[Paper Review] Extensive/nonextensive statistics for $p_T$ distributions of various charged particles produced in p+p and A+A collisions in a wide range of energies

Abdel Nasser Tawfik, Hayam Yassin+1 more|arXiv (Cornell University)|May 29, 2019

Statistical Mechanics and Entropy4 citations

TL;DR

This paper systematically compares extensive (Boltzmann-Gibbs) and nonextensive (Tsallis and generic axiomatic) statistical models to describe transverse momentum ($p_T$) distributions of charged pions, kaons, and protons in p+p and A+A collisions across energies from 7.7 to 2670 GeV. It finds that Tsallis statistics better fits p+p data, especially at low $p_T$, while Boltzmann statistics performs better for A+A systems, and proposes analytical expressions for fit parameters that depend on energy, system size, and statistical approach, highlighting inherent biases in ad hoc extensivity assumptions.

ABSTRACT

We present a systematic study for the statistical fits of the transverse momentum distributions of charged pions, Kaons and protons produced at energies ranging between 7.7 and 2670 GeV to the extensive Boltzmann-Gibbs (BG) and the nonextensive statistics (Tsallis as a special type and the generic axiomatic nonextensive approach). We also present a comprehensive review on various experimental parametrizations proposed to fit the transverse momentum distributions of these produced particles. The inconsistency that the BG approach is to be utilized in characterizing the chemical freezeout, while the Tsallis approach in determining the kinetic freezeout is elaborated. The resulting energy dependence of the different fit parameters largely varies with the particle species and the degree of (non)extensivity. This manifests that the Tsallis nonextensive approach seems to work well for p+p rather than for A+A collisions. Drawing a complete picture of the utilization of Tsallis statistics in modeling the transverse momentum distributions of several charged particle produced at a wide range of energies and accordingly either disprove or though confirm the relevant works are main advantages of this review. We propose analytical expressions for the dependence of the fit parameters obtained on the size of the colliding system, the energy, as well as the types of the statistical approach applied. We conclude that the statistical dependence of the various fit parameters, especially between Boltzmann and Tsallis approaches could be understood that the statistical analysis ad hoc is biased to the corresponding degree of extensivity (Boltzmann) or nonextensivity (Tsallis). Alternatively, the empirical parameterizations, the other models, and the generic (non)extensive approach seem to relax this biasness.

Motivation & Objective

• To investigate the applicability of extensive and nonextensive statistical models in describing $p_T$ distributions of charged particles in high-energy collisions.
• To resolve the inconsistency in the literature where Boltzmann statistics is used for chemical freezeout while Tsallis is used for kinetic freezeout.
• To derive analytical expressions for fit parameters (temperature, volume, nonextensivity) as functions of energy, system size, and statistical approach.
• To assess whether empirical parametrizations or generic nonextensive statistics reduce bias from assuming fixed extensivity (Boltzmann) or nonextensivity (Tsallis).
• To provide a comprehensive, unified picture of $p_T$ spectrum modeling across a wide energy range and particle species.

Proposed method

• Fits $p_T$ spectra of charged pions, kaons, and protons in p+p and A+A collisions from 7.7 to 2670 GeV using three statistical models: extensive Boltzmann-Gibbs, nonextensive Tsallis, and generic axiomatic nonextensive statistics.
• Applies the Tsallis $q$-exponential function and the generic $d$-exponential function to model nonextensive behavior, with $q>1$ indicating nonextensivity.
• Derives analytical expressions for the energy and system-size dependence of fit parameters (temperature $T$, volume $V$, nonextensivity $q$ or $d$) for each particle type and collision system.
• Compares results from statistical models with empirical parametrizations (e.g., Tsallis-type, Lévy-type) and other theoretical models to validate consistency and reduce model bias.
• Uses the Lambert-W function to characterize entropic equivalence classes and delayed relaxation in nonextensive systems.
• Performs systematic comparisons between Boltzmann, Tsallis, and generic axiomatic fits across particle species, collision types, and energies to assess model performance and parameter trends.

Experimental results

Research questions

• RQ1 How do the fit parameters (temperature, volume, nonextensivity) from Boltzmann, Tsallis, and generic axiomatic statistics vary with collision energy and system size in p+p and A+A collisions?
• RQ2 Why does the Tsallis nonextensive approach perform better for p+p than for A+A collisions, and how does this compare to the performance of Boltzmann statistics?
• RQ3 To what extent do empirical parametrizations and other models agree with the results from the proposed analytical expressions for fit parameters?
• RQ4 Can the generic axiomatic nonextensive approach reduce the bias inherent in assuming fixed extensivity (Boltzmann) or nonextensivity (Tsallis) a priori?
• RQ5 What is the physical significance of the entropic equivalence class parameter $d$ in the generic axiomatic framework, and how does it relate to relaxation dynamics?

Key findings

• The Tsallis nonextensive approach fits $p_T$ spectra in p+p collisions significantly better than in A+A collisions, particularly in the low-$p_T$ region.
• Boltzmann-Gibbs statistics performs better for A+A collisions, with temperature increasing with energy except at 200 GeV, where it drops.
• For pions, the Tsallis $q$ parameter increases with energy except at 62.4 and 200 GeV, while for other particles, $q$ increases monotonically with energy.
• The generic axiomatic nonextensive parameter $d$ decreases with increasing energy for all particle types, indicating a trend toward more extensive behavior at higher energies.
• The volume $V$ obtained from Boltzmann statistics decreases with increasing energy for all particles, while Tsallis $V$ is nearly constant for protons and kaons and slightly decreasing for pions.
• Analytical expressions for fit parameters derived from all three statistical models show good agreement with empirical parametrizations and other models, especially for $T$ and $q$ at specific energies like 62.4, 200, and 900 GeV.