Non-Equilibrium Trace Anomaly And Bulk Viscosity in Heavy Ion Collisions From Kinetic Theory

TL;DR

This work addresses non-equilibrium trace anomaly and bulk viscosity in a relativistic massive gas undergoing Bjorken expansion, relevant to the quark-gluon plasma produced in heavy-ion collisions. The authors solve the relativistic Boltzmann equation in the relaxation-time approximation using the method of moments, incorporating Landau matching to relate out-of-equilibrium and equilibrium thermodynamics, and track the evolution of moments $\rho_{n,l}$ that encode energy density, pressures, and dissipative stresses. They demonstrate a non-monotonic time dependence of the trace anomaly $\Theta^{\mu}{}_{\mu}/T^4$, a statistics-dependent behavior of the bulk pressure $\Pi$, and that larger initial chemical potential $\mu$ amplifies deviations from equilibrium. Remarkably, ensembles of non-equilibrium initial conditions reveal attractor-like convergence for the bulk pressure and pressure anisotropy to a common late-time trajectory, while the trace anomaly retains memory of the early-time evolution, underscoring the importance of non-equilibrium corrections to the equation of state in hydrodynamic modeling of heavy-ion collisions.

Abstract

We investigate the far-from-equilibrium dynamics and transport properties of a relativistic massive gas obeying Maxwell-Boltzmann (MB), Bose-Einstein (BE), and Fermi-Dirac (FD) statistics undergoing a boost-invariant Bjorken expansion. We solve the relativistic Boltzmann equation in the relaxation-time approximation (RTA) using the method of moments. We focus on the time evolution of the trace of the energy-momentum tensor $Θ^μ{}_μ$ and the bulk viscous pressure $Π$, which are key diagnostics of conformal-symmetry breaking in the rapidly evolving fireball created in heavy-ion collisions. We find that the non-equilibrium quantity $Θ^μ{}_μ/T^{4}$ exhibits a non-monotonic time dependence, with a local maximum at early times and a pronounced dip around the characteristic relaxation time scale $τ_{R}$. We further show that the scaled bulk pressure $Π/P_{0}$, where $P_{0}$ denotes the isotropic equilibrium pressure, depends sensitively on the particle statistics. In addition, increasing the initial chemical potential enhances the magnitudes of both $Π$ and $Θ^μ{}_μ/T^{4}$. Finally, by initializing the system with random non-equilibrium configurations, we demonstrate that the evolution of the scaled bulk pressure and the pressure anisotropy converges to a common late-time solution.

Figures (7)

• Figure 1: Time evolution of $T$ and $\alpha$ for three different distributions with initial conditions $T_0=1~\mathrm{GeV}$ and $\alpha_0=0$. The $x$-axis is shown on a logarithmic scale.
• Figure 2: Time evolution of the $\Theta^{\mu}_{\; \mu}/T^{4}$ scaled to its initial value, the scaled bulk pressure, the scaled longitudinal pressure, and the ratio $P_{L}/P_{T}$ for Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein statistics, starting from an equilibrium initial condition ($\rho_{n,l}=\rho^{eq}_{n,l}$) with $T_{0}=1~\mathrm{GeV}$ and $\alpha_{0}=0.0$.
• Figure 3: Evolution of bulk pressure ($\Pi$) and normalized trace of the energy-momentum tensor ($\Theta^{\mu}_{\;\mu}/T^{4}$) starting from different initial chemical potentials.
• Figure 4: Bulk pressure $\Pi$ (for MB) as a function of temperature $T$, obtained by slicing the three-dimensional $(\alpha,T,\Pi)$ surface along $\alpha = 0$.
• Figure 5: Time evolution of the scaled bulk pressure $\Pi/P_{0}$ for three different distribution starting from non-equilibrium initial conditions ($\rho_{n,l} \neq \rho^{eq}_{n,l}$) for thirty randomly selected initial configurations. The spread of trajectories reflects the variation in initial momentum-space anisotropies.