Heavy Ion Collision evolution modeling with ECHO-QGP

Heavy Ion Collision evolution modeling with ECHO-QGP

V. Rolando G. Inghirami A. Beraudo L. Del Zanna F. Becattini V. Chandra A. De Pace M. Nardi Dipartimento di Fisica e Scienze della Terra, Università di Ferrara, Via Saragat 1, I-44100 Ferrara, Italy INFN - Sezione di Ferrara, Via Saragat 1, I-44100 Ferrara, Italy Dipartimento di Fisica e Astronomia, Università di Firenze, Via G. Sansone 1, I-50019 Sesto F.no (Firenze), Italy INFN - Sezione di Firenze, Via G. Sansone 1, I-50019 Sesto F.no (Firenze), Italy INFN - Sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italy INAF - Osservatorio Astrofisico di Arcetri, L.go E. Fermi 5, I-50125 Firenze, Italy
Abstract

We present a numerical code modeling the evolution of the medium formed in relativistic heavy ion collisions, ECHO-QGP. The code solves relativistic hydrodynamics in D, with dissipative terms included within the framework of Israel-Stewart theory; it can work both in Minkowskian and in Bjorken coordinates. Initial conditions are provided through an implementation of the Glauber model (both Optical and Monte Carlo), while freezeout and particle generation are based on the Cooper-Frye prescription. The code is validated against several test problems and shows remarkable stability and accuracy with the combination of a conservative (shock-capturing) approach and the high-order methods employed. In particular it beautifully agrees with the semi-analytic solution known as Gubser flow, both in the ideal and in the viscous Israel-Stewart case, up to very large times and without any ad hoc tuning of the algorithm.

keywords:
QGP, Hydrodynamics, Heavy-ion, Heavy-ion collision
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1 Introduction

During the past years hydrodynamics has been aknowledeged as the most powerful tool to study the evolution of the medium formed in high-energy nuclear collisions. For this reason a variety of calculations and numerical codes have been developed, starting from the simple longitudinal boost-invariant Bjorken D calculation Bjorken:1982qr (), passing through D and D codes solving ideal hydrodynamics Kolb:2000sd (); Kolb:2002ve (); Kolb:2003dz (), up to more complex and structured codes simulating a full D viscous evolution for the QGP fluidLuzum2009 (); Baier2006 (); Romatschke:2009im (); Ryu:2012at (); Gale:2012rq (); Schenke:2010nt ().
In order to solve dissipative hydrodynamics, one needs to deal with a set of non-linear partial differential equations, which includes the conservation of the energy momentum tensor () and of conserved charges like the baryon number (), together with the time evolution of the bulk viscosity () and the shear stress tensor (). The system is closed by the choice of a suitable Equation of State (EoS). While the naive relativistic extension of the Navier-Stokes equations is affected by a well known causality problem, a consistent theoretical setup to include dissipative effects is the one formulated by Israel and Stewart Israel:1979wp () .

In this context ECHO-QGP was presented last year DelZanna:2013eua (), equipped with second-order treatment of causal relativistic viscosity effects and unique features such as the possibility of choosing between two different metric tensors and of solving either purely ideal or viscous hydrodynamics equations. As the other analogous tools, ECHO-QGP is characterized by a modular structure, allowing one to address the modeling of the initial conditions, the solution of the hydrodynamic evolution of the medium – representing the core of the code – and, eventually, the simulation of the final particle decoupling. ECHO-QGP is highly customizable, allowing the user to choose among a variety of initial conditions (including both optical and Monte Carlo Glauber model) as well as the possibility of defining an energy density (or entropy density) profile from scratch. The same principle applies to the equation of state: ECHO-QGP can handle both analytical and tabulated equations of state. The decoupling stage exploits the Cooper-Frye prescriptionCooper1974 (), allowing the calculation of a mean spectrum of or the Monte Carlo generation of a discrete set of particles. The hypersurface detection has been recently upgraded with the embedding of a refined algorithm which creates a smooth meshHuovinen:2012is ().

In order to ensure the reliability of the solving algorithm for the evolution stage, in our original publication DelZanna:2013eua () we showed how ECHO-QGP is able to overcome several numerical tests, displaying a beautiful agreement with some special analytic solutions. In the present contribution the above efforts in validationg the code will be briefly reminded and extended to the case of the so called Gubser-flow (both ideal and viscous), meanwhile appeared in the literature Gubser:2010ui (); Gubser:2010ze (); Marrochio:2013wla (). The latter represents a very important test, since the code has to reproduce a very non-trivial flow in (2+1)D in the presence of non-vanishing viscosity and relaxation time. ECHO-QGP turns out to be able to overcome such a test.

Finally, having at our disposal such a validated tool, we will outline our future programs.

2 Testing ECHO-QGP

ECHO-QGP has been widely tested, displaying agreement with analytic and semi-analytic solutions and other publicly available numerical codes.
Among the various test performed, an important role is played by the (2+1)-D shock-tube problem. Shock-capturing numerical schemes are designed to handle and evolve discontinuous quantities invariably arising due to the nonlinear nature of the fluid equations. In order to validate these codes, typical tests are the so-called shock-tube problems, performed in the heavy-ion field also by Molnar:2009tx (). The test gives a good esteem of the behavior of the fluid in presence of non-vanishing . In figure 1, we compare the energy density profile for the viscid and unviscid case, where the high accuracy of the results and the absence of numerical spurious oscillations near the shock front in the ideal case can be appreciated.

(a) Shock tube test: velocity profile
(b) Shock tube test: energy density profile
Figure 1: Shock tube test for ECHO-QGP. The test is performed placing an initial diaphragm along the diagonal of a square box (201 points and size 10 fm along x and y) adopting Minkowskian Cartesian coordinates, and letting ECHO-QGP evolve from t=1 up to t=4 fmc. The temperatures in the sides of the diaphragm are chosen to be GeV and GeV. It can be appreciated how the viscosity affects the evolution, and how ECHO-QGP handles shocks in the ideal case without the introduction of dumps.

Another fundamental qualifying test which numerical hydrodynamic codes have recently started addressing is the so called Gubser-flow. The latter is a (2+1)D solution for a conformal fluid (with an EoS ) characterized by longitudinal boost invariance and non-trivial azymuthally symmetric radial expansion. It was derived first by Gubser et al. Gubser:2010ze (); Gubser:2010ui () for an ideal fluid and extended only last year to the viscous case, within the Israel-Stewart setup, by Marrochio et al. Marrochio:2013wla (). The derivation of the solution is based on a rescaling of the metric (Weyl rescaling) and on a change of coordinates, which allows one to exploit at best the symmetries of the system arising from conformal invariance. In particular, one applies to the metric the rescaling .

In addition, one has to perform the coordinate transformation where the new coordinates are given by and , where is an arbitrary energy scale. In the new space (with quantities labeled by a hat) the fluid is at rest, , and the only equations to solve are:

(1)

where . Physical quantities in Minkowski space can be obtained from the above equations through the mapping

(2)

and compared to the numerical solution provided by ECHO-QGP, as displayed in Fig. 2.

The results have been obtained without any fine tuning of the parameters or modifications of the reconstruction algorithm. Nonetheless, ECHO-QGP reproduces with very high accuracy (the discrepancy is on average of the order of 0.1%) and up to late times (we verified the agreement up to 10 fm/c) the temperature (see Fig. (d)d), all the shear-stress tensor components (see Figs. (a)a,(b)b,(c)c) and obviously the flow (not shown).

(a)
(b)
(c)
(d)
Figure 2: Comparison between the semi-analytic solution, eq. 1 and an ECHO-QGP D simulation within the Bjorken coordinate metrics. The test is performed letting the fluid evolution be initialized with a symmetric energy density profile together, created ad hoc. The simulation has been performed with a grid of 0.025 fm in space and 0.001 fm in time. The shear viscosity to entropy density ratio is set to , while the shear relaxation time is . The energy scale is fixed at fm  while the equation of state must be the one of a conformal fluid, we choose .
The excellent agreement between the two can be appreciated for every component of the shear stress tensor (we show here just , and , but all the components present the same agreement), and for the thermodynamic variables (again we just show the temperature). For the sake of clearness, we show here just two close time steps ( and ), but the quality of the result is maintained up to much higher times. We remark then ECHO-QGP reproduces the semi-analytic solution without any need of fine-tuning.

3 Conclusions

ECHO-QGP guarantees stability and precision, as well as completeness in documentation and ease in the usage. We presented all those ECHO-QGP features in DelZanna:2013eua () and we remark its suitablity to model heavy-ion collisions showing how well it reproduces the flow and shear stress tensor in the viscous relativistic frame proposed in ref. Gubser:2010ui (); Gubser:2010ze (); Marrochio:2013wla (). Moreover, ECHO-QGP has been positively used for the investigation fluctuation propagationsFloerchinger:2013tya (); Floerchinger:2014fta (), and it is currently used to study vorticity effects on the directed flow (in preparation).

4 Aknowledgements

We would like to thank G. Denicol and the authors of the reference Marrochio:2013wla () for interesting discussions and advices about the semi-analytic solution. V.R. would like to thank prof. R. Tripiccione, G. Pagliara and A. Drago for their help in many interesting discussions about the decoupling stage.
This work has been supported by the Italian Ministry of Education and Research, grant n. 2009WA4R8W.

References

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