Viscous Gubser flow with conserved charges to benchmark fluid simulations

TL;DR

VGCC provides $T$ and $\mu_Y$ evolution for viscous Gubser flow with conserved charges, enabling precise benchmarks for relativistic hydrodynamics codes via semi-analytical solutions and a defined freeze-out hypersurface. The authors explore two conformal EOSs, derive de Sitter/Milne transformed ODEs for $\hat{T}(\rho)$, $\hat{\mu}_Y(\rho)$, and $\hat{\bar{\pi}}(\rho)$, and examine how finite $\mu_Y$ enhances viscous effects and alters freeze-out characteristics. They benchmark the SPH-based CCake code against VGCC, finding strong agreement in $\mathcal{E}$, $n_Y$, $u^x$, and $\pi^{\mu\nu}$ with small discrepancies in $\pi^{xy}$ and a late-time breakdown due to far-from-equilibrium dynamics. The work delivers a concrete, physics-based test suite for codes handling nonzero chemical potentials and paves the way for extensions to more realistic EoS and higher-dimensional simulations.

Abstract

We present semi-analytical solutions for the evolution of both the temperature and chemical potentials for viscous Gubser flow with conserved charges. Such a solution can be especially useful in testing numerical codes intended to simulate relativistic fluids with large chemical potentials. The freeze-out hypersurface profiles for constant energy density are calculated, along with the corresponding normal vectors and presented as a new unit test for numerical codes. We also compare the influence of the equation of state on the semi-analytical solutions. We benchmark the newly developed Smoothed Particle Hydrodynamics (SPH) code CCAKE that includes both shear viscosity and three conserved charges. The numerical solutions are in excellent agreement with the semi-analytical solution and also are able to accurately reproduce the hypersurface at freeze-out.

Figures (5)

• Figure 1: (Color online) We plot the semi-analytical solutions for Gubser flow with conserved charges in the plane $y=0$ (except $\pi^{xy}$, which is plotted in the plane $y=x$). Included are (a) the temperature, (b) chemical potential, (c) the dimensionless quantity $\pi^{yy}/w$, where $w=\mathcal{E} + \mathcal{P}$ is the enthalpy density, for a diagonal entry of the shear stress tensor, and (d) the dimensionless quantity $\pi^{xy}/w$ for an off-diagonal entry. The three different colors correspond to different initial conditions for the ratio of initial temperature and initial chemical potential: $\hat{\mu}_{Y,0}/\hat{T}_0=0$ (black), $10$ (red), $15$ (blue). The different types of lines correspond to selected times $\tau=1.0$ fm/$c$ (solid lines), $1.2$ fm/$c$ (dashed lines) and $2.0$ fm/$c$ (dotted lines) during the evolution. See text for discussion.
• Figure 2: (Color online) We plot (a) the energy density, (b) number density, and (c) entropy density corresponding to the semi-analytical solutions for viscous Gubser flow with conserved charges shown in \ref{['fig:evolution comparison']} in the plane $y=0$. We include the results for differential values of the initial conditions and at different times (see the caption of \ref{['fig:evolution comparison']} for details). The number density plot does not feature black lines because at $\hat{\mu}_{Y,0}/\hat{T}_0=0$ the system has exactly vanishing number densities such that there is no evolution of the number density. Most notably, an increase in chemical potential can significantly decrease spatial variations of the energy density and entropy. The entropy density is most sensitive to the details of the temperature profile, as can be seen by the presence of the shoulders for the blue lines ($\hat{\mu}_{Y,0}/\hat{T}_0 = 15$).
• Figure 3: (Color online) We plot (a) the freeze-out hypersurface and normal vectors, and $T$ versus $\mu_Y$ trajectories for two initial conditions, (b) $\hat{\mu}_{Y,0}/ \hat{T}_0 = 10$ and (c) $\hat{\mu}_{Y,0}/ \hat{T}_0 = 15$ with $\hat{T}_0 = 1.2$. Included in the freeze-out hypersurface plot is a color bar which indicates the entropy density for a freeze-out cell at a given $(\tau_\mathrm{FO}, r)$. As expected, at zero chemical potential, the freeze-out hypersurface is isentropic. At non-vanishing chemical potential, the entropy in the core is lower (i.e., lower temperature) than the entropy at the tails. To allow the normal vectors and freeze-out hypersurface on the screen, they have been rescaled by a factor of $0.075$.
• Figure 4: (Color online) We plot the numerical solution from the c c a k e code (dots) and semi-analytical solution (solid line) using EoS2 in Eq. \ref{['eq:eos2']} as a function of distance from the origin and for various time steps (indicated by the color bar) within the first fm/$c$ of evolution. The comparison is made between (a) energy density, (b) number density, (c) the $x$-component of the fluid velocity, (d) the $xx$-component of the shear stress tensor, (e) the inverse Reynolds number defined in Eq. \ref{['eq:rey-pi']}, and (f) the $xy$-component of the shear stress tensor. We see excellent agreement for all shown variables with a very mild deviation of around a percent around the core for the energy density and number density. For the $xx$-component of the shear stress tensor, we also observe an increasing deviation of the numerical solution from the semi-analytical one for increasing time, specifically between $r=1$ fm and $r=2$ fm, with the maximum relative deviations at $\tau = 1.6$ fm/$c$ getting large since the numerical solution is very close to zero. The largest deviations are seen on the $xy$-component of the shear stress tensor between $r=1$ fm and $r=2$ fm, and also present to a lesser extent on the diagonal component.
• Figure 5: (Color online) A comparison between the freeze-out hypersurface and normal vectors obtained from c c a k e (red) and the semi-analytical solution for VGCC (black). We see a slight disagreement on the magnitude of the (close to) divergent normal vectors. This disagreement is likely a result of the SPH particles freezing-out a time-step after the actual freeze-out time within the simulation.