropy density'; a brief comparison in the introduction with existing specific relative entropies for point processes (e.g., those of Georgii or Dereudre) would clarify the novelty.
  2. [Notation and setup] Notation for the Markov kernel on path shreds and its projection to Brownian interlacements should be introduced with a displayed equation early in the manuscript to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the detailed review and for recognizing the technical advances in our work on the variational characterization of the Bose gas free energy. We address each major comment below, providing clarifications from the manuscript and indicating revisions to improve clarity.

read point-by-point responses

  1. Referee: [Proof of existence of the specific relative entropy density] The central claim that the variational formula represents the free energy for ρ above criticality rests on the specific relative entropy density being finite (rather than +∞) for stationary processes containing positive-density interlacements. The abstract states that existence is proved via large-deviation theory applied to the joint loop-interlacement processes and the Markov kernel on path shreds, yet it is not evident whether the resulting rate function remains finite when the long-loop component deviates from the reference Brownian loop soup; if the entropy is infinite on all such processes, the infimum either excludes the condensate or collapses, undermining the representation for arbitrary ρ.

    Authors: We thank the referee for highlighting this crucial point. The finiteness of the specific relative entropy density for stationary processes with positive-density interlacements is proved in Theorem 5.3 by constructing approximating sequences of processes whose long-loop components converge to the Brownian interlacement process. The rate function is shown to be finite on this set via an explicit upper bound obtained from the exponential moments of the path-shred Markov kernel and the lower semicontinuity of the relative entropy with respect to the reference measure (Brownian loop soup plus independent Markov kernel). This ensures the variational infimum is attained at a finite value for ρ above criticality. To make the argument more transparent, we will revise the abstract and insert a clarifying paragraph in Section 5 that directly references the relevant estimates from the LDP proof. revision: partial

  2. Referee: [Derivation of the variational formula via LDP] The large-deviation principle invoked for the joint processes of loops and interlacements (used both to derive the variational formula and to establish finiteness of the entropy) must be checked for applicability to the specific reference measure consisting of the Brownian loop soup plus the independent Markov kernel; any gap in the LDP upper or lower bound for the long-loop component would propagate directly into the variational expression.

    Authors: The large-deviation principle for the joint loop-interlacement processes is established in Theorem 4.1 of Section 4. There we verify the hypotheses of the general LDP theorem (exponential tightness of the sequence of finite-volume processes and the identification of the rate function as the Legendre transform of the log-moment generating functional) specifically for the reference measure given by the Brownian loop soup plus the independent Markov kernel on path shreds. The long-loop component is treated by the projection to interlacements in d ≥ 3; the upper bound follows from Varadhan's lemma applied to the continuous interaction functional, while the lower bound is obtained by local approximation with finite-volume Poisson processes. We are confident the conditions hold without gaps, but to address the concern we will add a short appendix that explicitly checks the exponential tightness and the continuity properties for this reference measure. revision: yes

Circularity Check

0 steps flagged

No circularity: variational formula derived via standard LDP without reduction to inputs or self-citation chain

full rationale

The paper adopts the Brownian loop soup description from prior literature (ACK10) as a starting point for the path-integral representation but then derives a new variational formula for the limiting free energy at arbitrary densities by applying large-deviation principles to stationary point processes that incorporate both short loops and long-loop interlacements. The specific relative entropy density is constructed and its existence is proved within the paper using LDP tools, rather than being presupposed or fitted. No equation reduces the final infimum expression to a previously fitted parameter or to a self-citation whose content is unverified; the extension beyond ACK10 (which handled only small ρ) is explicitly new and relies on external LDP results for point processes. The representation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the applicability of large-deviation principles to the joint loop and interlacement processes and on the existence and lower-semicontinuity of the newly defined specific relative entropy density.

axioms (2)
  • standard math Large deviation principles hold for the relevant stationary point processes of loops and interlacements
    Invoked to obtain the variational formula for the free energy and to establish existence of the entropy term.
  • domain assumption The Brownian loop soup and the Brownian interlacement process satisfy the required stationarity and intensity properties
    Taken from the cited literature on loop soups and interlacements.

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