Bose-Einstein Condensation and Dissipative Dynamics in a Relativistic Pion Gas

TL;DR

This paper investigates how Bose-Einstein condensation of pions in ultra-relativistic hadronic matter affects dissipative transport and the equation of state. Using the relativistic Boltzmann transport equation in the relaxation time approximation, they compute shear viscosity $η$, bulk viscosity $ζ$, entropy density $s$, and the speed of sound $c_s$ for a pion gas with BE condensation, explicitly accounting for condensate and excited-state populations and finite-size effects, with expressions such as $η = (1/(15T)) ∫ d^3p τ(E_p) (p^4)/(E_p^2) f_p^0$ and $ζ = (1/T) ∫ d^3p τ(E_p) f_p^0 [E_p c_s^2 - p^2/(3E_p)]^2$. Principal results show strong suppression of $η/s$ and $ζ/s$ as the condensate fraction grows, while $c_s^2$ decreases, implying EOS softening; finite-size effects enhance BEC signatures, especially for larger systems. The work informs hydrodynamic modeling of the hadronic phase in heavy-ion collisions and motivates extensions to include interactions and two-fluid dynamics.

Abstract

Pion condensation in ultra-relativistic collisions presents a compelling theoretical phenomenon with significant implications for the dynamics of hadronic matter. Various theoretical frameworks offer insight into the nature of high-temperature Bose-Einstein condensation (BEC). The present study investigates the dissipative behavior of a relativistic pion gas undergoing Bose-Einstein condensation (BEC) in ultra-relativistic heavy-ion collisions. Further, we obtain viscosity ($η$), bulk viscosity ($ζ$), and speed of sound ($c_s$) by employing the Boltzmann transport equation with the relaxation time approximation. Findings show a substantial drop in $η/s$ and $ζ/s$ with the fractional increase in condensation. This effect is becoming more evident in larger systems approaching the thermodynamic limit. Alongside the reduction in viscosities, the speed of sound also decreases with increasing condensation, indicating a softening of the equation of state. The analysis of finite-size effects reveals that larger systems exhibit more pronounced signatures of BEC. These results suggest that pion condensation can influence the hydrodynamic evolution of the hadronic phase in heavy-ion collisions, with consequential implications for interpreting collective flow observables and the underlying equation of state.

Figures (6)

• Figure 1: (Colour Online) The left panel shows the condensate fraction as a function of temperature, and the right panel shows the temperature dependence of chemical potential, $\mu$, in the case of the thermodynamic limit as well as for different systems with size $R$ = 3, 5, and 7 fm. This is obtained by considering a fixed density of $n_{\rm tot}$ = 0.1 fm$^{-3}$ which corresponds to a critical temperature, $T_c$ = 0.076 GeV.
• Figure 2: Condensate fraction of pions as a function of temperature for different total number densities. The red curve in the $T-n_{\rm tot}$ plane is the BEC critical line ($T = T_c$) that represents the critical temperature, $T_c$, of the phase transition for different fixed densities. For $T<T_c$, the BEC phase is described by Eq. (\ref{['n_finite']}).
• Figure 3: The scaled entropy density, $s/n_{\rm tot}$ (left panel) and specific heat, $c_v/n_{\rm tot}$ (right panel) as a function of temperature for various system sizes with $R$ = 3, 5, and 7 fm.
• Figure 4: Shear viscosity (left panel) and bulk viscosity (right panel) as functions of the ratio of the number of pions in the condensate to the total number of pions for various system sizes with $R$ = 3, 5, and 7 fm.
• Figure 5: Shear viscosity to entropy density ratio (left panel) and bulk viscosity to entropy density ratio (right panel) of a pion gas under BEC as functions of temperature for various system sizes with $R$ = 3, 5, and 7 fm.