Code checking list

Huichao Song, 05/07/2008 (edited by U. Heinz 06/06/2008)

Here is the code-checking list that Kevin and I discussed last week at the BNL hydro workshop. I just wrote it down for discussion. If you feel that we need to check more details or change the checking procedures, please send an email with your suggestions to all parties involved.

To facilitate presentation of code verification checks, anyone posting a figure should also provide a link where the data files can be downloaded from which the figure was created. Then others can later plot their curves together with the already existing ones and update the old plot with a new version that includes their curves. Each figure should be accompanied by a few lines of text explaining what is plotted and, if applicable, how it was calculated.

All unpublished figures should be stamped "TECHQM preliminary, not to be shown" to avoid wholesale pirating.

Here is the plan:

We will compare results from ideal and viscous hydro for 2 different equations of state:

• A) ideal EoS (EOS I, $p=e/3$)

• B) an EoS connecting QGP and HRG phases with a first order phase transition (EoS Q).

For simplicity we begin with EoS I, $p=e/3$. EOS Q will be tabulated, with a link to the table to be posted here at a later time.

Initial and final conditions:

• We study Au+Au (A=197) at $b=0$ fm and $b=7$ fm.

• We use identical initial conditions for ideal viscous hydro.

• We start the evolution at time $\tau _{0}=0.6$ fm/c with zero transverse flow velocity.

• The initial transverse energy density profile is taken to be proportional to the transverse density of wounded nucleons calculated from the Glauber model according to Eq. (11) in arXiv:0712.3715. We use $\sigma =40$ mb for the nucleon-nucleon cross section. The nuclear thickness function $T_{A}$ is calculated with a Saxon-Woods nuclear density profile with saturation density $\rho _{0}=0.17$/fm**3, radius $R_{Au}=6.37$ fm, and surface thickness $\xi =0.54$ fm.

• The constant of proportionality is chosen such that in central collisions (b=0) the peak energy density in the fireball center $e(0,0;0)=e_{0}=30$ GeV/fm$^{3}$. (This corresponds roughly to RHIC collisions at ${\sqrt {s}}=200\,A\,{\rm {GeV}}$, and to a peak temperature in the fireball center of $T_{0}=359$ MeV.)

• For the viscous pressure tensor we assume Navier-Stokes initial conditions corresponding to boost-invariant longitudinal expansion with vanishing transverse flow: $\pi ^{xx}=\pi ^{yy}=-\pi ^{\eta \eta }/2=2\eta /(3\tau _{0})$.

• We decouple on a surface of constant temperature $T_{dec}=130$ MeV. For EOS I, we assume 2.5 massless quark flavor degrees of freedom (with degeneracy 2x2x3 for spin x (particle+antiparticle) x color) and 16 gluon degrees of freedom, so $T_{dec}=130$ MeV corresponds to $e_{dec}=0.516$ GeV/fm$^{3}$ for EOS I.

• We compute spectra and elliptic flow for directly emitted pions, kaons and protons, neglecting resonance decay contributions. (We will change this later when we test the resonance decay algorithm.)

• We assume zero baryon and isospin (charge) chemical potentials and (in EOS Q) assume chemical equilibrium at all temperatures in the hadron gas phase.

• The viscous hydro runs are done with $\eta /s=0.08$ and $\tau _{\pi }=3\eta /(sT)$. In order to avoid instability problems caused by relatively large viscous effects in the very dilute tails of the energy density distribution, we cut off the shear viscosity smoothly at large $r$ by multiplying it with the function $f_{\eta }(r)=1/[1+exp((r-R_{cut})/\delta )]$ with $R_{cut}=10$ fm and $\delta =0.6$ fm.

• We neglect bulk viscosity.

Things to be graphed for comparison:

1. Quantities closely related to final observables:

• time evolution of momentum anisotropy (note definition: $\langle \dots \rangle =\int \dots dxdy$ (standard integral over transverse plane)):

-- total momentum anisotropy: $\varepsilon '_{p}={\frac {\langle T_{0}^{xx}-T_{0}^{yy}+\pi ^{xx}-\pi ^{yy}\rangle }{\langle T_{0}^{xx}+T_{0}^{yy}+\pi ^{xx}+\pi ^{yy}\rangle }}$

-- ideal fluid (flow) contribution to momentum anisotropy: $\varepsilon _{p}={\frac {\langle T_{0}^{xx}-T_{0}^{yy}\rangle }{\langle T_{0}^{xx}+T_{0}^{yy}\rangle }}$
-- (here $T_{0}^{mn}=(e+p)u^{m}u^{n}-pg^{mn}$ is the ideal fluid part of the energy momentum tensor)

• surfaces of constant temperature $T=164,\ 150,\ 130\ {\rm {MeV}}$ in the $r-\tau$ plane ($b=0$) or in the $x-\tau$ and $y-\tau$ planes ($b=7$ fm).

• $p_{T}$-spectra for gluons (EoS I, 16 gluon degrees of freedom) or directly emitted $\pi ^{+},\ K^{+},\ p$ (EoS Q)). Gluon and pion spectra are evaluated using the Bose-Einstein distribution, all other hadron spectra are calculated with Boltzmann distributions.

• total final multiplicity of gluons (EOS I) or directly emitted $\pi ^{+},\ K^{+},\ p$ (EOS Q)

• differential elliptic flow $v_{2}(p_{T})$ and $p_{T}$-integrated $v_{2}$ for gluons (EoS I) or directly emitted $\pi ^{+},\ K^{+},\ p$ (EoS Q)

2. Other quantities:

• time evolution of the average transverse velocity $\langle \!\langle v_{r}\rangle \!\rangle$ (note definition: $\langle \!\langle \dots \rangle \!\rangle =\int \dots e(x,y)\gamma _{\perp }(x,y)dxdy$ (integral over transverse plane with Lorentz contracted energy density $\gamma _{\perp }e$ as weight function))

• velocity profiles along $r$, $x$ and $y$ directions at different times $\tau$ = 1, 4, 8 fm/c

-- $b=0$ fm: $v_{r}(r)$

-- $b=7$ fm: $v_{x}(x)$ and $v_{y}(y)$

• profiles of the shear pressure components $\pi ^{xx},\ \pi ^{yy},\ \pi ^{xy},\ \pi ^{\eta \eta }$ along the $r$, $x$ and $y$ di rections at different times $\tau$ = 1, 4, 8 fm/c

-- $b=0$ fm: $\pi ^{xx}(x)$ and $\pi ^{xx}(y)$, etc.

-- $b=7$ fm: $\pi ^{xx}(x)$ and $\pi ^{xx}(y)$, etc.

• velocity and shear pressure tensor profiles along the freeze-out surface:

-- $b=0$ fm: $v_{r}(r)$, $\pi ^{xx}(x)$, $\pi ^{xx}(y)$, etc.

-- $b=7$ fm: $v_{x}(x)$, $v_{y}(y)$, $\pi ^{xx}(x)$ and $\pi ^{xx}(y)$, etc.

-- (Note that along the freeze-out surface x, y and r are functions of freeze-out time, shown in the freeze-out surface above)

Question: Do different 2nd order theories converge to the same Navier-Stokes limit?

To address this question and the sensitity of viscous hydro results on parameters of the 2nd order theories such as the relaxation time $\tau _{\pi }$, let us follow the procedure introduced in arXiv:0805.1756: we compute the time evolution of the total momentum anisotropy $\varepsilon '_{p}$ from viscous hydro for different relaxation times $\tau _{\pi }$ and plot the value of $\varepsilon '_{p}$ at a fixed time, say $\tau =4$ fm/c, against $\tau _{\pi }$. In arXiv:0805.1756 it was found that for the "simplified" and "full" Israel-Stewart equations these plots were linear (although with opposie slopes) and, for $\tau _{\pi }\to 0$, extrapolated to the same values. Similar plots should be done for all other 2nd order formulations, such as Oettinger-Grmela used by KD+DT, to check sensitivity to the parameters in front of the 2nd order terms and convergence of the theory to the same Navier-Stokes limit.

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