QCD phase transition at finite temperature and chemical potential with the non-extensive statistics

TL;DR

The paper addresses how non-equilibrium, non-extensive dynamics alter the QCD phase transition at finite $T$ and $\mu$ by implementing Tsallis statistics in the HRG/QGP equation of state and connecting the two phases with a smooth crossover. It builds a non-extensive EOS using a $q$-dependent grand partition function $Z_q$, derives $P_q$, $s_q$, and related observables, and interpolates between hadronic and QGP regimes with $s_q(T)=f(T)s_q^{H}+(1-f(T))s_q^{QGP}$ where $f(T)=\tfrac12[1-\tanh((T-T_c)/\Gamma)]$. The results show strong $q$-dependence in dimensionless thermodynamic quantities and the speed of sound near the critical region, with finite $\mu$ amplifying these effects; when $q=1$, the results are consistent with LQCD and HRG data in the hadronic region, validating the framework while highlighting significant non-equilibrium modifications for $q>1$. Overall, the study provides a quantitative link between non-extensive statistics and QCD thermodynamics across the phase transition under finite density, offering insights into non-equilibrium effects in heavy-ion collisions.

Abstract

The intrinsic fluctuations, memory effects and long-range color interactions in high energy nuclear collisions imply the presence of non-Markovian processes in the fireball evolution, which affects the thermalization process towards equilibrium and produces a non-extensive behavior. In order to investigate the non-equilibrium effect on the quantum chromodynamics (QCD) phase transition at finite temperature ($T$) and chemical potential ($μ$), we apply a non-extensive correction to the equation of state in the parton (hadron resonance) gas at high (low) temperature and interpolate these two equation of states with a smooth crossover. The non-extensive statistics is characterized by a non-extensivity parameter $q$, which measures the degrees of deviation from the thermal equilibrium. It is found that the dimensionless thermodynamic quantities such as the entropy density, the pressure, the energy density, the specific heat at constant volume and the trace anomaly are sensitive to the deviation of $q$ from unity and they become large both in the hadronic and quark-gluon plasma phases with the increase of $q$. Moreover, this deviation leads to nontrivial corrections of the squared speed of sound ($(c_s^2)_q$) in the vicinity of the critical point ($T_c$) and at lower temperatures. Additionally, these thermodynamic quantities are sensitive to the deviation of $μ$ from zero. With increasing $μ$, they become enhanced in both phases. Specifically, for $(c_s^2)_q$, the value increases near $T_c$ but decreases at lower temperatures. Finally, we observe that our results with $q=1$ agree well with those from the Lattice QCD, the hadron resonance gas model, and the Thermal-Fist fit to the hadron yields in high energy nuclear collisions in the low temperature region up to $T\sim 150$ MeV.

Figures (3)

• Figure 1: (a) ((b), (c), (d), (e), and (f)): $s_q/T^3$ ($P_q/T^4$, $\varepsilon_q/T^4$, $\Delta_q$, $(C_V)_q/T^3$, and $(c_s^2)_q$) as a function of $T/T_c$. The solid and dashed black lines correspond to $q=$ 1.05 and $q=$ 1 with zero chemical potential, respectively. The solid and dashed blue lines correspond to $q=$ 1.05 and $q=$ 1 with $\mu_B=$ 398.2 MeV, $\mu_S=$ 89.5 MeV, and $\mu_Q=$ 0 MeV, respectively.
• Figure 2: (a) ((b), (c), (d), (e), and (f)): $s_q/T^3$ ($P_q/T^4$, $\varepsilon_q/T^4$, $\Delta_q$, $(C_V)_q/T^3$, and $(c_s^2)_q$) as a function of $T/T_c$ at zero chemical potential. The dashed (solid) black and blue curves, respectively, represents the temperature dependence of these quantities with $\it \Gamma=\textrm{0.05 }T_c$ and $\it \Gamma=\textrm{0.25 } T_c$ for the extensive (non-extensive) case.
• Figure 3: Upper panel: $P_q/T^4$ as a function of $T$ with zero chemical potential for $q=$ 1.05 (the solid line) and 1 (the dashed line). The results from the HRG model, the LQCD, and the Thermal-Fist fit are, respectively, shown as the dotted line, the solid black circles, and the empty markers. The inset is the temperature dependence of $P_q/T^4$ in the region with 136 MeV $<T<174$ MeV. Lower panel: $P_q/T^4$ as a function of $T$ with finite chemical potential for $q=$ 1.05 (the solid line) and $q=$ 1 (the dashed line). The results from the HRG model and the Thermal-Fist fit are, respectively, shown as the dotted line and the empty markers.