# Thermodynamical Aspects of the LGGR Approach for Hadron Energy Spectra

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The LGGR Model

## 3. A Simple Approach to Hadronization Statistics

#### 3.1. General Framework

#### 3.2. Phase Space Volumes

#### 3.3. Calculating the PDF for a Single Particle

## 4. LGGR Scenario for Hadronization

#### 4.1. Constant Growth and Reset Rates

#### 4.2. Preferential Growth and Reset Rates

#### 4.3. A Non-LGGR Approach

## 5. Equilibrium Thermodynamics Consideration

## 6. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

LGGR | Local Growth Global Reset |

QGP | Quark Gluon Plasma |

QCD | Quantum Chromodynamics |

## References

- Biró, T.S.; Néda, Z. Unidirectional random growth with resetting. Phys. A Stat. Mech. Appl.
**2018**, 499, 335. [Google Scholar] [CrossRef] - Biró, T.S.; Néda, Z.; Telcs, A. Entropic Divergence and Entropy Related to Nonlinear Master Equations. Entropy
**2019**, 21, 993. [Google Scholar] [CrossRef] - Néda, Z.; Varga, L.; Biró, T.S. Science and Facebook: The same popularity law! PLoS ONE
**2017**, 12, 0179656. [Google Scholar] [CrossRef] [PubMed] - Gere, I.; Kelemen, S.; Tóth, G.; Biró, T.S.; Néda, Z. Wealth distribution in modern societies: Collected data and a master equation approach. Phys. A Stat. Mech. Appl.
**2021**, 581, 126094. [Google Scholar] [CrossRef] - Néda, Z.; Gere, I.; Biró, T.S.; Tóth, G.; Derzsy, N. Scaling in income inequalities and its dynamical origin. Phys. A Stat. Mech. Appl.
**2020**, 549, 124491. [Google Scholar] [CrossRef] - Foka, P.; Janik, M.A. An overview of experimental results from ultra-relativistic heavy-ion collisions at the CERN LHC: Bulk properties and dynamical evolution. Rev. Phys.
**2016**, 1, 154. [Google Scholar] [CrossRef] - Rafelski, J.; Letessier, J. Hadronization of expanding QGP. Eur. Phys. J. A—Hadron. Nucl.
**2006**, 29, 107. [Google Scholar] [CrossRef] - Shen, K.; Barnaföldi, G.G.; Biró, T.S. Hadronization within the non-extesnive approach and the evolution of parameters. EPJ A
**2019**, 55, 126. [Google Scholar] [CrossRef] - Shen, K.; Barnaföldi, G.G.; Biró, T.S. Hadron spectra parameters within the non-extensive approach. Universe
**2019**, 5, 122. [Google Scholar] [CrossRef] - Bíró, G.; Barnaföldi, G.G.; Biró, T.S.; Ürmössy, K.; Takács, Á. Systematic analysis of the non-extensive statistical approach to high energy particle collisions. Entropy
**2017**, 19, 88. [Google Scholar] [CrossRef][Green Version] - Bíró, G.; Barnaföldi, G.; Biró, T.S. Tsallis-thermometer: A QGP indicator for large and small collisional systems. J. Phys. G
**2020**, 47, 105002. [Google Scholar] [CrossRef] - Biró, T.S.; Csillag, L.; Néda, Z. Transient Dynamics in the Random Growth and Reset Model. Entropy
**2021**, 23, 306. [Google Scholar] [CrossRef] [PubMed] - Inácio, I.; Velhinho, J. Comments on Mathematical Aspects of the Biró-Néda Model. Mathematics
**2022**, 10, 644. [Google Scholar] [CrossRef] - Biró, T.S.; Néda, Z. Equilibrium distributions in entropy driven balanced processes. Phys. A Stat. Mech. Its Appl.
**2017**, 474, 355. [Google Scholar] [CrossRef] - Adare, A.; Afanasiev, S.; Aidala, C.; Ajitanand, N.N.; Akiba, Y.; Al-Bataineh, H.; Alexander, J.; Al-Jamel, A.; Angerami, A.; Aoki, K.; et al. PHENIX Collaboration. Phys. Rev. C
**2008**, 78, 044902. [Google Scholar] [CrossRef] - Ang, H.W.; Rybczynski, M.; Wilk, G.; Wlodarczyk, Z. Sub-Poissonian multiplicity distributions in jets produced in hadron collisions. Phys. Rev. D
**2022**, 105, 054003. [Google Scholar] [CrossRef] - Tsallis, C. Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys.
**1988**, 52, 479. [Google Scholar] [CrossRef] - Bíró, G. The Application of the New Generation of Detector Simulations in High Energy Physics for the Investigation of Identified Hadron Spectra. Master’s Thesis, Roland Eötvös University, Budapest, Hungary, 2016. [Google Scholar]
- Zheng, H.; Zhu, L. Can Tsallis Distribution Fit All the Particle Spectra Produced at RHIC and LHC? Adv. High Energy Phys.
**2015**, 2015, 180491. [Google Scholar] [CrossRef] - Navarra, F.S.; Utyuzh, O.V.; Wilk, G.; Wlodarczyk, Z. Single particle spectra from information theory point of view. Nukleonika
**2004**, 49, S19. [Google Scholar] - Rybczynski, M.; Wlodarczyk, Z. Tsallis statistics approach to the transverse momentum distributions in p–p collisions. Eur. Phys. J. C
**2014**, 74, 2785. [Google Scholar] [CrossRef] - Jena, S.; Gupta, R. A unified formalism to study transverse momentum spectra in heavy-ion collision. Phys. Lett. B
**2020**, 807, 135551. [Google Scholar] [CrossRef]

**Figure 1.**Schematic illustration of the growth and reset process: (

**a**) The general mechanism of the process when all reset rates are positive; (

**b**) the case when the reset rate can be both positive and negative.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Biró, T.S.; Néda, Z. Thermodynamical Aspects of the LGGR Approach for Hadron Energy Spectra. *Symmetry* **2022**, *14*, 1807.
https://doi.org/10.3390/sym14091807

**AMA Style**

Biró TS, Néda Z. Thermodynamical Aspects of the LGGR Approach for Hadron Energy Spectra. *Symmetry*. 2022; 14(9):1807.
https://doi.org/10.3390/sym14091807

**Chicago/Turabian Style**

Biró, Tamás S., and Zoltán Néda. 2022. "Thermodynamical Aspects of the LGGR Approach for Hadron Energy Spectra" *Symmetry* 14, no. 9: 1807.
https://doi.org/10.3390/sym14091807