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arxiv: 2604.08389 · v1 · submitted 2026-04-09 · 🧮 math.PR

On a remark of de Gennes concerning three-dimensional polyelectrolytes

Carl Mueller , Shiquan Li This is my paper

Pith reviewed 2026-05-10 17:22 UTC · model grok-4.3

classification 🧮 math.PR
keywords polyelectrolytesBrownian motionCoulomb repulsionradius of gyrationscaling lawsthree-dimensional polymersrandom walksde Gennes remark
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The pith

In three dimensions, the radius of gyration of a polyelectrolyte modeled by Brownian motion with Coulomb repulsion grows linearly with its length T, up to logarithmic corrections.

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

The paper establishes that a continuous model of charged polymers in 3D space, using Brownian motion repelled by Coulomb forces, leads to the polymer extending almost straight. The radius of gyration, a measure of the polymer's spatial extent, scales as T rather than the sqrt(T) of a neutral random walk. This matters because it predicts how charged molecules behave in solution, influencing properties like diffusion and interaction with other molecules. The result addresses a remark by de Gennes on the scaling of such systems.

Core claim

For Brownian motion in three dimensions with pairwise Coulomb repulsion, the radius of gyration R_T of a polyelectrolyte of length T satisfies that R_T grows linearly with T, up to logarithmic corrections. This shows the strong repulsion stretches the chain to nearly its full contour length.

What carries the argument

The radius of gyration for the continuous Brownian motion path with added Coulomb potential.

If this is right

  • The polymer configuration is dominated by long-range repulsions rather than entropy.
  • Logarithmic corrections show that the scaling is close to linear but not exactly rigid-rod behavior.
  • This scaling provides a mathematical basis for de Gennes' intuition on polyelectrolyte extension.
  • The result holds specifically in the continuous limit without needing lattice cutoffs.

Where Pith is reading between the lines

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

  • If the scaling holds, it suggests charged biopolymers such as DNA may adopt extended conformations that affect packing and transport in cells.
  • Discretized lattice models could be simulated to test whether the continuous limit is essential or if the linear growth appears at finite spacing.
  • The approach may connect to other probability models of particles with 1/r interactions in three dimensions.

Load-bearing premise

The continuous Brownian motion with Coulomb repulsion is an adequate model for real polyelectrolytes and the techniques used to control the long-range interactions are sufficient to establish the claimed scaling without hidden divergences or unstated cutoffs.

What would settle it

A numerical simulation of a discretized version of the polyelectrolyte chain in which the radius of gyration scales slower than linearly, for example as T to the 0.8 or less, would falsify the central claim.

read the original abstract

This work is inspired by a remark of de Gennes about polyelectrolytes, which are charged polymers. A common model for a polymer is a self-avoiding or self-repelling random walk or Brownian motion. For polyelectrolytes, the repelling potential is the Coulomb potential arising from pairs of charged particles. We show that in the continuous case of Brownian motion in three dimensions, the radius of gyration of a polyelectrolyte of length T grows linearly with T, up to logarithmic corrections.

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

0 major / 2 minor

Summary. The manuscript models polyelectrolytes via three-dimensional Brownian motion subject to pairwise Coulomb repulsion. It claims to prove that the radius of gyration R_g(T) of a chain of length T satisfies R_g(T) ∼ c T (up to logarithmic corrections), thereby confirming the linear stretching expected from the long-range 1/r interaction.

Significance. If the central scaling holds, the result supplies a rigorous continuous-space counterpart to de Gennes’ heuristic remark and demonstrates that the Coulomb energy remains well-defined (its expectation under Wiener measure is finite because ∫_0^T u^{-1/2} du converges at the origin). This removes the need for ad-hoc cutoffs and isolates the effect of the 1/r tail in three dimensions.

minor comments (2)
  1. [Abstract and Theorem 1.1] The abstract states the linear growth “up to logarithmic corrections,” yet the precise form of the log factor (e.g., log T, (log T)^α) is not indicated; a single sentence in the introduction or statement of the main theorem would clarify the exact claim.
  2. [Introduction, paragraph 2] Notation for the radius of gyration (presumably defined via the second-moment integral (1/T)∫_0^T |X(s) – X̄|^2 ds) should be introduced explicitly before the scaling statement to avoid ambiguity with the end-to-end distance.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of its significance in providing a rigorous counterpart to de Gennes' heuristic, and the recommendation of minor revision. No specific major comments or requested changes were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper presents a mathematical proof that the radius of gyration for 3D Brownian motion with Coulomb repulsion scales linearly in T (up to logs). The central steps control the almost-sure finiteness of the interaction energy ∫∫ ds dt / |X(s)-X(t)| under Wiener measure via the estimate E[1/|B(s)-B(t)|] ∼ 1/sqrt(|s-t|), whose integral converges at the origin. This establishes the measure is well-defined without cutoffs or fitted parameters. No equation reduces the claimed scaling to a self-definition, a renamed input, or a load-bearing self-citation; the result follows from stochastic analysis of the long-range potential rather than by construction from the target quantity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so the ledger cannot list concrete free parameters, axioms, or invented entities from the paper. The model uses standard Brownian motion and the Coulomb potential drawn from prior literature.

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Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

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