Early collective expansion: Relativistic hydrodynamics and the transport properties of QCD matter

TL;DR

Relativistic hydrodynamics provides a framework to model the space-time evolution of QCD matter created in high-energy collisions and to extract transport properties from final hadron spectra. The paper reviews both ideal and dissipative formulations, including Navier-Stokes and Israel-Stewart theories, and discusses how initialization, the equation of state, and freeze-out influence observables like radial and elliptic flow. It contrasts Glauber and CGC-based (KLN) initializations, analyzes central and non-central collision dynamics, and explains how the QCD equation of state and non-equilibrium hadronic chemistry shape the evolution and final spectra. The work emphasizes RHIC-era evidence for fast thermalization and strong collective expansion, while outlining limitations of ideal hydrodynamics and the necessity for improved pre-equilibrium descriptions and hydro+cascade hybrids to enhance quantitative understanding.

Abstract

Relativistic hydrodynamics for ideal and viscous fluids is discussed as a tool to describe relativistic heavy-ion collisions and to extract transport properties of the quark-gluon plasma from experimentally measured hadron momentum spectra.

Figures (8)

• Figure 1: Left: Number of wounded nucleons and binary collisions as a function of impact parameter, for Au+Au collisions $\sqrt{s}=130\,A$ GeV ($\sigma_0=40$ mb). Right: Density of wounded nucleons and binary collisions in the center of the collision as a function of impact parameter.
• Figure 2: Left: Density of binary collisions in the transverse plane for a Au+Au collision with impact parameter $b=7$ fm. Shown are contours of constant density together with the projection of the initial nuclei (dashed lines). Right: Spatial eccentricity $\epsilon$ as a function of the impact parameter Hirano:2005xf, calculated with Eq. (\ref{['equ:epsilonx']}) using the initial energy density as weight function, for four different models as described in the text.
• Figure 3: Chemical and thermal freeze-out points extracted from heavy-ion collisions at the GSI SIS, BNL AGS, CERN SPS and RHIC. The shaded area indicates the likely location of the quark-hadron phase transition as extracted from lattice QCD and theoretical models. An updated version adding many more chemical freeze-out points can be found in Ref. Cleymans:2007kj.
• Figure 4: Left: Abundance ratios of stable hadrons from central $200\,A$ GeV Au+Au collisions at RHIC Adams:2005dq. The blue lines show predictions from a thermal model fit with $T_\mathrm{chem}=163\pm4$ MeV, $\mu_B=24\pm4$ MeV, and a strangeness saturation factor $\gamma_s=0.99\pm0.07$Adams:2005dq. The inset shows the centrality dependence of $\gamma_s$. Right: Centrality dependence (with centrality measured by charged hadron rapidity density $dN_\mathrm{ch}/d\eta$) of (a) the thermal freeze-out temperature $T_\mathrm{kin}{\equiv}T_\mathrm{dec}$ (open triangles), the chemical freeze-out temperature $T_\mathrm{chem}$ (open circles), and the square root of the transverse areal density of pions $(dN_\pi/d\eta)/S$ (solid stars), and (b) the average transverse flow velocity $\langle\beta\rangle{\equiv}\langle v_\perp\rangle$ (solid triangles), for the same collision system STAR_Tdec.
• Figure 5: Left: The equation of state for baryon-free QCD matter. The upper plot shows the pressure $p$ as a function of energy density $e$ and (in the inset) the squared speed of sound $c_s^2=\frac{\partial p}{\partial e}$ as a function of temperature $T$. The lower panel shows $c_s^2$ as a function of energy density $e$Song:2008si. The solid red line (SM-EOS Q) is a slightly smoothed version of EOS Q (green dashed line). Right: Energy density (top) and temperature (bottom) of the central cell as a function of longitudinal proper time from a (2+1)-d ideal fluid dynamical simulation with of Au+Au collisions at RHIC HT02, for an EOS with a first-order quark-hadron transition at $T_c=170$ MeV and three choices of the chemical composition of the HRG below $T_c$: CE (dashed) assumes full hadronic chemical equilibrium at all temperatures (this case corresponds to the green dashed lines in the left panel); CFO (dotted) assumes chemical freeze-out of all hadronic species (stable and unstable) at $T_c$; PCE (solid) makes the realistic assumption that unstable resonances continue to re-equilibrate in the HRG phase via resonance scattering, but that the final yields of all stable decay products remain unchanged below $T_c$. The $p(e)$ curves for all three choices are almost identical HT02, resulting in identical time evolutions of the energy density $e(\tau)$.