arxiv: 2605.12333 · v1 · submitted 2026-05-12 · ❄️ cond-mat.stat-mech · cond-mat.soft

Recognition: no theorem link

Link length and energy fluctuations in extensible freely jointed chains

Michael R. Buche

Authors on Pith no claims yet

Pith reviewed 2026-05-13 03:04 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.soft

keywords extensible freely jointed chainpolymer fluctuationslink lengthenergy fluctuationsthermal fluctuationsanalytic approximationsprobability distributions

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The pith

Asymptotically exact analytic formulas now describe the fluctuations of link lengths and energies in extensible freely jointed chains.

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

The extensible freely jointed chain adds a stretching potential to each link so that lengths can vary under thermal motion and applied force. The paper supplies closed-form asymptotic expressions for the mean link length, its standard deviation, the mean energy stored in each link, the corresponding energy deviation, and the probability distributions of both quantities. These expressions become exact in the relevant limits up to corrections smaller than any power of the expansion parameter. Because polymer models often trigger dissociation when a link reaches a critical length or energy, the width of these fluctuations directly controls the rate at which such events occur by chance. In many regimes the distributions are close to normal, simplifying further use of the results.

Core claim

Within the extensible freely jointed chain held at constant force, the thermal fluctuations of each link's length and stored energy are governed by exact integral expressions that reduce to simple asymptotic formulas. These formulas give the mean link length, the mean energy, the corresponding standard deviations, and the probability density functions, all correct to within transcendentally small errors. Direct numerical checks confirm the accuracy of the approximations across the full range of forces.

What carries the argument

The single-link partition function under constant force combined with a link-stretching potential, from which all averages, variances, and distributions follow by differentiation or integration.

If this is right

Models that break chains when a link reaches a fixed length or energy threshold must integrate over the probability that thermal fluctuations exceed that threshold.
Network-level rupture calculations become more accurate once they replace mean link values with the full fluctuation statistics.
The asymptotic formulas allow rapid evaluation of fluctuation effects inside larger simulations without repeated sampling of chain configurations.
Approximately normal distributions in many regimes let fluctuation effects be added to existing models with simple Gaussian noise terms.

Where Pith is reading between the lines

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

The same partition-function approach could be applied to other extensible models such as the worm-like chain to obtain comparable fluctuation statistics.
Inserting the derived distributions into Monte Carlo simulations of polymer networks would allow direct numerical tests of dissociation rates driven by fluctuations.
The formulas could be extended to time-varying forces to track how length and energy fluctuations evolve during loading or unloading.

Load-bearing premise

Each link is assumed to stretch according to a fixed potential energy function, usually harmonic, while the whole chain remains in thermal equilibrium under constant applied force.

What would settle it

A Monte Carlo sampling of an extensible chain of 100 links at moderate force that yields a standard deviation for link length differing from the analytic prediction by more than a few percent.

Figures

Figures reproduced from arXiv: 2605.12333 by Michael R. Buche.

**Figure 3.** Figure 3: FIG. 3. Probability density [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 5.** Figure 5: to be quite accurate compared to the exact relation ⟨υ⟩ = κ 2 ⟨λ 2 ⟩ − 2⟨λ⟩ + 1 , (21) where ⟨λ⟩ and ⟨λ 2 ⟩ are given by Eqs. (A1) and (A4). The limit ⟨υ⟩/κ → η 2/2κ 2 as κ → ∞ is also shown in [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Probability density [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. Relative [PITH_FULL_IMAGE:figures/full_fig_p006_9.png] view at source ↗

read the original abstract

The freely jointed chain is often applied to model the thermodynamics of single polymer chains, but the traditional formulation of the model lacks internal energy changes due to bond stretching. For this reason, the extensible freely jointed chain model includes a potential energy function, typically harmonic, that governs the length of each link in the chain. Among the other quantities of interest that are subject to thermal fluctuations, these link lengths and energies too fluctuate about their ensemble average values. Since a plethora of models for polymer chains and networks incorporate chain dissociation as a function of either link length or energy, these fluctuations are crucial to understand and quantify. Motivated by this fact, fluctuations in link length and energy are analyzed within a freely jointed chain under an applied force. These fluctuations are quantified through their average values, standard deviations, and probability distributions. Across all values, asymptotically correct analytic relations and their less ergonomic exact counterparts are introduced. The asymptotic relations are verified to be accurate through direct comparison and to be correct within transcendentally small terms through error analysis. In certain cases, the fluctuations are shown to be approximately normally distributed. Hereafter, model components predicated on link length or energy ought to account for these fluctuations.

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.

Desk Editor's Note private letter to a colleague

Buche has worked out clean asymptotic expressions for link length and energy fluctuations in the extensible FJC, with exact forms and numerical checks that make them usable right away.

read the letter

The main point is that this paper supplies asymptotic analytic relations for the means, standard deviations, and probability distributions of link length and energy fluctuations in the extensible freely jointed chain under constant force. These are paired with the less convenient exact integral or sum expressions, and the asymptotics are shown to match direct numerical evaluation while differing by transcendentally small terms according to the error analysis. That combination is the concrete addition here. The derivations start from the usual partition function for the model with a stretching potential and move to the fluctuation statistics without fitted parameters or circular steps, which keeps the logic straightforward. Numerical comparisons across regimes back up the accuracy claims, and the note that some distributions are approximately normal in certain limits is a practical bonus. This kind of explicit fluctuation data has not been available in closed form for the extensible version, so it directly addresses a need in models that use length or energy thresholds for dissociation or failure. The work stays within standard assumptions: thermal equilibrium in the constant-force ensemble and a fixed potential for link stretching, typically harmonic. Those choices are conventional, but they mean the specific formulas are tied to the potential form; a different potential would require re-deriving the asymptotics, though the overall method should transfer. The focus remains on single-chain statistics, so network-level or non-equilibrium extensions would need separate effort. These are normal scope limits rather than gaps in the argument itself. The paper is aimed at people in polymer physics and soft matter who build chain or network models that depend on fluctuation-corrected thresholds. A reader in that area can plug the expressions in and get immediate value without re-deriving everything. The reasoning is internally consistent and engages the existing FJC literature properly. I would send it to peer review; the analytic work is grounded enough to deserve referee time even if revisions are needed on presentation or extensions.

Referee Report

0 major / 3 minor

Summary. The manuscript derives exact (integral or sum) expressions and asymptotically correct closed-form approximations for the means, standard deviations, and probability distributions of link-length and energy fluctuations in the constant-force ensemble of an extensible freely jointed chain with a stretching potential (typically harmonic). These relations are verified by direct numerical comparison to the exact forms, with error analysis showing the discrepancies are transcendentally small; in certain regimes the distributions are shown to be approximately normal. The work is motivated by the need to account for such fluctuations in polymer dissociation models.

Significance. If the derivations hold, the paper supplies practical analytic tools for fluctuation statistics in a standard polymer-physics model, directly relevant to network and single-chain dissociation calculations. A notable strength is the explicit pairing of exact and asymptotic results together with numerical verification and error analysis, yielding reproducible, parameter-free predictions that can be falsified by direct computation.

minor comments (3)

A dedicated paragraph or table summarizing the asymptotic parameter (e.g., force or temperature regime) and the explicit order of the transcendentally small error term would make the validity range of the closed-form expressions immediately clear to readers.
The notation for the single-link partition function, the stretching potential, and the constant-force ensemble averages should be introduced in a single early section with consistent symbols to improve readability across the derivations.
Figure captions for the numerical comparisons should report the specific parameter values used and include a quantitative measure (e.g., maximum relative deviation) to support the claim of transcendentally small discrepancies.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful and positive evaluation of our manuscript. The summary and significance assessment accurately capture the derivations of exact and asymptotic expressions for link-length and energy fluctuations in the extensible freely jointed chain, along with the numerical verifications and error analysis. We note the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity; derivations start from standard partition function

full rationale

The paper derives mean values, standard deviations, and probability distributions for link length and energy fluctuations directly from the partition function of the extensible freely jointed chain in the constant-force ensemble. Asymptotic closed-form expressions are obtained via standard analysis techniques and verified by explicit comparison to exact integral/sum forms plus error bounds showing transcendentally small discrepancies. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the central claims remain independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard statistical mechanics framework for the extensible freely jointed chain with an added stretching potential; no new entities are postulated.

axioms (2)

domain assumption Links obey Boltzmann-weighted statistics in the canonical ensemble under constant force
Standard assumption for thermal equilibrium models of polymer chains.
domain assumption Link stretching is governed by a potential energy function, typically harmonic
Explicitly stated as the basis for the extensible model.

pith-pipeline@v0.9.0 · 5504 in / 1245 out tokens · 76326 ms · 2026-05-13T03:04:40.540841+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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