Causal Viscous Hydrodynamics for Central Heavy-Ion Collisions

TL;DR

This work develops and tests a causal viscous hydrodynamics framework based on Israel-Stewart theory for central, longitudinally expanding, radially symmetric heavy-ion collisions. A straightforward finite-difference algorithm advances the hydrodynamic fields, and correlation-function fluctuations provide stringent numerical tests. It analyzes how shear viscosity characterized by $\eta/s$ alters the temperature evolution and particle spectra, showing that higher $\eta/s$ slows cooling and flattens spectra, with mass-dependent sensitivity. The study also shows that, for sufficiently large $\eta/s$, viscous corrections become large, signaling a possible breakdown of hydrodynamics and highlighting the need for improved modeling, while initial-condition adjustments can partially compensate some viscous effects.

Abstract

We study causal viscous hydrodynamics in the context of central relativistic heavy-ion collisions and provide details of a straightforward numerical algorithm to solve the hydrodynamic equations. It is shown that correlation functions of fluctuations provide stringent test cases for any such numerical algorithm. Passing these tests, we study the effects of viscosity on the temperature profile in central heavy-ion collisions. Also, we find that it is possible to counter-act the effects of viscosity to some extent by re-adjusting the initial conditions. However, viscous corrections are strongest for high-mass particles, signaling the breakdown of hydrodynamic descriptions for large eta/s.

Figures (6)

• Figure 1: Comparison between numerical results and the analytic approximation Eq.(\ref{['Baymres']}) (full and dashed lines, respectively) for the evolution of the temperature (left figure) and the velocity $v=\frac{u^r}{u^\tau}$ (right figure). The slight disagreement between numerical and analytical results is due to the approximations involved in the analytic solution and has the same sign and size as in the original work Baym:1984sr.
• Figure 2: Left figure: The correlation function $f(\kappa,\tau,\tau_0)$ from solving hydrodynamic equations on a lattice (symbols, see text for details) for $\tau_0=1$ fm/c and different $\tau$ compared to the analytic result (full lines). As one can see, the agreement between the analytic result and the measured correlation function is very good in general (a slight difference e.g. in the speed of sound would be clearly visible by a shift in the minima/maxima of $f$). Note that at later times, a discrepancy at low momenta $\kappa$ develops. This is probably a lattice artifact since increasing the simulated volume reduces the discrepancy (right figure).
• Figure 3: Comparison for $f(\kappa,\tau,\tau_0)$ from solving hydrodynamic equations on a lattice (symbols, $\eta/s=0.1$ (left) and $\eta/s=0.3$ (right), respectively) with 4096 sites and lattice spacing $a=0.25\, {\rm GeV}^{-1}$ and "analytic" solution (full lines) of Eqs.(\ref{['ansol2a']}).
• Figure 4: Temperature profile for calculations with different $\eta/s$ (dashed, dotted and solid lines, respectively) for three different times (see text for details). As expected, for larger values of $\eta/s$, differences to ideal hydrodynamics are biggest and viscous hydrodynamics initially cools slower than ideal hydrodynamics. However, note that in certain regions and at later times, viscous hydrodynamics turns out to give temperatures slightly smaller than the corresponding ideal hydrodynamic calculation.
• Figure 5: Inverse slope parameter $T_{\rm slope}$ for $T_f=0.135$ GeV and $\tau_0=1$ fm/c. Two initial conditions for $\Pi^{\mu\nu}$, corresponding to pressure isotropy (full line) and vanishing longitudinal pressure (dashed line) at $\tau=\tau_0$ are shown. Choosing $T_0=0.36$ GeV (left), the spectra become increasingly flatter when raising $\eta/s$, while this effect can be compensated by lowering $T_0$ (right, shown for $\eta/s=0.16$).