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arxiv: 2605.11147 · v3 · pith:I2IYMHMYnew · submitted 2026-05-11 · ✦ hep-ph

An approximate formula for the entropy of the negative binomial distribution

S\'andor L\"ok\"os This is my paper

Pith reviewed 2026-05-20 21:34 UTC · model grok-4.3

classification ✦ hep-ph
keywords negative binomial distributionShannon entropyapproximate formulacharged particle multiplicitieshigh energy physicsmultiplicity distributionsentropy approximationparticle production
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The pith

An approximate formula estimates the Shannon entropy of the negative binomial distribution within 20 percent for extreme parameter values.

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

The paper investigates series and integral representations of the Shannon entropy for the negative binomial distribution, which has no known closed form. It derives a practical approximation that stays within roughly 20 percent of the exact value when the distribution parameters reach extreme regimes. This matters for studies of charged particle multiplicities in high-energy collisions, where entropy quantifies the spread of multiplicity data and exact calculations can be costly. A sympathetic reader would see it as a tool to handle limiting cases without repeated numerical work.

Core claim

By approximating one of the existing series or integral representations, the authors obtain a formula for the entropy of the negative binomial distribution that deviates by at most about 20 percent from the true value when the parameters are taken to their extreme limits.

What carries the argument

An approximate closed-form expression obtained by simplifying a series or integral representation of the Shannon entropy for the negative binomial distribution.

If this is right

  • Entropy estimates become feasible for multiplicity data in limiting regimes without full numerical summation.
  • Model comparisons that rely on entropy differences can be performed more rapidly across wide parameter ranges.
  • Theoretical predictions for particle production in extreme kinematic conditions gain a lightweight entropy observable.
  • Parameter fitting routines for negative binomial distributions can incorporate entropy constraints at lower computational cost.

Where Pith is reading between the lines

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

  • The same approximation strategy could be tested on related distributions such as the Poisson or binomial to see if similar accuracy holds.
  • In experimental analyses, the formula might allow real-time entropy monitoring during multiplicity unfolding procedures.
  • If the 20 percent tolerance proves acceptable in practice, it opens the door to analytic derivatives of entropy with respect to the NBD parameters.

Load-bearing premise

One of the existing series or integral representations of the NBD entropy admits an approximation accurate to within 20 percent for extreme parameter values.

What would settle it

Compute the exact entropy numerically from the full series representation at several extreme parameter points and check whether the proposed formula stays inside 20 percent of those values.

Figures

Figures reproduced from arXiv: 2605.11147 by S\'andor L\"ok\"os.

Figure 1
Figure 1. Figure 1: FIG. 1. The [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
read the original abstract

Recent theoretical developments revived the interest in charged particle multiplicities and their wide-spread parametrization, the negative binomial distribution (NBD). The central observable of the studies is the Shannon entropy of the NBD. A closed form is not known, however, there are representations with special series and integrals. In this note, we will investigate one of these and give an approximate formula for the entropy that is valid up to $\sim$20\% deviation from the exact value for extreme values of the NBD parameters.

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 / 1 minor

Summary. The manuscript investigates one of the series or integral representations of the Shannon entropy of the negative binomial distribution (NBD) and derives an approximate formula claimed to remain within ∼20% of the exact value for extreme values of the NBD parameters.

Significance. If the approximation and its error bound are established, the result would provide a practical tool for entropy calculations in high-energy physics analyses that employ the NBD to parametrize charged-particle multiplicity distributions, where closed-form expressions are unavailable.

major comments (2)
  1. [Abstract] Abstract: the central claim that the approximation is 'valid up to ∼20% deviation from the exact value for extreme values of the NBD parameters' does not define the extreme regime (e.g., explicit ranges or limiting behavior for parameters r and p, or equivalently the mean and variance). This definition is load-bearing for assessing whether the stated accuracy holds throughout the claimed domain.
  2. [Derivation] Derivation section: the step that produces the approximate formula from the chosen series/integral representation must be accompanied by an explicit error bound or asymptotic analysis demonstrating that the truncation error remains ≤20% for all extreme parameter values; numerical checks at isolated points are insufficient to establish the bound over the full regime (e.g., r→0 at fixed mean or r→∞).
minor comments (1)
  1. Add a plot or table of relative error versus the NBD parameters across the extreme regime to make the 20% claim visually verifiable.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and outline the revisions we intend to implement.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the approximation is 'valid up to ∼20% deviation from the exact value for extreme values of the NBD parameters' does not define the extreme regime (e.g., explicit ranges or limiting behavior for parameters r and p, or equivalently the mean and variance). This definition is load-bearing for assessing whether the stated accuracy holds throughout the claimed domain.

    Authors: We agree that a precise definition of the 'extreme' regime is necessary. In the revised version we will explicitly state the parameter domain, for example 0.01 ≤ r ≤ 1 with mean multiplicity μ ≥ 10 (corresponding to p = r/(r+μ) approaching the relevant limits), and confirm that the stated 20% deviation holds throughout this region by additional targeted checks. revision: yes

  2. Referee: [Derivation] Derivation section: the step that produces the approximate formula from the chosen series/integral representation must be accompanied by an explicit error bound or asymptotic analysis demonstrating that the truncation error remains ≤20% for all extreme parameter values; numerical checks at isolated points are insufficient to establish the bound over the full regime (e.g., r→0 at fixed mean or r→∞).

    Authors: The approximation arises from truncating the integral representation after the leading contribution that dominates when r is small or large at fixed mean. While the original manuscript relied on numerical verification at representative points, we acknowledge that this is not a substitute for an explicit bound. In the revision we will add a short asymptotic analysis of the truncation error in the limits r→0 (fixed mean) and r→∞, together with a denser numerical scan confirming that the relative deviation stays below 20% across the full extreme regime. revision: yes

Circularity Check

0 steps flagged

No circularity: approximation derived from existing representations without reduction to fitted inputs or self-citations

full rationale

The paper starts from known series and integral representations of the NBD entropy (external to the present work), investigates one of them, and produces an approximate closed-form expression whose claimed accuracy is checked against the exact entropy. No step equates the output formula to a parameter fitted from the target result itself, renames a known pattern, or relies on a load-bearing self-citation or uniqueness theorem imported from the authors' prior work. The derivation chain remains independent of the final approximation and is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of series and integral representations for NBD entropy and on the premise that one such representation admits a useful approximation at extreme parameters.

axioms (1)
  • domain assumption The Shannon entropy of the negative binomial distribution admits representations via special series and integrals.
    Invoked in the abstract as the starting point for the investigation.

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Reference graph

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