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arxiv: 1907.03172 · v1 · pith:ZGRAGGCFnew · submitted 2019-07-06 · ❄️ cond-mat.stat-mech

Quantum implications of non-extensive statistics

Nana Cabo Bizet , C\'esar Dami\'an Ascencio , Octavio Obreg\'on , Roberto Santos-Silva This is my paper

Pith reviewed 2026-05-25 01:27 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords non-extensive statisticsQuantropy functionalquantum propagatorTsallis entropysemiclassical approximationq-wave functionnon-additive entropymodified Schrödinger equation
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The pith

An integrated Quantropy functional yields propagators for non-additive statistics that match modified q-Schrödinger wave functions and extend to interacting systems.

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

The paper extends the quantum-statistical mechanics analogy by defining an integrated Quantropy functional that applies to non-additive entropies S± and Tsallis S_q. This functional is used to derive semiclassical propagators, including power-series solutions for S± that continue analytically into the complex plane. The resulting q-propagator reproduces the known q-wave function for the free particle. The same construction supplies generalized wave functions for interacting problems and isolates the non-linear effects that follow from non-additive statistics.

Core claim

Formulating an integrated Quantropy functional for non-additive statistics S± and S_q permits calculation of their associated propagators in the semiclassical approximation. For S± a power series solution for probability versus energy is obtained and continued to the complex plane. The modified q-propagator leads to the same q-wave function for the free particle as the modified q-Schrödinger equation. The procedure allows calculation of q-wave functions in problems with interactions and determines the corresponding generalized wave functions for S±, with explicit corrections given for the free particle and the harmonic oscillator.

What carries the argument

The integrated Quantropy functional, which generalizes the link between classical action and entropy to non-additive cases and yields semiclassical propagators.

If this is right

  • A power series solution for probability versus energy for S± statistics that is analytically continued to the complex plane to obtain propagators.
  • Explicit corrections to the original propagator for the free particle and the harmonic oscillator.
  • Calculation of q-wave functions for interacting systems under Tsallis statistics.
  • Construction of generalized wave functions associated with S± statistics.
  • Determination of non-linear quantum implications arising from non-additive statistics.

Where Pith is reading between the lines

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

  • The same functional construction could be applied to other non-additive entropies not examined in the paper.
  • The method supplies a route to check consistency between the derived wave functions and exact solutions in additional solvable models.
  • It suggests that non-additivity may induce measurable modifications to interference or transition amplitudes in semiclassical regimes.

Load-bearing premise

The analogy between quantum mechanics and statistical mechanics extends to an integrated Quantropy functional that applies to non-additive statistics S± and S_q.

What would settle it

Explicit computation of the free-particle propagator from the integrated Quantropy functional that fails to recover the q-exponential wave function obtained from the modified q-Schrödinger equation.

Figures

Figures reproduced from arXiv: 1907.03172 by C\'esar Dami\'an Ascencio, Nana Cabo Bizet, Octavio Obreg\'on, Roberto Santos-Silva.

Figure 1
Figure 1. Figure 1: FIG. 1. Probability versus [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Real parts of the modified propagator (blue line) [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Real parts of the modified propagator (blue line) [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Real parts of the modified propagator (blue line) vs. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Real parts of the modified propagator (blue line) vs. [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. We set the units [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Numerical check for the series expansion of the mod [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Numerical check for the harmonic oscillator propaga [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
read the original abstract

Exploring the analogy between quantum mechanics and statistical mechanics we formulate an integrated version of the Quantropy functional [1]. With this prescription we compute the propagator associated to Boltzmann-Gibbs statistics in the semiclassical approximation as $K=F(T) \exp\left(i S_{cl}/\hbar\right)$. We determine also propagators associated to different non-additive statistics; those are the entropies depending only on the probability $S_{\pm}$ [2] and Tsallis entropy $S_q$ [3]. For $S_{\pm}$ we obtain a power series solution for the probability vs. the energy, which can be analytically continued to the complex plane, and employed to obtain the propagators. Our work is motivated by [4] where a modified q-Schr\"odinger equation is obtained; that provides the wave function for the free particle as a q-exponential. The modified q-propagator obtained with our method, leads to the same q-wave function for that case. The procedure presented in this work allows to calculate q-wave functions in problems with interactions; determining non-linear quantum implications of non-additive statistics. In a similar manner the corresponding generalized wave functions associated to $S_{\pm}$ can also be constructed. The corrections to the original propagator are explicitly determined in the case of a free particle and the harmonic oscillator for which the semi-classical approximation is exact.

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

3 major / 2 minor

Summary. The manuscript formulates an integrated Quantropy functional extending the quantum-statistical mechanics analogy. It derives the semiclassical propagator K = F(T) exp(i S_cl / ħ) for Boltzmann-Gibbs statistics and obtains analogous propagators for the non-additive entropies S± and Tsallis S_q. For S± a power-series solution for probability versus energy is obtained, analytically continued to the complex plane, and used to construct the propagator. The resulting q-propagator is stated to reproduce the free-particle q-wave function from the modified q-Schrödinger equation of reference [4]; the procedure is claimed to extend to interacting systems, yielding generalized wave functions and explicit corrections for the free particle and harmonic oscillator (where the semiclassical approximation is exact).

Significance. If the derivations are valid, the work supplies a concrete route from non-extensive entropies to associated propagators and wave functions, potentially allowing non-linear quantum corrections to be computed for interacting problems. The explicit reproduction of the known free-particle q-wave function and the exactness for the harmonic oscillator constitute verifiable strengths that could be cited if the intermediate steps are supplied.

major comments (3)
  1. [Abstract] Abstract: the central consistency claim—that the modified q-propagator reproduces the q-wave function of the modified q-Schrödinger equation—is asserted without any derivation, explicit functional form, or comparison of the two expressions being shown, rendering the claim unverifiable from the given material.
  2. [Abstract] Abstract: the power-series solution for the probability-versus-energy relation under S±, its analytic continuation to the complex plane, and the subsequent construction of the propagator are described only at the level of results; no explicit series coefficients, radius of convergence, or continuation procedure is supplied, which are load-bearing for the extension beyond Boltzmann-Gibbs statistics.
  3. [Abstract] Abstract: the statement that the procedure 'allows to calculate q-wave functions in problems with interactions' is presented without any worked interacting example or indication of how the semiclassical approximation is controlled when interactions are present, leaving the scope of the claimed non-linear implications unsupported.
minor comments (2)
  1. [Abstract] Abstract: the prefactor F(T) in the Boltzmann-Gibbs propagator is introduced without definition or derivation; its explicit form should be stated.
  2. [Abstract] Abstract: the phrase 'non-linear quantum implications' is used without a precise definition of the non-linearity being introduced by the non-additive statistics.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful review and constructive comments on our manuscript. We address each major comment point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central consistency claim—that the modified q-propagator reproduces the q-wave function of the modified q-Schrödinger equation—is asserted without any derivation, explicit functional form, or comparison of the two expressions being shown, rendering the claim unverifiable from the given material.

    Authors: The abstract summarizes the result. The explicit derivation of the modified q-propagator, its functional form, and the direct comparison showing reproduction of the q-wave function from the modified q-Schrödinger equation are provided in the main text. We will revise the abstract to reference the relevant section containing this derivation and comparison. revision: partial

  2. Referee: [Abstract] Abstract: the power-series solution for the probability-versus-energy relation under S±, its analytic continuation to the complex plane, and the subsequent construction of the propagator are described only at the level of results; no explicit series coefficients, radius of convergence, or continuation procedure is supplied, which are load-bearing for the extension beyond Boltzmann-Gibbs statistics.

    Authors: The abstract summarizes these elements at a high level. The explicit power series coefficients for the probability-versus-energy relation under S±, the radius of convergence, the analytic continuation procedure to the complex plane, and their employment in the propagator construction are derived and presented in the main text. We will revise the abstract to note that these details appear in the body of the paper. revision: partial

  3. Referee: [Abstract] Abstract: the statement that the procedure 'allows to calculate q-wave functions in problems with interactions' is presented without any worked interacting example or indication of how the semiclassical approximation is controlled when interactions are present, leaving the scope of the claimed non-linear implications unsupported.

    Authors: The manuscript demonstrates the procedure explicitly for the free particle and harmonic oscillator (where the semiclassical approximation is exact). For interacting systems the same semiclassical framework is used, with the approximation controlled in the conventional manner; the non-extensive statistics enter through the modified propagator. No specific interacting example is worked in this paper. We will revise the text to clarify the scope of the claim and note that explicit interacting applications are left for future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The abstract formulates an integrated Quantropy functional to compute semiclassical propagators for Boltzmann-Gibbs, S± and Sq statistics, then notes that the resulting q-propagator reproduces the free-particle q-wave function already obtained in cited reference [4]. This reproduction is presented as a consistency check rather than the load-bearing claim; the central assertion is the extension of the method to interacting problems. No equations, fitted parameters renamed as predictions, or self-citation chains that reduce the derivation to its inputs by construction are exhibited in the supplied text. The procedure is therefore self-contained against the external benchmark of the known free-particle case.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the Quantropy functional extends to non-extensive entropies and that the semiclassical approximation remains valid for deriving propagators and wave functions.

axioms (1)
  • domain assumption The analogy between quantum mechanics and statistical mechanics extends to non-extensive statistics through formulation of an integrated Quantropy functional.
    The paper begins with this analogy to compute propagators for Boltzmann-Gibbs, S± and S_q.

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Reference graph

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