Quasi-Power Law Ensembles: Nonextensive Statistics or Superstatistics

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The authors parametrize the experimental data for multiparticle production using Tsalis statistics. The parameters are obtained using RHIC A-A collisions data.They also consider the statistics due to local temperature fluctuations.They show that  temperature-fluctuation mechanismscapable of generating Tsallis-like power-law distributions in multiparticle production necessarilyinduce nontrivial inter-particle correlations among the emitted hadrons. The authors then outline a strategy to discriminate fluctuations realized on an event-by-event basis from thosearising predominantly through event-to-event variability. Such a separation may be particularlypertinent for the characterization of high-multiplicity (high-density) final states produced at the LargeHadron Collider. Then I have the questions to the autors. It seems that the "small systems"i.e. high multiplicity p-p collisions will be an ideal example for illustration of the authors work. I think the work must include the study of these collisions, as well as the pA ones.

 

Author Response

Authors replies to the Referees comments

Paper: Quasi-power law ensembles: non-extensive statistics or superstatistics, entropy-4100244

We thank the Reviewers for their helpful feedback on the strengths of our paper. Below, we respond to the comments and questions in the reports.

Referee 1.

The authors parametrize the experimental data for multiparticle production using Tsalis statistics. The parameters are obtained using RHIC A-A collisions data. They also consider the statistics due to local temperature fluctuations. They show that  temperature-fluctuation mechanisms capable of generating Tsallis-like power-law distributions in multiparticle production necessarily induce nontrivial inter-particle correlations among the emitted hadrons. The authors then outline a strategy to discriminate fluctuations realized on an event-by-event basis from those arising predominantly through event-to-event variability. Such a separation may be particularly pertinent for the characterization of high-multiplicity (high-density) final states produced at the Large Hadron Collider. Then I have the questions to the autors. It seems that the "small systems" i.e. high multiplicity p-p collisions will be an ideal example for illustration of the authors work. I think the work must include the study of these collisions, as well as the pA ones.

The most important are p+p interactions, for which there are no uncertainties in N_coll. Such an analysis will be conducted in the future. Due to the available ALICE data, a detailed analysis requires careful attention.

Referee 2:

This article is dedicated to the long-standing puzzle on sudden success of the Tsallis statistics in description of the transverse momentum spectrum observed in relativistic collisions of hadrons and nuclei. Authors suggest to look deeply into not only inclusive spectra but on fluctuations of the mean pT as well for a deeper insight on the origin of the Tsallis (or Tsallis-like) statistics. The advantage of this article is the detailed explanation given in Sections 1 and 2 on how to properly select constraints in the entropy maximization procedure in order to obtain the desired Tsallis distribution. In Section 3, in turn, it is explained how sensitive are pT fluctuations to different scenarios of temperature fluctuations.

A few point are remained to be clarified:

1) In lines 192-198 and Fig.1 authors compare values of nonextensivity parameter q extracted from the p+p data with results from A+A collisions. Why? It does not seem to be correct to extract conclusion from such a comparison.

 Based on the A+A data, we calculate pT fluctuations and attribute them to temperature fluctuations. Using the number of binary collisions, we estimate fluctuations in internucleon interactions. The relative temperature fluctuations are described by q, which we compare with those of p+p interactions.

2) values of pT fluctuations in STAR data were extracted using certain pT cuts. Is it justified to compare q-value extracted from it with q-value extract from inclusive spectra from other experiments with different pT intervals?

The Tsallis distribution at high pT follows a power law. pT cuts do not greatly change the distribution's slope.

3) values of parameter q presented in Fig.1 are slightly different from values published e.g. in reference [J.Phys.G 50 (2023) 12, 125105; DOI: 10.1088/1361-6471/acffe1]. Can some explantion be found?

In Fig. 1, we present the q parameters derived from experimental data on transverse momentum distributions, fitted using the Tsallis distribution (1). Different versions of the Tsallis distribution produce slightly different q values. In J. Phys. G 50 (2023) 125105, transverse momentum distributions were modeled using the color-string model. The q-Gaussian distribution was employed to describe string tension fluctuations. The q-parameter in J. Phys. G 50 (2023) 125105 is not exactly the same as the q-parameter in the q-exponential Tsallis distribution (1).

4) In Conclusions authors claim that measurements of pT fluctuations in p+p would be helpful because there are no fluctuations in number of participating nucleons in this reaction. First of all, such measurements exist, see e.g. ALICE results in [Eur.Phys.J.C 85 (2025) 7, 776; DOI:10.1140/epjc/s10052-025-14325-4].

The ALICE results are very interesting. Extracting the relative pT fluctuations from them requires detailed analysis, which we plan to do in the future. We have added a relevant reference to the ALICE paper in the revised manuscript.

Second, in p+p interactions one can, nevertheless, expect fluctuations in number of particle-emerging sources, e.g. quark-gluon strings, which can overlap into different color ropes a.k.a. sources with different temperatures.

Certainly, internal fluctuations in p+p interactions can occur. Unlike solely event-to-event variability, which eliminates the possibility of non-extensive statistics, intra-event variability does not. In a single p+p interaction, fluctuations in the number of particle sources, such as quark-gluon strings, can take place. These strings may overlap, forming ropes of different colors that are recognized as sources of different temperatures. We have added this explanation in the revised manuscript (new footnote 3).

Given the extremely short period (5 days) the Editor gave us to make corrections to the text, we hope the above explanations and the revisions in the updated version of the paper meet the Referees' requirements.

Yours truly,

The Authors

Summary of the changes made in the manuscript:

1. As per the Editor's recommendation, four self-cited references have been removed,
2. Two new references have been added,
3. A new footnote was added (see footnote 3 in the revised text),
4. A new sentence was added at the end of the conclusion section.

Reviewer 2 Report

Comments and Suggestions for Authors

This article is dedicated to the long-standing puzzle on sudden success of the Tsallis statistics in description of the transverse momentum spectrum observed in relativistic collisions of hadrons and nuclei. Authors suggest to look deeply into not only inclusive spectra but on fluctuations of the mean pT as well for a deeper insight on the origin of the Tsallis (or Tsallis-like) statistics. The advantage of this article is the detailed explanation given in Sections 1 and 2 on how to properly select constraints in the entropy maximization procedure in order to obtain the desired Tsallis distribution. In Section 3, in turn, it is explained how sensitive are pT fluctuations to different scenarios of temperature fluctuations.

A few point are remained to be clarified:

1) In lines 192-198 and Fig.1 authors compare values of nonextensivity parameter q extracted from the p+p data with results from A+A collisions. Why? It does not seem to be correct to extract conclusion from such a comparison.

2) values of pT fluctuations in STAR data were extracted using certain pT cuts. Is it justified to compare q-value extracted from it with q-value extract from inclusive spectra from other experiments with different pT intervals?

3) values of parameter q presented in Fig.1 are slightly different from values published e.g. in reference [J.Phys.G 50 (2023) 12, 125105; DOI: 10.1088/1361-6471/acffe1]. Can some explantion be found?

4) In Conclusions authors claim that measurements of pT fluctuations in p+p would be helpful because there are no fluctuations in number of participating nucleons in this reaction. First of all, such measurements exist, see e.g. ALICE results in [Eur.Phys.J.C 85 (2025) 7, 776; DOI:10.1140/epjc/s10052-025-14325-4]. Second, in p+p interactions one can, nevertheless, expect fluctuations in number of particle-emerging sources, e.g. quark-gluon strings, which can overlap into different color ropes a.k.a. sources with different temperatures.

Author Response

Authors replies to the Referees comments

Paper: Quasi-power law ensembles: non-extensive statistics or superstatistics, entropy-4100244

We thank the Reviewers for their helpful feedback on the strengths of our paper. Below, we respond to the comments and questions in the reports.

Referee 1.

The authors parametrize the experimental data for multiparticle production using Tsalis statistics. The parameters are obtained using RHIC A-A collisions data. They also consider the statistics due to local temperature fluctuations. They show that  temperature-fluctuation mechanisms capable of generating Tsallis-like power-law distributions in multiparticle production necessarily induce nontrivial inter-particle correlations among the emitted hadrons. The authors then outline a strategy to discriminate fluctuations realized on an event-by-event basis from those arising predominantly through event-to-event variability. Such a separation may be particularly pertinent for the characterization of high-multiplicity (high-density) final states produced at the Large Hadron Collider. Then I have the questions to the autors. It seems that the "small systems" i.e. high multiplicity p-p collisions will be an ideal example for illustration of the authors work. I think the work must include the study of these collisions, as well as the pA ones.

The most important are p+p interactions, for which there are no uncertainties in N_coll. Such an analysis will be conducted in the future. Due to the available ALICE data, a detailed analysis requires careful attention.

Referee 2:

This article is dedicated to the long-standing puzzle on sudden success of the Tsallis statistics in description of the transverse momentum spectrum observed in relativistic collisions of hadrons and nuclei. Authors suggest to look deeply into not only inclusive spectra but on fluctuations of the mean pT as well for a deeper insight on the origin of the Tsallis (or Tsallis-like) statistics. The advantage of this article is the detailed explanation given in Sections 1 and 2 on how to properly select constraints in the entropy maximization procedure in order to obtain the desired Tsallis distribution. In Section 3, in turn, it is explained how sensitive are pT fluctuations to different scenarios of temperature fluctuations.

A few point are remained to be clarified:

1) In lines 192-198 and Fig.1 authors compare values of nonextensivity parameter q extracted from the p+p data with results from A+A collisions. Why? It does not seem to be correct to extract conclusion from such a comparison.

 Based on the A+A data, we calculate pT fluctuations and attribute them to temperature fluctuations. Using the number of binary collisions, we estimate fluctuations in internucleon interactions. The relative temperature fluctuations are described by q, which we compare with those of p+p interactions.

2) values of pT fluctuations in STAR data were extracted using certain pT cuts. Is it justified to compare q-value extracted from it with q-value extract from inclusive spectra from other experiments with different pT intervals?

The Tsallis distribution at high pT follows a power law. pT cuts do not greatly change the distribution's slope.

3) values of parameter q presented in Fig.1 are slightly different from values published e.g. in reference [J.Phys.G 50 (2023) 12, 125105; DOI: 10.1088/1361-6471/acffe1]. Can some explantion be found?

In Fig. 1, we present the q parameters derived from experimental data on transverse momentum distributions, fitted using the Tsallis distribution (1). Different versions of the Tsallis distribution produce slightly different q values. In J. Phys. G 50 (2023) 125105, transverse momentum distributions were modeled using the color-string model. The q-Gaussian distribution was employed to describe string tension fluctuations. The q-parameter in J. Phys. G 50 (2023) 125105 is not exactly the same as the q-parameter in the q-exponential Tsallis distribution (1).

4) In Conclusions authors claim that measurements of pT fluctuations in p+p would be helpful because there are no fluctuations in number of participating nucleons in this reaction. First of all, such measurements exist, see e.g. ALICE results in [Eur.Phys.J.C 85 (2025) 7, 776; DOI:10.1140/epjc/s10052-025-14325-4].

The ALICE results are very interesting. Extracting the relative pT fluctuations from them requires detailed analysis, which we plan to do in the future. We have added a relevant reference to the ALICE paper in the revised manuscript.

Second, in p+p interactions one can, nevertheless, expect fluctuations in number of particle-emerging sources, e.g. quark-gluon strings, which can overlap into different color ropes a.k.a. sources with different temperatures.

Certainly, internal fluctuations in p+p interactions can occur. Unlike solely event-to-event variability, which eliminates the possibility of non-extensive statistics, intra-event variability does not. In a single p+p interaction, fluctuations in the number of particle sources, such as quark-gluon strings, can take place. These strings may overlap, forming ropes of different colors that are recognized as sources of different temperatures. We have added this explanation in the revised manuscript (new footnote 3).

Given the extremely short period (5 days) the Editor gave us to make corrections to the text, we hope the above explanations and the revisions in the updated version of the paper meet the Referees' requirements.

Yours truly,

The Authors

Summary of the changes made in the manuscript:

1. As per the Editor's recommendation, four self-cited references have been removed,
2. Two new references have been added,
3. A new footnote was added (see footnote 3 in the revised text),
4. A new sentence was added at the end of the conclusion section.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

there is also CMS and ATLAS very detailed data

Author Response

Authors replies to the Referees’ second round comments

Paper: Quasi-power law ensembles: non-extensive statistics or superstatistics, entropy-4100244

We thank the Reviewers again for their helpful feedback on the strengths of our paper. Below, we respond to the comment of Reviewer 1.

Referee 1.

there is also CMS and ATLAS very detailed data

Published data mainly focus on correlators in heavy-ion collisions. The most significant are p+p interactions, which have no uncertainties in N_coll. Estimates of event-by-event mean transverse momentum ( ) fluctuations are presented through the integral correlator,  Fluctuations in  are usually characterized by the second moment of its distribution, providing a way to measure momentum fluctuations by assessing how much a particle’s  deviates from the event-wise . Such an analysis will be carried out in the future. Given the current data, a detailed analysis requires careful consideration. We have revised the last sentence of the conclusion section.

 

We hope that the above explanations and the revision in the updated version of the paper meet the Referees' requirements.

Yours truly,

The Authors

 

Summary of the changes made in the manuscript:

1. The last sentence of the conclusion section has been modified.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

My questions were answered and I have no further comments. The paper can be published in the present form.

Author Response

Authors replies to the Referees’ second round comments

Paper: Quasi-power law ensembles: non-extensive statistics or superstatistics, entropy-4100244

We thank the Reviewers again for their helpful feedback on the strengths of our paper. Below, we respond to the comment of Reviewer 1.

Referee 1.

there is also CMS and ATLAS very detailed data

Published data mainly focus on correlators in heavy-ion collisions. The most significant are p+p interactions, which have no uncertainties in N_coll. Estimates of event-by-event mean transverse momentum ( ) fluctuations are presented through the integral correlator,  Fluctuations in  are usually characterized by the second moment of its distribution, providing a way to measure momentum fluctuations by assessing how much a particle’s  deviates from the event-wise . Such an analysis will be carried out in the future. Given the current data, a detailed analysis requires careful consideration. We have revised the last sentence of the conclusion section.

We hope that the above explanations and the revision in the updated version of the paper meet the Referees' requirements.

 

Yours truly,

The Authors

 

Summary of the changes made in the manuscript:

1. The last sentence of the conclusion section has been modified.

Author Response File: Author Response.pdf