[Paper Review] Towards precise calculation of transport coefficients in the hadron gas. The shear and the bulk viscosities

Oleg N. Moroz|arXiv (Cornell University)|Dec 1, 2011

High-Energy Particle Collisions Research1 references3 citations

TL;DR

This paper presents a precise calculation of shear and bulk viscosities in a relativistic hadron gas using the Boltzmann equation with constant elastic cross sections and Boltzmann statistics, yielding analytical closed-form expressions. It finds that bulk viscosity is significantly higher in the hadron gas than in the pion gas, while shear viscosity is less sensitive to the hadron mass spectrum, offering a benchmark for hydrodynamic modeling in heavy-ion collisions and cosmology.

ABSTRACT

The shear and the bulk viscosities of the hadron gas at low temperatures are studied in the model with constant elastic cross sections being relativistic generalization of the hard spheres model. One effective radius ${r=0.4 fm}$ is chosen for all elastic collisions. Only elastic collisions are considered which are supposed to be dominant at temperatures ${T\leq 120-140 MeV}$. The calculations are done in the framework of the Boltzmann equation with the Boltzmann statistics distribution functions and the ideal gas equation of state. The applicability of these approximations is discussed. It's found that the bulk viscosity of the hadron gas is much larger than the bulk viscosity of the pion gas while the shear viscosity is found to be less sensitive to the mass spectrum of hadrons. The constant cross sections and the Boltzmann statistics approximation allows one not only to conduct precise numerical calculations of transport coefficients in the hadron gas but also to obtain some relatively simple relativistic analytical closed-form expressions. Namely, the correct single-component first-order shear viscosity coefficient is found. The single-component first-order nonequilibrium distribution function, some analytical results for the binary mixture and expressions for mean collision rates, mean free paths and times are presented. Comparison with some previous calculations for the hadron gas and the pion gas is done too. This paper is the first step towards calculations with inelastic processes included.

Motivation & Objective

• To develop a precise, analytically tractable framework for computing transport coefficients in the hadron gas at low temperatures.
• To assess the relative contributions of shear and bulk viscosities in the hadron gas compared to the pion gas, particularly near phase transitions.
• To establish a foundation for future inclusion of inelastic processes and quantum statistics by first validating the model with elastic collisions only.
• To provide closed-form analytical expressions for transport coefficients under the constant cross section and Boltzmann statistics approximations.
• To enable accurate numerical calculations of viscosities in the hadron-resonance gas for hydrodynamic modeling in heavy-ion collisions and cosmology.

Proposed method

• Uses the relativistic Boltzmann equation with classical Boltzmann statistics and ideal gas equation of state.
• Applies constant elastic differential cross sections (equivalent to a relativistic hard-sphere model) for all hadronic collisions.
• Derives analytical expressions for the first-order shear viscosity coefficient and the single-component nonequilibrium distribution function.
• Computes mean collision rates, mean free paths, and mean free times using the collision integral formalism.
• Performs numerical calculations for the hadron-resonance gas and compares results with the pion gas under identical approximations.
• Employs asymptotic expansions for collision integrals in high- and low-temperature limits to validate analytical expressions.

Experimental results

Research questions

• RQ1 How do the shear and bulk viscosities of the hadron gas compare quantitatively to those of the pion gas under the same approximations?
• RQ2 To what extent does the bulk viscosity depend on the hadron mass spectrum, and why is it significantly larger in the hadron gas than in the pion gas?
• RQ3 Can constant cross sections and Boltzmann statistics yield reliable analytical expressions for transport coefficients in a relativistic many-component system?
• RQ4 What are the analytical forms of the mean free path and collision rate in the hadron gas under the constant cross section model?
• RQ5 How do the results compare with previous calculations that used different statistics (e.g., Bose-Einstein) or dynamic cross sections?

Key findings

• The bulk viscosity of the hadron gas is substantially larger than that of the pion gas due to the broader mass spectrum and non-zero pion mass effects.
• The shear viscosity is less sensitive to the hadron mass spectrum, indicating that it is more robust across different hadronic compositions.
• A closed-form analytical expression for the single-component first-order shear viscosity coefficient is derived, valid in the constant cross section and Boltzmann statistics approximation.
• The mean free path for the single-component gas is given by $ l^{el}_{1'} = rac{raket{|m{v}_{1'}|}}{R^{el}_{1'1'}} = rac{ au e^{-z_1}(z_1+1)}{g_1 4 au ho_{11} T^3 z_1^3 K_3(2z_1)} $, matching nonrelativistic and ultrarelativistic limits.
• The model reproduces the nonrelativistic mean free path $ l^{el}_1 = rac{1}{4 au ho_{11} n_1 au} $ and ultrarelativistic limit $ l^{el}_1 = rac{1}{4 au ho_{11} n_1} $, validating consistency across limits.
• Discrepancies with previous works like prakash arise due to the constant cross section approximation, with differences in the bulk viscosity peak position and magnitude.