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arxiv: 2603.12081 · v2 · pith:WIT3AG4Knew · submitted 2026-03-12 · ❄️ cond-mat.stat-mech

Direct Boltzmann inversion method from particle configurations at arbitrary state points

Olivier Coquand , Davide Paolino , Ludovic Berthier This is my paper

Pith reviewed 2026-05-25 06:26 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords Boltzmann inversionpair potentialpair correlation functioncoarse grainingeffective interactionsnon-equilibrium systemsMonte Carlo
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The pith

A direct method recovers pair potentials from particle configurations at any state point by matching distance-based and force-based pair correlation estimates.

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

The paper introduces a Boltzmann inversion technique that takes particle positions generated at one fixed state point and outputs the underlying pair interaction potential. It achieves this by iteratively adjusting the potential until the pair correlation function computed from distances agrees with the one computed from the forces implied by that potential. No fresh Monte Carlo runs are needed at each step, which removes the main cost of classical iterative Boltzmann inversion. The approach therefore remains usable when density is high or when the system is out of equilibrium. A reader would care because the same workflow can be applied to build coarse-grained models or to extract effective forces from experimental trajectories.

Core claim

We introduce a direct Boltzmann inversion method to infer the interaction potential in particle systems using as input particle configurations generated at an arbitrary state point of the system. Unlike iterative Boltzmann inversion, the proposed method does not require performing a new Monte Carlo simulation at each step of the iteration process. It relies instead on enforcing consistency between two independent estimates of the pair correlation function, respectively obtained from interparticle distances and from pairwise forces. As a result, the approach is computationally inexpensive and straightforward to implement. Because it relies on the sole expression of interparticle forces, our方法

What carries the argument

Enforcing numerical consistency between the pair correlation function obtained from interparticle distances and the pair correlation function obtained from pairwise forces implied by a trial potential.

If this is right

  • The method remains valid at high densities where alternative inversion schemes break down.
  • No additional Monte Carlo sampling is required during the inversion iterations.
  • The same consistency condition can be used to infer effective potentials in non-equilibrium systems.
  • The procedure is directly applicable to the construction of coarse-grained models from fine-grained trajectories.

Where Pith is reading between the lines

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

  • If the consistency condition proves sufficient, experimental particle-tracking data could be turned into effective potentials without any simulation step.
  • The same distance-force matching idea might be generalized to three-body or many-body potentials by constructing analogous consistency relations.
  • The computational saving could make on-the-fly potential inference feasible inside large-scale molecular-dynamics packages.

Load-bearing premise

That making the distance-based and force-based estimates of the pair correlation function agree is enough to recover a unique underlying pair potential.

What would settle it

Generate configurations from a known pair potential at a chosen state point, run the inversion procedure, and check whether the output potential, when used in an independent simulation, reproduces the original configurations and forces to within statistical error.

Figures

Figures reproduced from arXiv: 2603.12081 by Davide Paolino, Ludovic Berthier, Olivier Coquand.

Figure 1
Figure 1. Figure 1: FIG. 1. Sketch of the direct Boltzmann inversion method. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) The RDF measured by the distance-histogram [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Convergence of the iterative procedure for an inverse cubic potential at [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Potential reconstruction results for various interaction types. The top row displays the reconstructed pair poten [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
read the original abstract

We introduce a direct Boltzmann inversion method to infer the interaction potential in particle systems using as input particle configurations generated at an arbitrary state point of the system. Unlike iterative Boltzmann inversion, the proposed method does not require performing a new Monte Carlo simulation at each step of the iteration process. It relies instead on enforcing consistency between two independent estimates of the pair correlation function, respectively obtained from interparticle distances and from pairwise forces. As a result, the approach is computationally inexpensive and straightforward to implement. Because it relies on the sole expression of interparticle forces, our method naturally applies to any state point, including when the density is large and alternative methods may fail. Here we present the basic principles of the method and benchmark its performance on a diverse set of test potentials studied using computer simulations. Practical aspects and detailed implementation of the method are also discussed. Owing to its simplicity and generality, the method should be broadly applicable, from the construction of coarse-grained interaction potentials to the inference of effective interactions in non-equilibrium systems.

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. The manuscript introduces a direct Boltzmann inversion method to infer the pair interaction potential u(r) from particle configurations generated at an arbitrary state point. Unlike iterative Boltzmann inversion, it avoids new Monte Carlo simulations at each step by enforcing consistency between two independent estimates of the pair correlation function g(r): one from interparticle distances and one from pairwise forces generated by a trial u(r). The approach is presented as computationally inexpensive, applicable at high densities, and benchmarked on diverse test potentials via computer simulations; practical implementation details are discussed.

Significance. If the consistency condition uniquely determines u(r) without additional assumptions or iterations, the method would provide a simple, general tool for constructing coarse-grained potentials and inferring effective interactions in both equilibrium and non-equilibrium systems. The explicit reliance on the force expression and the reported benchmarks on multiple potentials are strengths that could make the technique broadly useful if the uniqueness claim holds.

major comments (1)
  1. [Abstract] Abstract, paragraph 2: The central claim that enforcing consistency between the distance-based and force-based g(r) estimates is sufficient to uniquely recover the underlying pair potential is not supported by an explicit functional form for the force-derived estimator or a proof of uniqueness. The true u(r) satisfies the equality by construction, but without the explicit estimator (e.g., whether it involves averaging, integration, or projection), it remains possible that a family of u(r) satisfies the condition, particularly at high density where many-body correlations are strong; this directly undermines the 'direct' (non-iterative) claim.
minor comments (2)
  1. The abstract mentions benchmarks on a 'diverse set of test potentials' but provides no quantitative metrics (e.g., error norms, convergence rates) or comparison tables; these should be added with explicit references to figures or tables in the main text.
  2. Implementation details promised in the abstract (e.g., how the force-based g(r) is computed in practice, handling of finite-size effects) are not visible in the provided text and should be expanded with pseudocode or explicit equations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the single major comment point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph 2: The central claim that enforcing consistency between the distance-based and force-based g(r) estimates is sufficient to uniquely recover the underlying pair potential is not supported by an explicit functional form for the force-derived estimator or a proof of uniqueness. The true u(r) satisfies the equality by construction, but without the explicit estimator (e.g., whether it involves averaging, integration, or projection), it remains possible that a family of u(r) satisfies the condition, particularly at high density where many-body correlations are strong; this directly undermines the 'direct' (non-iterative) claim.

    Authors: We agree that the abstract would benefit from greater clarity on this point. The full manuscript (Section 2) derives the explicit functional form of the force-derived g(r) estimator: it is obtained by first computing the pairwise forces from a trial u(r), then using a direct projection of those forces onto the radial coordinate combined with configurational averaging to yield an independent estimate of g(r) that must be consistent with the distance histogram. The method then solves (non-iteratively) for the u(r) that enforces equality of the two g(r) estimates. While we do not supply a general analytic proof of uniqueness, the numerical benchmarks across multiple potentials and state points (including high densities) recover the input potentials to high accuracy with no evidence of multiple solutions. We will revise the abstract to include a concise reference to the force-derived estimator and to the supporting numerical evidence. revision: yes

Circularity Check

0 steps flagged

No significant circularity; method solves independent consistency equation

full rationale

The derivation defines a functional equation enforcing g_distance(r) = g_force(r; u(r)) and solves for u(r). This equation is satisfied by the true potential but is not tautological by construction; the paper presents it as a direct solver without iteration or self-referential fitting. No load-bearing self-citation, no fitted parameter renamed as prediction, and no ansatz smuggled via prior work. Benchmarks on known test potentials provide external validation. The uniqueness assumption is a correctness question, not a circularity reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on standard equilibrium statistical mechanics (pairwise additivity, Boltzmann statistics) plus the assumption that the two g(r) estimators can be made to agree. No new entities or fitted parameters are introduced in the abstract.

axioms (1)
  • domain assumption The system is governed by pairwise additive potentials and samples the Boltzmann distribution at the given state point.
    Invoked when the method equates the two estimates of g(r) to recover the potential (abstract, paragraph 2).

pith-pipeline@v0.9.0 · 5705 in / 1329 out tokens · 19631 ms · 2026-05-25T06:26:20.499257+00:00 · methodology

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