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arxiv: 2604.14681 · v1 · submitted 2026-04-16 · 🧮 math-ph · math.MP· math.PR

An inversion formula for the 2-body interaction given the correlation functions

Fabio Frommer , Tobias Kuna , Dimitrios Tsagkarogiannis This is my paper

Pith reviewed 2026-05-10 09:58 UTC · model grok-4.3

classification 🧮 math-ph math.MPmath.PR
keywords classical gasestruncated correlation functionspair potentialsinversion formulasinfinite volume limitcluster expansionsstatistical mechanics
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The pith

The pair interaction of a classical gas can be recovered from its truncated correlation functions of all orders via a convergent expansion at infinite volume.

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

The paper proves that if all truncated correlation functions are known for a classical gas, then the two-body part of the interaction potential can be written as a series in those functions. The series is shown to converge when the system is taken to infinite volume. A reader would care because this supplies the missing inverse map from measured correlations back to the unknown forces that produced them, rather than the usual forward calculation from forces to correlations.

Core claim

Given the truncated correlation functions of all orders for a classical gas, the pair interaction component of the potential admits a convergent expansion in terms of those same functions that remains valid in the infinite-volume limit.

What carries the argument

The inversion expansion of the pair potential, which assembles the two-body interaction from the entire hierarchy of truncated correlation functions and establishes convergence of the resulting series at infinite volume.

If this is right

  • The two-body potential is uniquely recoverable from complete knowledge of all truncated correlations.
  • The recovery formula remains valid after the thermodynamic limit is taken.
  • Any classical gas whose correlations are fully characterized can have its pair interaction reconstructed by summing the series.
  • The result supplies a rigorous justification for using correlation data to determine effective interactions in the infinite-volume setting.

Where Pith is reading between the lines

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

  • Simulation outputs that yield truncated correlations could be post-processed to infer the effective pair potential without additional fitting.
  • The formula offers a consistency check: given measured correlations, one can test whether they are compatible with an assumed pair potential by seeing if the inverted expression matches.
  • Similar inversion strategies might apply to higher-body potentials or to lattice gases once the appropriate truncated functions are supplied.

Load-bearing premise

The truncated correlation functions of every order must exist and be known exactly, and the underlying potential must be such that the infinite-volume limit exists and the series remains controllable.

What would settle it

Take a solvable model such as hard spheres whose pair potential and truncated correlations are known independently, insert the correlations into the claimed expansion, and verify whether the output recovers the original potential without divergence; failure to recover it would refute the claim.

Figures

Figures reproduced from arXiv: 2604.14681 by Dimitrios Tsagkarogiannis, Fabio Frommer, Tobias Kuna.

Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
read the original abstract

Given a classical gas described by the truncated correlation functions of all orders, we prove convergence of an expansion of the pair interaction part of the (unknown) potential in terms of the truncated correlation functions of all orders, at infinite volume.

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

1 major / 2 minor

Summary. Given a classical gas described by the truncated correlation functions of all orders, the paper proves convergence of an expansion of the pair interaction part of the (unknown) potential in terms of the truncated correlation functions of all orders, at infinite volume.

Significance. If the convergence holds, the result supplies a rigorous inversion formula recovering the two-body potential from the full hierarchy of truncated correlations in the thermodynamic limit. This addresses an inverse problem in classical statistical mechanics and could support reconstruction methods when correlation data are available. The explicit focus on infinite-volume convergence is a positive feature of the approach.

major comments (1)
  1. [Main convergence argument (proof of the inversion formula)] The central convergence claim (abstract and main theorem) requires a volume-independent majorant for the series terms built from the truncated correlations ρ_n^T of all orders. The manuscript assumes existence of the correlations in the infinite-volume limit but does not supply or cite the necessary uniform bounds (e.g., |ρ_n^T| ≤ C^n n! or equivalent growth control independent of volume) that would guarantee the series remains convergent after the limit is taken. This step is load-bearing for the stated result.
minor comments (2)
  1. [Introduction] Notation for the truncated correlations and the pair-potential expansion should be introduced with a single consistent set of symbols in the introduction rather than re-defined later.
  2. [Statement of main result] The statement of the main theorem would benefit from an explicit list of the technical assumptions (e.g., decay of correlations, regularity of the potential) placed immediately before the theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive assessment of its significance. We address the major comment below.

read point-by-point responses

  1. Referee: The central convergence claim (abstract and main theorem) requires a volume-independent majorant for the series terms built from the truncated correlations ρ_n^T of all orders. The manuscript assumes existence of the correlations in the infinite-volume limit but does not supply or cite the necessary uniform bounds (e.g., |ρ_n^T| ≤ C^n n! or equivalent growth control independent of volume) that would guarantee the series remains convergent after the limit is taken. This step is load-bearing for the stated result.

    Authors: We appreciate the referee's identification of this key technical point. The proof constructs the majorant series directly from the infinite-volume truncated correlations ρ_n^T, whose existence is assumed in the thermodynamic limit under the standard conditions on the pair potential (stability and temperedness). These conditions imply, via the convergence of the Mayer cluster expansion in infinite volume, that the truncated correlations obey the uniform bound |ρ_n^T| ≤ C^n n! with C independent of volume; this is a standard result in the literature (e.g., Ruelle's book on statistical mechanics and subsequent works on cluster expansions). The volume-independent majorant then follows immediately, allowing the series to converge after the limit. To make the argument fully explicit, we will insert a short clarifying paragraph before the main theorem citing this standard growth control and explaining its role in the majorant. We do not view this as requiring a new proof, but rather as improved exposition of an existing prerequisite. revision: partial

Circularity Check

0 steps flagged

No circularity: direct proof of convergence for inversion expansion

full rationale

The manuscript establishes a mathematical theorem proving convergence of a series expansion that recovers the pair potential from the full hierarchy of truncated correlation functions in the infinite-volume limit. This is a self-contained derivation under stated assumptions on existence and decay of the correlations; no step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional tautology. The central claim is an existence-and-convergence statement whose validity rests on analytic estimates rather than on re-expressing the input data.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard domain assumptions of classical statistical mechanics (existence of correlation functions, suitable decay at infinity) with no free parameters or invented entities visible in the abstract.

axioms (2)
  • domain assumption Existence and sufficient decay of truncated correlation functions of all orders for the classical gas
    Invoked to define the input data for the expansion.
  • domain assumption The infinite-volume limit exists and permits term-by-term control of the series
    Required for the convergence statement.

pith-pipeline@v0.9.0 · 5327 in / 1162 out tokens · 38147 ms · 2026-05-10T09:58:57.030944+00:00 · methodology

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Reference graph

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