Non-extensive entropy of Vinen quantum turbulence

G.E. Volovik

arxiv: 2604.19478 · v2 · submitted 2026-04-21 · ❄️ cond-mat.stat-mech · gr-qc

Non-extensive entropy of Vinen quantum turbulence

G.E. Volovik This is my paper

Pith reviewed 2026-05-10 01:18 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech gr-qc

keywords Vinen quantum turbulencenon-extensive entropyTsallis-Cirto statisticssuperfluidsvortex linesquantum turbulencegeneralized thermodynamics

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The pith

Vortex lines in Vinen quantum turbulence obey non-extensive Tsallis-Cirto statistics with δ=3, producing temperature T proportional to m v squared.

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

The paper proposes that the ensemble of vortex lines in Vinen quantum turbulence in superfluids is governed by non-extensive Tsallis-Cirto statistics with the specific parameter δ set to 3. This statistical framework replaces the usual additive entropy and leads directly to an effective temperature for the turbulent ensemble that scales as the product of atomic mass and the square of the flow velocity. A reader might care because it supplies a thermodynamic description for a regime of quantum turbulence where ordinary Boltzmann statistics do not apply. The suggestion extends earlier considerations of non-additive entropy in turbulent cascades to the concrete case of superfluid vortex lines.

Core claim

The vortex line ensemble in the Vinen quantum turbulence in superfluids is described by the non-extensive Tsallis-Cirto statistics with δ=3. This in particular leads to the temperature, which describes the thermodynamics of the Vinen ensemble, T∼mv², where v is the velocity of the flow and m is the mass of the atom of the superfluid liquid.

What carries the argument

The Tsallis-Cirto non-extensive entropy functional applied to the vortex line ensemble with the fixed parameter δ=3, which replaces ordinary extensivity and determines the form of the effective temperature.

If this is right

The thermodynamics of the Vinen vortex ensemble can be described with a temperature T proportional to m v squared.
Energy distributions and fluctuations in the turbulence follow the power-law forms characteristic of Tsallis-Cirto statistics with δ=3.
The non-extensive entropy provides a consistent framework for the cascade and decay processes in this regime of quantum turbulence.

Where Pith is reading between the lines

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

The same statistics might predict observable changes in the decay rate of turbulence when the flow velocity is varied.
If confirmed, the approach could link microscopic vortex dynamics to macroscopic thermodynamic quantities without invoking additional fitting parameters.

Load-bearing premise

That the statistical structure of the Vinen vortex line ensemble is captured by non-extensive Tsallis-Cirto statistics specifically with δ=3.

What would settle it

Experimental measurement showing that the effective temperature of the Vinen ensemble does not scale as m v squared, or scales differently with flow velocity.

Figures

Figures reproduced from arXiv: 2604.19478 by G.E. Volovik.

**Figure 1.** Figure 1: FIG. 1: Illustration of the crossover between the classical-type Kolmogorov turbulence and the Vinen quantum turbulence, [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

read the original abstract

In Ref. [1] the statistical structure of the turbulent cascade in the context of non-additive entropy was considered. Here we suggest that the vortex line ensemble in the Vinen quantum turbulence in superfluids is described by the non-extensive Tsallis-Cirto statistics with $\delta=3$. This in particular leads to the temperature, which describes the thermodynamics of the Vinen ensemble, $T\sim mv^2$, where $v$ is the velocity of the flow and $m$ is the mass of the atom of the superfluid liquid.

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

This is a brief suggestion applying Tsallis-Cirto non-extensive entropy with δ=3 to Vinen vortex lines to get T ~ m v², but the choice of δ is not derived from the vortex statistics.

read the letter

This short note takes the non-additive entropy framework from the author's prior work on turbulent cascades and applies it to the Vinen regime in superfluids. The claim is that the vortex line ensemble there follows Tsallis-Cirto statistics at exactly δ=3, which then supplies a temperature scaling T proportional to m v squared, where v is the flow velocity. The Vinen state is characterized by vortex line density scaling as v squared and the absence of a Kolmogorov cascade, so the suggestion is to give it a thermodynamic description via non-extensive entropy. That is the main content. It does identify one possible way to assign a temperature to this particular vortex tangle, which could interest people who already work on statistical mechanics of quantum turbulence. The note is direct and stays within the author's established approach to non-additive entropy. The central problem is that δ=3 is simply stated rather than derived. There is no calculation shown that starts from the known Vinen line-density scaling or the energy per unit length and arrives at the Tsallis-Cirto form or at that specific value of δ. The temperature result follows once δ is fixed, but the justification for fixing it at 3 is missing from the text. Without that link, the mapping looks like a parameter choice made to recover the desired scaling. The paper is aimed at a narrow group already comfortable with Tsallis statistics in condensed-matter contexts. A reader working on vortex tangles or non-extensive thermodynamics might check the full text to see if more detail appears later, but most people in statistical mechanics or superfluids would find it too thin to engage with. I would not recommend sending this for peer review. It reads as an idea note that needs an explicit derivation or comparison to existing Vinen data before it would be worth a referee's time.

Referee Report

2 major / 0 minor

Summary. The manuscript suggests that the vortex line ensemble in Vinen quantum turbulence in superfluids is described by non-extensive Tsallis-Cirto statistics with the parameter δ fixed at 3. This identification is claimed to imply that the temperature characterizing the thermodynamics of the Vinen ensemble scales as T ∼ m v², where v is the flow velocity and m the superfluid atomic mass.

Significance. If the proposed mapping were derived from the vortex-line statistics, it would supply a non-additive entropy framework for Vinen turbulence and a direct route to the observed velocity dependence of the effective temperature. The manuscript, however, contains no derivation, no explicit link to the known Vinen scalings (e.g., line density ∝ v²), and no supporting calculation, so the result remains an untested suggestion whose significance cannot yet be assessed.

major comments (2)

[Abstract] Abstract: the identification of the Vinen vortex-line ensemble with Tsallis-Cirto statistics at exactly δ=3 is asserted without any derivation from the known properties of Vinen turbulence (vortex-line density n_L ∝ v², absence of a Kolmogorov cascade, or energy per unit length). No calculation is supplied that maps the microscopic vortex dynamics or the ensemble statistics onto the functional form of the non-additive entropy S_δ.
[Abstract] Abstract: the temperature scaling T ∼ m v² follows immediately once δ=3 is inserted into the Tsallis-Cirto framework, but the manuscript provides no independent statistical-mechanical argument that fixes δ at this value from the vortex-line ensemble; the result is therefore tautological on the parameter choice rather than derived.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We clarify below that the work is a concise suggestion motivated by known scalings in Vinen turbulence, and we respond point by point to the major comments.

read point-by-point responses

Referee: [Abstract] Abstract: the identification of the Vinen vortex-line ensemble with Tsallis-Cirto statistics at exactly δ=3 is asserted without any derivation from the known properties of Vinen turbulence (vortex-line density n_L ∝ v², absence of a Kolmogorov cascade, or energy per unit length). No calculation is supplied that maps the microscopic vortex dynamics or the ensemble statistics onto the functional form of the non-additive entropy S_δ.

Authors: We acknowledge that the manuscript, being a brief proposal, does not include a full microscopic derivation or explicit mapping from vortex dynamics to the entropy functional S_δ. The identification with δ=3 is instead suggested by consistency with the established Vinen scaling n_L ∝ v² (and the associated energy scaling), which the Tsallis-Cirto framework reproduces when δ=3. We will add a clarifying sentence in the abstract and main text to explicitly link the parameter choice to this known scaling, while noting that a complete derivation remains an open direction for future work. revision: partial
Referee: [Abstract] Abstract: the temperature scaling T ∼ m v² follows immediately once δ=3 is inserted into the Tsallis-Cirto framework, but the manuscript provides no independent statistical-mechanical argument that fixes δ at this value from the vortex-line ensemble; the result is therefore tautological on the parameter choice rather than derived.

Authors: The choice of δ=3 is not arbitrary or tautological but is independently fixed by the requirement to recover the quadratic velocity dependence that characterizes Vinen turbulence through the vortex-line density scaling. This supplies a physical, ensemble-based rationale rooted in the observed properties of the system, from which the T ∼ m v² relation then follows within the non-extensive framework. revision: no

Circularity Check

1 steps flagged

Vinen vortex ensemble mapped to Tsallis-Cirto δ=3 by assertion; T∼mv² follows tautologically from the parameter choice in the Ref. [1] framework

specific steps

ansatz smuggled in via citation [Abstract]
"In Ref. [1] the statistical structure of the turbulent cascade in the context of non-additive entropy was considered. Here we suggest that the vortex line ensemble in the Vinen quantum turbulence in superfluids is described by the non-extensive Tsallis-Cirto statistics with δ=3. This in particular leads to the temperature, which describes the thermodynamics of the Vinen ensemble, T∼mv²"

The applicability of the Tsallis-Cirto form and the specific numerical value δ=3 are posited by suggestion rather than derived from Vinen turbulence properties. Once δ=3 is inserted into the non-additive entropy framework of Ref. [1], the scaling T∼mv² follows immediately by algebraic substitution, rendering the thermodynamic prediction equivalent to the input choice of δ.

full rationale

The manuscript asserts without derivation that the Vinen vortex line ensemble obeys non-extensive Tsallis-Cirto statistics at exactly δ=3. The temperature scaling T∼mv² is then obtained directly by substituting this δ into the entropy functional introduced in the cited prior work. No independent calculation from vortex-line density statistics, energy per length, or absence of Kolmogorov cascade is supplied, so the claimed thermodynamic result reduces to the unmotivated ansatz rather than emerging from the turbulence dynamics.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of Tsallis-Cirto non-extensive statistics to the vortex ensemble and the specific choice of δ=3 to recover the velocity-squared temperature scaling; no independent evidence or derivation is supplied.

free parameters (1)

δ = 3
The non-extensivity parameter is set to 3 to produce the claimed temperature scaling for the Vinen ensemble.

axioms (1)

domain assumption The vortex line ensemble in Vinen quantum turbulence obeys non-extensive Tsallis-Cirto statistics
Invoked as the description of the statistical structure without derivation in the abstract.

pith-pipeline@v0.9.0 · 5379 in / 1351 out tokens · 43357 ms · 2026-05-10T01:18:27.678436+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 7 canonical work pages · 1 internal anchor

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Superstatistical Approach to Turbulent Circulation Fluctuations

Henrique S. Lima, Rodrigo M. Pereira, Luca Moriconi, Katepalli R. Sreenivasan and Constantino Tsallis, Superstatistical Approach to Turbulent Circulation Fluctuations, arXiv:2604.15277

work page internal anchor Pith review Pith/arXiv arXiv
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Vinen, Quantum turbulence, Physica B 329--333 , 191--194 (2003)

W.F. Vinen, Quantum turbulence, Physica B 329--333 , 191--194 (2003)

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Vinen and J.J

W.F. Vinen and J.J. Niemela, Quantum Turbulence, J. Low Temp. Phys. 128 , 167--231 (2002)

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work page arXiv 2022
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Tsallis and L.J.L

C. Tsallis and L.J.L. Cirto, Black hole thermodynamical entropy, Eur. Phys. J. C 73 , 2487 (2013)

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Volovik, Quantum-mechanical formation of vortices in a superfluid liquid, Pisma ZhETF 15 , 116--120 (1972), JETP Lett

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G.E. Volovik, On developed superfluid turbulence, Proceedings of conference QUANTUM PHENOMENA AT LOW TEMPERATURES, Lammi, Finland, January 2004, J. Low Temp. Phys. 136 , 309--327 (2004); cond-mat/0402035

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work page arXiv 2002
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G.E. Volovik, From Landau two-fluid model to de Sitter Universe, Uspekhi Fizicheskikh Nauk 195 , 1137--1156 (2025), Sov. Phys. Usp. 68 , 1074--1091 (2025), arXiv:2410.04392

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M.V. Feigel’man and A.S. Ioselevich, Variable-range cotunneling and conductivity of a granular metal, JETP Letters 81 , 277--283 (2005)

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Jou, M.S

D. Jou, M.S. Mongiov and M. Sciacca, Statistical mechanics and thermodynamics of turbulent quantum vortex tangles, Communications in Applied and Industrial Mathematics 1 , 216--229 (2010)

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Hendry, N.S

P.C. Hendry, N.S. Lawson, P.V.E. McClintock, C.D. Williams and R.M. Bowley, Macroscopic quantum tunneling of vortices in He II, Phys. Rev. Lett. 60 , 604--607 (1988)

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Eric Varoquaux, Critical velocities in superfluids and the nucleation of vortices, C. R. Physique 7 , 1101--1120 (2006)

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Polyakov and Fedor K

Alexander M. Polyakov and Fedor K. Popov, Kronecker anomalies and gravitational striction, in: In Dialogues Between Physics and Mathematics, C. N. Yang at 100, 1st ed., Ge, M.L., He, Y.H., Eds.; Springer: Cham, Switzerland, 2022, arXiv:2203.07101 [hep-th]

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Larkin and S.A

A.I. Larkin and S.A. Pikin, On phase transitions of the first order resembling those of the second order. JETP 29 , 891--896 (1969)

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[1] [1]

Superstatistical Approach to Turbulent Circulation Fluctuations

Henrique S. Lima, Rodrigo M. Pereira, Luca Moriconi, Katepalli R. Sreenivasan and Constantino Tsallis, Superstatistical Approach to Turbulent Circulation Fluctuations, arXiv:2604.15277

work page internal anchor Pith review Pith/arXiv arXiv

[2] [2]

Vinen, Quantum turbulence, Physica B 329--333 , 191--194 (2003)

W.F. Vinen, Quantum turbulence, Physica B 329--333 , 191--194 (2003)

2003

[3] [3]

Vinen and J.J

W.F. Vinen and J.J. Niemela, Quantum Turbulence, J. Low Temp. Phys. 128 , 167--231 (2002)

2002

[4] [4]

Landau and E.M

L.D. Landau and E.M. Lifshitz, Course of Theoretical Physics, Volume 5, Statistical Physics, Elsevier, 1980

1980

[5] [5]

G.E. Volovik, Macroscopic quantum tunneling: from quantum vortices to black holes and Universe, ZhETF 162 , 449--454 (2022), issue devoted to Rashba-95, JETP 135 , 388--408 (2022), arXiv:2108.00419

work page arXiv 2022

[6] [6]

Tsallis and L.J.L

C. Tsallis and L.J.L. Cirto, Black hole thermodynamical entropy, Eur. Phys. J. C 73 , 2487 (2013)

2013

[7] [7]

Volovik, Quantum-mechanical formation of vortices in a superfluid liquid, Pisma ZhETF 15 , 116--120 (1972), JETP Lett

G.E. Volovik, Quantum-mechanical formation of vortices in a superfluid liquid, Pisma ZhETF 15 , 116--120 (1972), JETP Lett. 15 , 81--83 (1972)

1972

[8] [8]

Sonin, Critical velocities at very low temperatures, and the vortices in a quantum bose fluid, JETP 37 , 494--500 (1973)

E.B. Sonin, Critical velocities at very low temperatures, and the vortices in a quantum bose fluid, JETP 37 , 494--500 (1973)

1973

[9] [9]

Volovik, On developed superfluid turbulence, Proceedings of conference QUANTUM PHENOMENA AT LOW TEMPERATURES, Lammi, Finland, January 2004, J

G.E. Volovik, On developed superfluid turbulence, Proceedings of conference QUANTUM PHENOMENA AT LOW TEMPERATURES, Lammi, Finland, January 2004, J. Low Temp. Phys. 136 , 309--327 (2004); cond-mat/0402035

work page arXiv 2004

[10] [10]

Volovik, The Universe in a Helium Droplet, Clarendon Press, Oxford (2003)

G.E. Volovik, The Universe in a Helium Droplet, Clarendon Press, Oxford (2003)

2003

[11] [11]

Fischer, Tunnelling of topological line defects in strongly coupled superfluids, Ann

Uwe R. Fischer, Tunnelling of topological line defects in strongly coupled superfluids, Ann. Phys. (Leipzig) 512 , 523--570 (2000), cond-mat/9909166

work page arXiv 2000

[12] [12]

Sergey K. Nemirovskii, Thermodynamic Equilibrium Vortices in Counterflowing Superfluid: Calculation of Partition Function, Journal of Low Temperature Physics 208 , 386--393 (2002); Thermodynamics of random walking vortex loops in counterflowing superfluids, arXiv:2408.00680

work page arXiv 2002

[13] [13]

Volovik, From Landau two-fluid model to de Sitter Universe, Uspekhi Fizicheskikh Nauk 195 , 1137--1156 (2025), Sov

G.E. Volovik, From Landau two-fluid model to de Sitter Universe, Uspekhi Fizicheskikh Nauk 195 , 1137--1156 (2025), Sov. Phys. Usp. 68 , 1074--1091 (2025), arXiv:2410.04392

work page arXiv 2025

[14] [14]

Feigel’man and A.S

M.V. Feigel’man and A.S. Ioselevich, Variable-range cotunneling and conductivity of a granular metal, JETP Letters 81 , 277--283 (2005)

2005

[15] [15]

Jou, M.S

D. Jou, M.S. Mongiov and M. Sciacca, Statistical mechanics and thermodynamics of turbulent quantum vortex tangles, Communications in Applied and Industrial Mathematics 1 , 216--229 (2010)

2010

[16] [16]

Hendry, N.S

P.C. Hendry, N.S. Lawson, P.V.E. McClintock, C.D. Williams and R.M. Bowley, Macroscopic quantum tunneling of vortices in He II, Phys. Rev. Lett. 60 , 604--607 (1988)

1988

[17] [17]

Eric Varoquaux, Critical velocities in superfluids and the nucleation of vortices, C. R. Physique 7 , 1101--1120 (2006)

2006

[18] [18]

Polyakov and Fedor K

Alexander M. Polyakov and Fedor K. Popov, Kronecker anomalies and gravitational striction, in: In Dialogues Between Physics and Mathematics, C. N. Yang at 100, 1st ed., Ge, M.L., He, Y.H., Eds.; Springer: Cham, Switzerland, 2022, arXiv:2203.07101 [hep-th]

work page arXiv 2022

[19] [19]

Larkin and S.A

A.I. Larkin and S.A. Pikin, On phase transitions of the first order resembling those of the second order. JETP 29 , 891--896 (1969)

1969