Simulating heavy ion collisions with MUSIC Documentation. Jean-Francois Paquet (Duke University)

Transcription

1 Simulating heavy ion collisions with MUSIC Documentation Jean-Francois Paquet (Duke University) Summer 2018

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3 CONTENTS 1 A brief introduction to heavy ion physics MUSIC: software overview Hydrodynamics Before hydrodynamics: initial conditions, energy deposition & pre-equilibrium dynamics After hydrodynamics: Particlization with Cooper-Frye Final hadron dynamics: decays & afterburners Additional topics i

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5 Collisions of large nuclei at relativistic energies 1 are used to study the many-body properties of Quantum Chromodynamics. The theoretical tools that are the usual workhorses of quantum field theory, such as perturbative methods and lattice calculations, can only describe very specific features of relativistic heavy ion collisions. As such, a first principles description of the complex dynamics of heavy ion collisions is not yet possible. On the other hand, very successful multi-stage models of heavy ion collisions have been built by combining lattice and perturbative calculations with effective models such as hydrodynamics. This document is an introduction to the C++ code MUSIC, a relativistic viscous hydrodynamic simulation of heavy ion collisions. The aim of this document is to explain the physics that MUSIC is designed to describe, and discuss the corresponding code where this physics is implemented. 1 Relativistic energies mean that the energy of the nuclei is much larger than their mass. CONTENTS 1

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7 CHAPTER ONE A BRIEF INTRODUCTION TO HEAVY ION PHYSICS Quantum Chromodynamics (QCD) is the theory of the strong (nuclear) interaction. It is a quantum field theory whose elementary degrees of freedom are quarks and gluons, which carry a colour charge. These elementary degrees of freedom are not asymptotic states however: quarks and gluons are normally confined into colour-neutral states called hadrons (protons, neutrons, nuclei,... ). Heavy ion physics uses relativistic collisions of nuclei to deconfine, for a short amount of time, quarks and gluons from hadrons. This makes possible to study the many-body properties of deconfined matter. Producing deconfined matter with relativistic heavy ion collisions Nuclei are imparted with a huge amount of kinetic energy ( relativistic collisions ). When the nuclei collide, this energy is used to break up the nuclei and keep the produced coloured matter deconfined for a short period of time. Fig. 1.1: [Ref: MADAI collaboration, Hannah Petersen and Jonah Bernhard] The figure above depicts different stages of the spacetime evolution of a heavy ion collision. The unit that is used to measure the size of the medium, and the amount of time that the medium remains deconfined, is the fermi 1. Five different stages are pictured, corresponding to: 1 1 fermi = 1 fm = m. The radius of a proton is approximately 1 fm, while large nuclei have radii of order 5-10 fm. By dividing by the speed of light, c, we obtain a unit for time as well: 1 fm/c s. In natural units, where c=1, time can be written in fm/c or just fm. 3

8 Before the collisions The two nuclei are moving close to the speed of light along the beam axis (horizontal axis in the figure), which is the reason for their Lorentz contraction (i.e. the nuclei are flattened along that axis). Early stage dynamics After the nuclei collide, a soup of deconfined QCD matter is produced. Hydrodynamic phase After fm, the overall spacetime evolution of the deconfined QCD matter can begin to be described with relativistic hydrodynamics. This phase can be referred to as the strongly-coupled quark-gluon plasma (sqgp). Particlization As the medium expands and becomes more dilute, the strongly-coupled quark-gluon plasma begins to reconfine into a gas of hadrons (a small amount of energy is also released as photons, leptons,... ). This reconfinement begins at the edge of the plasma and usually ends in the central regions. After 10 fm, most of the plasma is reconfined into hadrons. Late stage hadronic dynamics The gas of hadrons that follows the hydrodynamic phase still undergoes a complex dynamic. Unstable hadrons decay, and hadrons undergo elastic and inelastic collisions that change the momentum distribution and species composition of the hadron gas. Once all unstable hadrons have decayed into stable ones and once the hadron gas is too dilute for even hadronic interactions to occur, hadrons continue their way unhindered toward the detectors. MUSIC is primarily designed to describe the hydrodynamic phase of heavy ion collision. As explained in more details later, the early stage dynamics is not described in MUSIC, but is provided as an initial condition for the hydrodynamics phase. On the other hand, particlization and hadronic decays (one of the forms of late stage hadronic dynamics) are included in MUSIC, allowing for hadronic observables to be computed directly within MUSIC for comparison with data. Since hydrodynamics forms the basis of MUSIC, it is preferable to describe it first. The early stage dynamics is then discussed, followed up particlization, late stage hadronic dynamics and a discussion of how to compute hadronic observables in MUSIC. Before entering into these discussions, a brief software-oriented overview of MUSIC is provided. 1.1 MUSIC: software overview MUSIC is a C++ code that should compile and run on any POSIX system (Linux, OS X; Unix in general). As with any code actively used for research, there exists multiple versions of MUSIC. The one discussed in these notes is the official public version, hosted on Sourceforge 2. The source code of this public version is managed with Git. This public version of the code was released in All the C++ source code and header files can be found in the src/ directory. MUSIC additionally relies on two external libraries: MPI (Message Passing Interface), which is used to parallelize the code, and GSL (GNU Scientific Library), used for interpolation and random number generation. A copy of the code can be obtained with the command: git clone music-hydro-code The code can be compiled with both make and CMake : ( README.md ). instructions can be found in the README file Running MUSIC Since the public version of MUSIC is compiled with MPI, it may need to be run using the mpirun command: 2 Sourceforge was chosen to host the code in part for its well-designed Discussion section, which allows users to ask questions about MUSIC and allows for our replies to these questions to be publicly available to everybody as well. We have a good track record of replying promptly to questions asked on this Discussion website, and I would highly recommend to ask any question about MUSIC there. 4 Chapter 1. A brief introduction to heavy ion physics

9 mpirun -n 1./mpihydro input_file where mpihydro is the name of the executable produced after MUSIC is compiled. The input file is discussed below. The -n option of mpirun controls how many CPU s are used to run MUSIC, since the code can run in parallel. Using mpirun -n 2./mpihydro input_file would run MUSIC on two CPUs, and so on. There is obviously a large number of parameters in MUSIC. Some are numerical and some are physical. Numerical parameters include those related to the spacetime grid on which hydrodynamics is solved, for example, or the momentum grid on which Cooper-Frye is evaluated. Physical parameters can be used to make MUSIC solve ideal hydrodynamics instead of viscous hydrodynamics, for example, or to change the value of shear viscosity in the simulations All the numerical and physical parameters have default values. Warning: While it is possible that you want to keep the default values for the numerical parameters, it is extremely unlikely that you want to do the same for the physical parameters. All the parameters in MUSIC can be changed using an input file. Input files are text file which take the format parameter value, one parameter value combination per line, and the file ends with the line EndOfData : mode 2 Grid_size_in_y 201 Grid_size_in_x 201 Grid_size_in_eta 1 EndOfData The input file is passed to MUSIC as command-line argument: mpirun -n 1./mpihydro input_file_name The default value is used for any parameter not specified in the input file. There is a large number of parameters in MUSIC, and they will be discussed in their respective sections (Hydrodynamics, Particlization,... ). The only parameter that is important to discuss here is the parameter mode. MUSIC can simulate the hydrodynamic evolution of a heavy ion collisions, the particlization and the post-particlization hadronic decays. These correspond to different modes in MUSIC: Input parameter mode Value of parameter What MUSIC does 1 Runs hydrodynamics, particlization (Cooper-Frye) and hadronic decays in succession 2 Runs hydrodynamics only 3 Runs particlization (Cooper-Frye) only (assume mode 2 was run before) 4 Runs hadronic decays only (assume mode 3 was run before) Note : input parameters & default values The function that reads the input file is ReadInParameters::read_in_parameters() in read_in_parameters.cpp. While the parameters are discussed and explained at various other places, the most accurate reference for the list of input parameters and default values is this function MUSIC: software overview 5

10 1.1.2 Software licence MUSIC is released under the GPLv2 software licence. More practical matters related to academic research are discussed in the README file ( README.md ). Here is an abbreviated version, for reference: Publishing results computed with MUSIC New results computed with MUSIC may necessitate modifying certain parts of the code (routines computing hadronic observables, transport coefficients, initial conditions,... ). To avoid any confusion about the physics behind MUSIC, we ask that publications containing results computed with MUSIC specify the initial conditions, equation of state, transport coefficients (first and second order), freeze-out criteria and other information necessary to reproduce the results. The following papers should be cited when referring to MUSIC: 1. Schenke, S. Jeon, C. Gale. 3+1D hydrodynamic simulation of relativistic heavy-ion collisions Phys.Rev.C 82, (2010) [arxiv: ] 2. Schenke, S. Jeon, C. Gale. Elliptic and triangular flow in event-by-event (3+1)D viscous hydrodynamics Phys.Rev.Lett. 106, (2011) [arxiv: ] 3. J.-F. Paquet, C. Shen, G. S. Denicol, M. Luzum, B. Schenke, S. Jeon and C. Gale. Production of photons in relativistic heavy-ion collisions Phys. Rev. C 93, (2016) [arxiv: ] Modifying MUSIC MUSIC users may need to modify various parts of the code. The GPL licence allows for such modifications to be made and distributed (subject to restrictions that can be found in the licence). Nevertheless, we ask and require that you state clearly any scientifically relevant modifications made to the code when you share this modified version of MUSIC. This is to avoid any misunderstanding about the physics behind the code. Contributing to MUSIC Should you make changes to MUSIC that you believe would benefit from being shared with the wider community, please contact the code s maintainers through the MUSIC website. Contributions will be considered whenever possible. While no promises can be made regarding the inclusion of these modifications in the main version of the code, they could be made available separately on the MUSIC website for other users convenience. Bug fixes will be considered separately and will be applied promptly Running MUSIC: a simple example Here is a simple example of how to run MUSIC for a first time # Downloading the code with Git git clone music-hydro-code # Go to the base directory of MUSIC cd music-hydro-code # The source code is in "src/" cd src # MUSIC requires the libraries GSL and MPI to run. # If you are on Fedora or a similar Linux flavour, you may be able # to install the libraries this way: # yum install gsl mpich mpich-devel 6 Chapter 1. A brief introduction to heavy ion physics

11 # This example assumes that you are using the MPICH implentation of MPI # On certain system (and on most supercomputers), MPI is installed as a module that must be loaded before compiling and running the code module load mpi/mpich-x86_64 # MUSIC can be compiled with both "CMake" and "make". I use "make" here make -f makefile_guillimin # The compiled code is called "mpihydro". We move it to the base directory and go there cp mpihydro../ cd.. # Examples of input files are available in the "examples_inputs" directory. # Here we copy one of them to the base directory cp examples_inputs/2d/music_input_mode_2. # Then the code can be run with "mpirun" as explained in previously mpirun -np 1./mpihydro music_input_mode_2 1.2 Hydrodynamics Hydrodynamics is used to described the space-time evolution of the energy-momentum tensor T μν of the stronglycoupled quark-gluon plasma. This makes possible to learn about certain general many-body properties of QCD without necessarily tracking the complicated dynamics of every quark, gluon and hadron in the system. Such properties of QCD include the equation of state and the shear and bulk viscosities of QCD. The basis of hydrodynamics is local conservation of energy, which can be written: μ T μν (X) = 0 Additional conserved charged If there are conserved charged in the fluid, these are written μ j μ (X) = 0 The public version of MUSIC does not have any charge conservation and as such, this equation is not solved in the code. The energy-momentum tensor T μν can be decomposed into four different fields: T μν (X) = ε(x)u μ (X)u ν (X) [P(X) + Π(X)] [g μν u μ (X)u ν (X)] + π μν (X) where ε(x) is the energy density 1.2. Hydrodynamics 7

12 u μ (X) is the flow velocity π μν (X) is the shear stress tensor Π(X) is the bulk pressure The fifth field, P(X), is the pressure and it is not independent: it is related to the energy density by the equation of state, as we will discuss below. The tensor g μν is the metric tensor. Warning: Beware of the metric convention. These notes are based on the mostly-minus convention: g μν = diag(1, 1, 1, 1) The metric convention in most if not all MUSIC publications is mostly-minus. On the other hand, the formulae implemented in the code use the mostly-plus metric. Unless you have to modify the core algorithms of MUSIC, this should not affect you in anyway. If you do need to modify the code, pay special attention to the metric. What the decomposition of T μν above means is that every spacetime point of the quark-gluon plasma has a value for T μν, which can be expressed in terms of ε, u μ, π μν and Π. In MUSIC, this information is saved in the Grid class: class Grid { public: double epsilon; double p; //! u[flag][mu]: flag=0 is the actual values. flag!=0 are for RK steps double **u; //! bulk part of the TJb with the rk_flag double ***Pimunu; //! bulk pressure */ double *pi_b; ; Note that the Grid class contains much more information, which was hidden here and replaced by [... ]. Most of this information is used to store intermediate numerical steps. Other is redundant information, for example the temperature corresponding to the energy density, as calculated with the equation of state Ideal hydrodynamics In ideal hydrodynamics, π μν and Π are zero, and the energy-momentum tensor reduces to: T μν (X) = ε(x)u μ (X)u ν (X) P(X) [g μν u μ (X)u ν (X)] The flow velocity u μ (X) is defined as the unit vector satisfying: T μ ν(x)u ν (X) ε(x)u μ (X); u μ (X)u μ (X) = 1 8 Chapter 1. A brief introduction to heavy ion physics

13 meaning that the flow velocity represents the flow of energy. This is the so-called Landau frame. Other choice of frame are possible, but the Landau frame is the one used in MUSIC. Because it is a normalised four-vector, u μ (X) has three degrees of freedom for each spacetime point. The energy density ε(x) is another degree of freedom. Since the pressure P(X) is related to the energy density, T μν (X) has four degrees of freedom per spacetime point, and the energy conservation equation μ T μν (X) = 0 provides four constraints per spacetime point: this forms a closed set of differential equations. Fig. 1.2: [Ref.: J.-F.P.] MUSIC can be run ideal hydrodynamics by turning off both shear and bulk viscosities in the input file: Include_Shear_Visc_Yes_1_No_0 0 Include_Bulk_Visc_Yes_1_No_0 0 Include_second_order_terms 0 # include shear viscous effect # include bulk viscous effect # include second order non-linear coupling terms Pressure, equation of state and the assumptions behind ideal hydrodynamics The pressure is related to the energy density because ideal hydrodynamics is built around the assumption of local thermal equilibrium: one can define a small region at each spacetime point X where the fluid is assumed to be in local equilibrium. The system evolves in space and time, but the fluid is assumed to always maintain equilibrium locally. Thus one can use thermodynamics locally to relate the energy density to the pressure, the temperature, the entropy: P(X) = P(ε(X)),... The equation of state is that of the strongly-coupled quark-gluon plasma. It is calculated with lattice QCD and matched to a hadron resonance gas 3. 3 The composition of this hadron resonance gas is important at the particlization stage, when the energy-momentum tensor of the fluid is converted into hadrons. This discussion is best postponed to the Particlization section Hydrodynamics 9

14 Since the equation of state calculated from lattice QCD and from the hadron resonance gas are numerical (no simple closed expression), and since the matching between the two may also be performed numerically, the equation of state must effectively be provided to MUSIC either in the form of a parametrization or a table. Tables are used for all current equations of state in MUSIC. The code related to the equations of state is in the file eos.cpp. There are multiple functions to allocate and load to memory the files containing the equation of state information. There are also functions to interpolate the equation of state tables. In general, there shouldn t be any need to modify eos.cpp, since the equation of state can be chosen from the input file with the parameter EOS_to_use. The different choices of equations of state are explained at various points in the code and in example input files. Here is a summary: EOS_to_use 2 # type of the equation of state # 0: ideal gas # 1: EOS-Q from azhydro # 2: lattice EOS s95p-v1 # (from Huovinen and Petreczky) # 3: lattice EOS s95p with partial # chemical equilibrium (PCE) at 150 MeV # (see # /index.php/qcd_equation_of_state) # 4: lattice EOS s95p with chemical freeze # out at 155 MeV # 5: lattice EOS s95p at 160 MeV # 6: lattice EOS s95p at 165 MeV # 7: lattice EOS s95p-v1.2 for UrQMD Warning: Many of the equation of states implemented in MUSIC are dated. Updated ones are being implemented and tested: they should be available soon Viscous hydrodynamics Almost all hydrodynamics models of heavy ion collisions use Israel-Stewart-type second-order viscous hydrodynamics. In these models, conservation of energy μ T μν (X) = 0 and the equation of state P = P(ε) are complemented by equations of motion for the shear stress tensor and the bulk pressure. The equation of motions implemented in the public version of MUSIC are based on these publications: Denicol, G. S.; Niemi, H.; Molnár, E.; Rischke, D. H. Derivation of transient relativistic fluid dynamics from the Boltzmann equation, PRD85: (2012); PRD91: (2015) Molnár, E.; Niemi, H.; Denicol, G. S.; Rischke, D. H. Relative importance of second-order terms in relativistic dissipative fluid dynamics PRD89: (2014) This derivation of second-order viscous hydrodynamics is sometimes referred to as DNMR, after its authors. The equations of motion are also summarised in this publication 4 : Denicol, G. S.; Jeon, S.; Gale, C. Transport coefficients of bulk viscous pressure in the 14-moment approximation PRC90: (2014) 4 There is a τ π missing in front of π μ α ω ν α in Equation 2 of Denicol, G. S.; Jeon, S.; Gale, C. PRC90: (2014) This is clarified in the erratum of Denicol, G. S.; Niemi, H.; Molnár, E.; Rischke, D. H. PRD85: (2012) : PRD91: (2015) 10 Chapter 1. A brief introduction to heavy ion physics

15 Explicitly, the equations of motions can be compactly written as: for the shear stress tensor and τ π π μν + π μν = 2ησ μν + 2τ π π μ α ω ν α δ ππ π μν θ + φ 7 π μ α π ν α τ ππ π μ α σ ν α +λ ππ Πσ μν + φ 6 Ππ μν τ Π Π + Π = ζθ δππ Πθ + φ 1 Π 2 + λ Ππ π μν σ μν + φ 3 π μν π μν for the bulk pressure. Please refer to Denicol, G. S.; Jeon, S.; Gale, C. PRC90: (2014) for more information about the notation used to write the equation of motions. Vorticity Although the vorticity term, 2τ π π μ α ω ν α is included in the code, it is turned off by default. Turning it on requires modifying the following variable in the code to 1 instead of 0 : int include_vorticity_term = 0; in function Diss::Make_uWSource() of dissipative.cpp These equations have two first-order transport coefficients, the shear viscosity η and the bulk viscosity ζ. There are also 11 second-order transport coefficients: τ π, δ ππ, φ 7, τ ππ, λ ππ, φ 6 τ Π, δ ΠΠ, φ 1, λ Ππ, φ 3 All these transport coefficients, first and second order, are characteristic properties of how the quark-gluon plasma responds to deviation from equilibrium. Transport coefficients from first principles In principle, the transport coefficient of the quark-gluon plasma are calculable from QCD. In practice, these are extremely challenging calculations which can only be performed under very specific conditions, such as very high temperature (possibly temperatures orders of magnitude higher than the typical temperatures of the quark-gluon plasma) or with effective models of QCD. In MUSIC, the second order transport coefficients are related to the first-order transport coefficients through relations derived in the three publications listed above: τ π = 5η (ε + P) δ ππ = 4 3 τ π φ 7 = 9 70P τ ππ = 10 7 τ π 1.2. Hydrodynamics 11

16 λ ππ = 6 5 τ Π = ζ 15 ( 1 3 c2 s) 2 (ε + P) δ ΠΠ = 2 3 τ Π λ Ππ = 8 ( ) c2 s τ Π Three second order transport coefficient have not been derived in the publications above and are set to zero: φ 1 = φ 3 = φ 6 = 0. The first order transport coefficient are parametrized, with the objectives that they will be extracted from measurements. The relevant part of the code can be found in functions Diss::Make_uWSource() and Diss::Make_uPiSource() in source file dissipative.cpp. Here is a abbriged overview of these functions: double Diss::Make_uWSource(...) { if (DATA->T_dependent_shear_to_s == 1) { if (T < Ttr) { shear_to_s= *t/ttr-0.544*(t/ttr)*(t/ttr); else { shear_to_s= *t/ttr *(t/ttr)*(t/ttr); else { shear_to_s = DATA->shear_to_s; int include_wwterm = 0; int include_vorticity_term = 0; int include_wsigma_term = 0; if (DATA->include_second_order_terms == 1) { include_wwterm = 1; include_wsigma_term = 1; if (DATA->Initial_profile == 0) { include_wwterm = 0; include_wsigma_term = 0; include_vorticity_term = 0; // ////////////////////////////////////////////////////////////////////// // // ////////////////////////////////////////////////////////////////////// // // Defining transport coefficients // // ////////////////////////////////////////////////////////////////////// // // ////////////////////////////////////////////////////////////////////// // //double s_den = eos->get_entropy(epsilon, rhob); shear = (shear_to_s)*(grid_pt->epsilon + grid_pt->p)/(t + 1e-15); tau_pi = 5.0*shear/(grid_pt->epsilon + grid_pt->p + 1e-15); // transport coefficient for nonlinear terms -- shear only terms -- 4Mar2013 double transport_coefficient, transport_coefficient2, transport_coefficient3, transport_coefficient4; 12 Chapter 1. A brief introduction to heavy ion physics

17 // transport coefficients of a massless gas of single component particles transport_coefficient = 9./70.*tau_pi/shear*(4./5.) ; transport_coefficient2 = 4./3.*tau_pi; transport_coefficient3 = 10./7.*tau_pi; transport_coefficient4 = 2.*tau_pi; // transport coefficient for nonlinear terms -- coupling to bulk viscous pressure -- 4Mar2013 double transport_coefficient_b, transport_coefficient2_b; // transport coefficients not yet known -- fixed to zero transport_coefficient_b = 6./5.*tau_pi ; transport_coefficient2_b = 0.; /* This source has many terms */ /* everything in the 1/(tau_pi) piece is here */ /* third step in the split-operator time evol use Wmunu[rk_flag] and u[rk_flag] with rk_flag = 0 */ // ////////////////////////////////////////////////////////////////////// /// // ////////////////////////////////////////////////////////////////////// /// // Wmunu + transport_coefficient2*wmunu*theta /// // ////////////////////////////////////////////////////////////////////// /// // ////////////////////////////////////////////////////////////////////// /// // full term is tempf = -(1.0 + transport_coefficient2*(grid_pt->theta_u[rk_flag]) )*(grid_pt-> Wmunu[rk_flag][mu][nu]); // FOR GUBSER ANALYTIC // tempf = -( transport_coefficient2*(grid_pt->theta_u[rk_flag]) )*(grid_pt-> Wmunu[rk_flag][mu][nu]); // ////////////////////////////////////////////////////////////////////// /// // ////////////////////////////////////////////////////////////////////// /// // Navier-Stokes Term *shear*sigma^munu /// // ////////////////////////////////////////////////////////////////////// /// // ////////////////////////////////////////////////////////////////////// /// // remember: dusup[m][n] = partial^n u^m /// // remember: a[n] = u^m*partial_m u^n /// for( a=0;a<4;a++ ) { for( b=0;b<4;b++ ) { sigma[a][b] = grid_pt->sigma[rk_flag][a][b]; /* Make_uWSource */ double Diss::Make_uPiSource() { 1.2. Hydrodynamics 13

18 // switch to include non-linear coupling terms in the bulk pi evolution int include_bbterm = 0; int include_coupling_to_shear = 0; if (DATA->include_second_order_terms == 1) { include_bbterm = 1; include_coupling_to_shear = 1; // T dependent bulk viscosity from Gabriel // /////////////////////////////////////////// // Parametrization 1 // // /////////////////////////////////////////// double Ttr=0.18/0.1973; double dummy=temperature/ttr; double A1=-13.77, A2=27.55, A3=13.45; double lambda1=0.9, lambda2=0.25, lambda3=0.9, lambda4=0.22; double sigma1=0.025, sigma2=0.13, sigma3=0.0025, sigma4=0.022; bulk = A1*dummy*dummy + A2*dummy - A3; if(temperature < 0.995*Ttr) { bulk = lambda3*exp((dummy-1)/sigma3)+ lambda4*exp((dummy-1)/sigma4)+0.03; if(temperature > 1.05*Ttr) { bulk = lambda1*exp(-(dummy-1)/sigma1)+ lambda2*exp(-(dummy-1)/sigma2)+0.001; bulk = bulk*(grid_pt->epsilon + grid_pt->p)/temperature; // defining bulk relaxation time and additional transport coefficients // Bulk relaxation time from kinetic theory Bulk_Relax_time = 1./14.55/(1./3.-cs2)/(1./3.-cs2)/(grid_pt->epsilon + grid_pt- >p)*bulk; transport_coeff1 = 2.0/3.0*(Bulk_Relax_time); // from kinetic theory, small mass limit transport_coeff2 = 0.; // not known; put 0 transport_coeff1_s = 8./5.*(1./3.-cs2)*Bulk_Relax_time; // from kinetic theory transport_coeff2_s = 0.; // not known; put 0 // Computing Navier-Stokes term (-bulk viscosity * theta) NS_term = -bulk*(grid_pt->theta_u[rk_flag]); // Computing relaxation term and nonlinear term: - Bulk - transport_ coeff1*bulk*theta tempf = -(grid_pt->pi_b[rk_flag]) - transport_coeff1*(grid_pt->theta_u[rk_ flag])*(grid_pt->pi_b[rk_flag]); // Computing nonlinear term: + transport_coeff2*bulk*bulk if (include_bbterm == 1) BB_term = transport_coeff2*(grid_pt->pi_b[rk_flag])*(grid_pt->pi_b[rk_flag]); else BB_term = 0.0; 14 Chapter 1. A brief introduction to heavy ion physics

19 // Computing terms that Couple with shear-stress tensor double Wsigma, WW, Shear_Sigma_term, Shear_Shear_term, Coupling_to_Shear; if (include_coupling_to_shear == 1) { // Computing sigma^mu^nu double sigma[4][4]; for(int a=0;a<4;a++ ) { for(int b=0;b<4;b++ ) { sigma[a][b] = grid_pt->sigma[rk_flag][a][b]; Wsigma = ( grid_pt->wmunu[rk_flag][0][0]*sigma[0][0] +grid_pt->wmunu[rk_flag][1][1]*sigma[1][1] +grid_pt->wmunu[rk_flag][2][2]*sigma[2][2] +grid_pt->wmunu[rk_flag][3][3]*sigma[3][3] -2.*( grid_pt->wmunu[rk_flag][0][1]*sigma[0][1] +grid_pt->wmunu[rk_flag][0][2]*sigma[0][2] +grid_pt->wmunu[rk_flag][0][3]*sigma[0][3] ) +2.*( grid_pt->wmunu[rk_flag][1][2]*sigma[1][2] +grid_pt->wmunu[rk_flag][1][3]*sigma[1][3] +grid_pt->wmunu[rk_flag][2][3]*sigma[2][3] ) ); // multiply term by its respective transport coefficient Shear_Sigma_term = Wsigma*transport_coeff1_s; else Coupling_to_Shear = 0.0; /* Make_uPiSource */ Note that the bulk viscosity is hard-coded. For the shear viscosity, there are two options: a hard-coded parametrization or a constant value of η/s, whose value is set by the input parameter Shear_to_S_ratio. The hard-code parametrizations for η/s and ζ/s are shown below: The input parameters that control shear and bulk viscosity are: # transport coefficients Viscosity_Flag_Yes_1_No_0 1 # turn on viscosity in the evolution 1.2. Hydrodynamics 15

20 Include_Shear_Visc_Yes_1_No_0 1 # include shear viscous effect Shear_to_S_ratio 0.08 # value of \eta/s T_dependent_Shear_to_S_ratio 0 # flag to use temperature dep. \eta/s(t) Include_Bulk_Visc_Yes_1_No_0 0 Include_second_order_terms 0 # include bulk viscous effect # include second order non-linear coupling terms The Include_second_order_terms input parameter turns on and off a subset of second-order terms: φ 7 π α μ π ν α, τ ππ π α μ σ ν α, λ ππ Πσ μν and φ 6 Ππ μν in the shear tensor equation of motion, and δ ΠΠ Πθ, λ Ππ π μν σ μν and φ 3 π μν π μν in the bulk pressure equation of motion Solving the hydrodynamic equations: Numerical parameters The previous section was a brief overview of the physics and the equations that are solved in MUSIC. Ultimately the equations of hydrodynamics are a set of coupled differential equations that must be solved numerically. Hyperbolic coordinates are used in MUSIC: ( τ = ) t 2 z 2, x, y, η s = arctanh(z/t) The spatial coordinates (x, y, η s ) are discretized on a grid, and the time coordinate τ is evolved. Having a grid in (x, y, η s ) implies that each grid points have a coordinates which must be specified in one way or another. The grid used in MUSIC is regular in (x, y, η s ). It is centered around the origin, (x, y, η s ) = 0, up to a small detail. Coordinate system of the grid used in MUSIC The (x, y, η s ) grid in MUSIC is centered at x = 0 and y = 0 in the sense that there is an equal number of grid points on each side of x = 0 and y = 0. This is achieved by having an odd number of grid points in x and y. The number of grid point specified by the user is even, but an extra grid point is added so that the total number of grid points is odd and there can be a grid point exactly at x = y = 0. The number of grid point in η s specified by the user is even, and no additional grid point is added by MUSIC. To make sure that there is a grid point at η s = 0, the grid is staggered to the left: there is one more grid point in η s < 0 than there is for η s > 0. There only time these subtleties should be of concern is when external initial conditions are loaded into MUSIC. 16 Chapter 1. A brief introduction to heavy ion physics

21 Since it is a regular grid, the grid s coordinates can be specified by the grid s total size in each directions (x/y/η s ) and the number of cells that each coordinate is divided in. For the τ coordinate, an initial time is specified, along with a maximum time and a time-step. The corresponding input parameters are (see also this file): Initial_time_tau_0 0.4 Delta_Tau 0.02 # starting time of the hydrodynamic # evolution (fm/c) # time step to use in the evolution [fm/c] Total_evolution_time_tau 300. # the maximum allowed running # evolution time (fm/c) # almost always needs to be set to some large number Eta_grid_size 12.8 Grid_size_in_eta 64 X_grid_size_in_fm 20.0 Y_grid_size_in_fm 20.0 Grid_size_in_y 200 Grid_size_in_x 200 # spatial rapidity range # [-Eta_grid_size/2, Eta_grid_size/2 - delta_eta] # number of the grid points in spatial # rapidity direction # Must have at least 4 cells per processor. # Must be an even number. # One cell is positioned at eta=0, # half the cells are at negative eta, # the rest (one fewer) are at positive eta # spatial range along x direction in the # transverse plane # [-X_grid_size_in_fm/2, X_grid_size_in_fm/2] # spatial range along y direction in the # transverse plane # [-Y_grid_size_in_fm/2, Y_grid_size_in_fm/2] # number of the grid points in y direction # number of the grid points in x direction The algorithm used in MUSIC to solve the equations of hydrodynamics is named Kurganov-Tadmor. It is an algorithm designed to handle large gradients when solving partial differential equations. Kurganov-Tadmor is itself a variation of a larger family of partial differential equation solver called MUSCL (MUSIC stands for MUSCL-based Ion Collider ). A small number of parameters control numerical details of how the equation are solved. These parameters are hardcoded in the public version of MUSIC, and you should not need to modify them unless you are an expert with numerical solution of hydrodynamics with MUSCL-based schemes Construction of the particlization hypersurface Hydrodynamics cannot describe a fluid of arbitrary low density, in particular if this fluid is expanding rapidly (as is the case in heavy ion collisions). Physically, this means that the hydrodynamics description of the plasma produced in heavy ion collisions must be stopped at some point, and that the matter being described with hydrodynamics must be converted into different degrees of freedom. This procedure has been given different names over the years, but particlization is probably the most accurate and descriptive, since the fluid is converted into particles. This conversion from fluid to particle is performed in MUSIC using the Cooper-Frye algorithm, as is the case for almost every simulation of heavy ion collisions Hydrodynamics 17

22 Particlization, Cooper-Frye, hadronization and freeze-out Particlization is not necessarily the most widely used term to describe the conversion of the fluid to particles. Cooper-Frye is probably used more often. It is important not to confuse particlization/cooper-frye with hadronization and freeze-out, which are sometimes used loosely to refer to particlization. Technically speaking, hadronization refers to the reconfinement of the strongly-coupled quark-gluon plasma into hadrons. However, hadronization and particlization do not have to happen at the same time. Cooper-Frye actually requires that hadronization happens before particlization. Freeze-out refers to the moment when interactions among particles stop completely (or, in practice, when these interactions become negligible). For a hadron gas, it implies that the gas is so dilute that hadrons do not interact with each other anymore. Freeze-out can also refer to two separate concepts: chemical freeze-out, when inelastic collisions stop, and kinetic freeze-out, when both inelastic and elastic collisions stop. When freeze-out is used alone, it usually refers to the kinetic freeze-out. The Cooper-Frye formula will be discussed in a later section. What is relevant to discuss here is the particlization criteria, the criteria that is used to stop the hydrodynamics and convert the fluid into hadrons. In MUSIC as in almost all other hydrodynamics simulation of heavy ion collisions, particlization is assumed to happen at a given temperature (or energy density, which is related to temperature through the equation of state). As the temperature of the hydrodynamic medium decreases, different parts of the medium reach the particlization temperature. Connecting all the points at which the local temperature of the medium cross the particlization temperature forms a 4D spacetime hypersurface. This hypersurface is complete once every point in the hydrodynamic medium is below this particlization criteria, and hydrodynamics can then be stopped. A two-dimensional view of a particlization hypersurface is shown on the figure below. The colour and the contours represent different temperatures. The horizontal axis is time and the vertical axis is in the transverse direction of the fluid (perpendicular to the collisions axis). The particlization criteria is T = 155 MeV in this example. Warning: All particlization parameters in MUSIC are labelled with freeze or freeze_out, since the initial studies performed with MUSIC did not distinguish between particlization and freeze-out. In MUSIC, this hypersurface is constructed at every timestep in τ of the numerical solver. There is a number of parameters controlling this feature of MUSIC: Do_FreezeOut_Yes_1_No_0 1 calculated or not will evolve specified by # Whether the particlization hypersurface is # If the hypersurface is not calculated, the code # in time :math:`\tau` until the maximum time # input parameter "Total_evolution_time_tau" average_surface_over_this_many_time_steps 5 tau # Coarse-grain the hypersurface in the use_eps_for_freeze_out 1 T_freeze epsilon_freeze 0.18 # flag to use energy density as criteria to # find particlization surface # 0: use temperature, 1: use energy density # particlization temperature (GeV) # the particlization energy density (GeV/fm^3) The hypersurface is saved in a file named surface.dat. The information that is saved is the vector normal to the 18 Chapter 1. A brief introduction to heavy ion physics

23 Fig. 1.3: [Ref.: J.-F.P.] hypersurface pointing outward from the fluid, the position of every hypersurface element, along with the value of all hydrodynamic fields at this spacetime point. This surface.dat file is then used the evaluate Cooper-Frye when MUSIC is run in mode Before hydrodynamics: initial conditions, energy deposition & pre-equilibrium dynamics In these notes we define the early stage of heavy ion collisions as everything that happens before the plasma can be described by hydrodynamics. Hydrodynamics describes the evolution of the strongly-coupled quark-gluon plasma in space and time. It can describe this evolution if it is provided the state of the fluid on a given spacetime surface 5. Recall that, with second-order viscous hydrodynamics, the energy-momentum tensor T μν (X) is decomposed into four independent hydrodynamic fields: The energy density ε(x) The flow velocity u μ (X) The shear stress tensor π μν (X) 5 That is, if hydrodynamics is taken as nothing more than a system of partial differential equations, boundary conditions or initial conditions must be provided before a numerical solution of the equations can be obtained Before hydrodynamics: initial conditions, energy deposition & pre-equilibrium dynamics 19

24 The bulk pressure Π(X) In almost every simulation of heavy ion collisions, the equations of hydrodynamics are initialized at a fixed value of τ: τ 0. That is, hydrodynamic initial conditions are provided as T μν (τ = τ 0, x, y, η s ). More typically the different fields are initialized separately: ε(τ = τ 0, x, y, η s ), u μ (τ = τ 0, x, y, η s ), π μν (τ = τ 0, x, y, η s ) and Π(τ = τ 0, x, y, η s )separately. Note that these initial conditions represents a considerable amount of information. To understand how much of a challenge it is to determine T μν (τ = τ 0, x, y, η s ), it is worth taking a step back and look at the initial moments of a heavy ion collisions The earliest stages of heavy ion collisions No two collisions of nuclei are the same, and one of the reasons can be understood from a simple geometric picture, illustrated in the following figure: Fig. 1.4: [Ref: ] The two nuclei (ions) can overlap completly or can have a small overlap, and the strongly-coupled quark-gluon plasma that is formed in these two cases will be very different. The geometric degree of overlap can be quantified with the impact parameter b, shown on the above figure. In general, the more the overlap (the more central the collision is, b small), the larger the plasma. Smaller overlap between the nuclei results in smaller and shorter-lived plasma, and while very peripheral heavy ion collisions that barely graze each other might not produce any quark-gluon plasma at all. Note: The geometric picture discussed above is a very intuitive way of picturing heavy ion collisions. However, it is important to remember that the impact parameter b is not a quantity that can be extracted from heavy ion collision data. In this sense, it can be preferable to think of it as a model parameter, not a measurable quantity. There are additional fluctuations aside from fluctuations in the degree of overlap of the nuclei: the plasma produced by two collisions with the same overlap will differ, since collisions of nuclei are quantum processes with a probabilistic rather than deterministic outcome. The other major factor that affects the size and the lifetime of the plasma is the center-of-mass energy, s NN. The higher the collision energy, the larger and longer-lived the produced quark-gluon plasma. Note: MUSIC and hydrodynamic models in general have been found to describe a large number of experimental measurements in collisions of gold nuclei (Au) at the top RHIC energy ( s NN = 200 GeV) and collision of lead ion 20 Chapter 1. A brief introduction to heavy ion physics

25 (Pb) at the LHC ( s NN = 2760, 5020 GeV,... ). It is likely that hydrodynamic models can still be used in collisions of smaller nuclei and/or lower s NN collisions, but one should proceed with additional caution as these systems can have additional complications Obtaining initial conditions for hydrodynamics From a purely practical point of view, we can ask the question Given the species of the colliding nuclei (e.g. Au or Pb) and a collision energy ( s NN ), what are the initial conditions T μν (τ = τ 0, x, y, η s ) of hydrodynamics? This is a pragmatic question that hides a lot of the complexity of the early stage of heavy ion collisions. Between the moment that the nuclei collide and the time at which hydrodynamics become applicable, there is a complex dynamical evolution of the deconfined matter. Moreover the fact that there are fluctuations in heavy ion collisions implies that there is no single initial condition corresponding to two nuclei colliding. There is rather a family, or a distribution, of initial conditions. Describing this pre-hydrodynamic stage of stage of heavy ion is out of the scope of MUSIC. Broadly speaking, there are two options to initialise the hydrodynamics in MUSIC at τ = τ 0 : An external model can be used to describe the early stage of to collision. The results of this external model can be read in and used as initial conditions for MUSIC. This is the precedure adopted when IP-Glasma initial conditions are used with MUSIC. The initial conditions of hydrodynamics can be parametrized with the Glauber ansatz The initialization of the hydrodynamics fields is performed in function Init::InitTJb() in file init.cpp. This function first allocates memory for the discretized hydrodynamic fields and then initialize the fields. The input parameter Initial_profile is used to choose which type of initial conditions are used to the subsequent hydrodynamic evolution. The most important options are ################################### # parameters for initial conditions ################################### Initial_profile 3 # type of initial condition # 1: Optical Glauber model # 3: MC-Glauber model # 8: Read in initial profile from a file # (e.g., IP-Glasma) Reading-in an external initial condition is a more flexible option, but in certain cases Glauber initial conditions are sufficient. Reading-in external hydrodynamic initial conditions Since there is no accepted universal format of hydrodynamic initial conditions, reading-in external initial conditions will almost certainly require modifications to the code. This should be straightforward, however, given examples already in MUSIC that show how to initialize hydrodynamics with external files. When the input parameter Initial_profile 8 is used, MUSIC will read an external file specified by another input parameter, Initial_Distribution_Filename. The format expected when using Initial_profile 8 is the following. First, there is a header which indicates the number of grid points in each directions and the size of the grid cells, in fermi for x/y (unitless in η): 1.3. Before hydrodynamics: initial conditions, energy deposition & pre-equilibrium dynamics 21

26 string dummy; double dummy2; profile >> dummy >> dummy >> dummy2 >> dummy >> neta >> dummy >> nx >> dummy >> ny >> dummy >> deta >> dummy >> dx >> dummy >> dy ; where neta, nx and ny are the number of grid points in each directions, and deta, dx and dy are the step size. An example of header line would be: # dummy 1 etamax= 1 xmax= 200 ymax= 200 deta= 0 dx= 0.17 dy= 0.17 The rest of the file is read as follow: for (ix = 0; ix <= DATA->nx; ix++) { for (iy = 0; iy <= DATA->ny; iy++) { profile >> dummy1 >> dummy2 >> dummy3 >> density >> utau >> ux >> uy >> dummy >> dummy >> dummy >> dummy; where density is the energy density in GeV/fm^3, utau is u^tau, ux is u^x and uy is u^y. The dummy variable contain either redundant or unused information. The information read from the external file is then used to initialize the discretized hydrodynamic fields: int Init::InitTJb() { if (DATA->Initial_profile == 8) { // read in the profile from file // - IPGlasma initial conditions with initial flow music_message.info(" information on initial distribution -----"); music_message << "file name used: " << DATA->initName; music_message.flush("info"); ifstream profile(data->initname.c_str()); string dummy; int nx, ny, neta; double dx, dy, deta; // read the information line profile >> dummy >> dummy >> dummy >> dummy >> neta >> dummy >> nx >> dummy >> ny >> dummy >> deta >> dummy >> dx >> dummy >> dy; music_message << "neta=" << DATA->neta << ", nx=" << nx << ", ny=" << ny << ", deta=" << DATA->delta_eta << ", dx=" << dx << ", dy=" << dy; music_message.flush("info"); [Allocates memory] //read the one slice for (ix = 0; ix <= DATA->nx; ix++) { 22 Chapter 1. A brief introduction to heavy ion physics

27 for (iy = 0; iy <= DATA->ny; iy++) { profile >> dummy1 >> dummy2 >> dummy3 >> density >> utau >> ux >> uy >> dummy >> dummy >> dummy >> dummy; temp_profile_ed[ix][iy] = density; temp_profile_utau[ix][iy] = utau; temp_profile_ux[ix][iy] = ux; temp_profile_uy[ix][iy] = uy; if (ix == 0 && iy == 0) { DATA->x_size = -dummy2*2; DATA->y_size = -dummy3*2; music_message << "eta_size=" << DATA->eta_size << ", x_size=" << DATA->x_size << ", y_size=" << DATA->y_size; music_message.flush("info"); profile.close(); [Allocates more memory] for(int ieta = 0; ieta < DATA->neta; ieta++) { double eta = (DATA->delta_eta*(ieta + DATA->neta*rank) - (DATA->eta_size)/2.0); double eta_envelop_ed = eta_profile_normalisation(data, eta); for(ix = 0; ix <= DATA->nx; ix++) { for(iy = 0; iy<= DATA->ny; iy++) { rhob = 0.0; epsilon = (temp_profile_ed[ix][iy]*eta_envelop_ed *DATA->sFactor/hbarc); if (epsilon< ) epsilon = ; // initial pressure distribution p = eos->get_pressure(epsilon, rhob); // 1/fm^4 // set all values in the grid element: (*arena)[ix][iy][ieta].epsilon = epsilon; (*arena)[ix][iy][ieta].epsilon_t = epsilon; (*arena)[ix][iy][ieta].prev_epsilon = epsilon; (*arena)[ix][iy][ieta].rhob = rhob; (*arena)[ix][iy][ieta].rhob_t = rhob; (*arena)[ix][iy][ieta].prev_rhob = rhob; (*arena)[ix][iy][ieta].p = p; (*arena)[ix][iy][ieta].p_t = p; (*arena)[ix][iy][ieta].t = eos->get_temperature(epsilon, rhob); (*arena)[ix][iy][ieta].mu = eos->get_mu(epsilon, rhob); [Allocate even more memory] 1.3. Before hydrodynamics: initial conditions, energy deposition & pre-equilibrium dynamics 23

28 /* for HIC */ u[0] = (*arena)[ix][iy][ieta].u[0][0] = temp_profile_utau[ix][iy]; u[1] = (*arena)[ix][iy][ieta].u[0][1] = temp_profile_ux[ix][iy]; u[2] = (*arena)[ix][iy][ieta].u[0][2] = temp_profile_uy[ix][iy]; u[3] = (*arena)[ix][iy][ieta].u[0][3] = 0.0; (*arena)[ix][iy][ieta].prev_u[0][0] = temp_profile_utau[ix][iy]; (*arena)[ix][iy][ieta].prev_u[0][1] = temp_profile_ux[ix][iy]; (*arena)[ix][iy][ieta].prev_u[0][2] = temp_profile_uy[ix][iy]; (*arena)[ix][iy][ieta].prev_u[0][3] = 0.0; (*arena)[ix][iy][ieta].pi_b[0] = 0.0; for(int mu=0; mu<4; mu++) { /* baryon density */ (*arena)[ix][iy][ieta].tjb[0][4][mu] = rhob*u[mu]; for(nu=0; nu<4; nu++) { (*arena)[ix][iy][ieta].tjb[0][nu][mu] = ( (epsilon + p)*u[mu]*u[nu] + p*(data->gmunu)[mu][nu]); (*arena)[ix][iy][ieta].wmunu[0][nu][mu] = 0.0; (*arena)[ix][iy][ieta].prevwmunu[0][nu][mu] = 0.0; (*arena)[ix][iy][ieta].pimunu[0][nu][mu] = 0.0; (*arena)[ix][iy][ieta].prevpimunu[0][nu][mu] = 0.0; /* nu */ /* mu */ /* ix, iy, ieta */ // clean up [Free up memory] music_message.info("initial distribution done."); return 1; To summarize, there are two main input parameters used in this case: Initial_profile 8 # 8: Read in initial profile from a file # (e.g., IP-Glasma) # external file from which the initial conditions are read Initial_Distribution_input_filename example_inputs/filename Extra parameters For the example above, there are two extra parameters: initialize_with_entropy 0 # 0: with energy density # 1: with entropy density s_factor 1.0 # normalization factor read in # initial data file 24 Chapter 1. A brief introduction to heavy ion physics

29 The code in which parameter initialize_with_entropy appears was hidden from the previous example (but can be seen in the original source code), as I think this parameter should not need to be used when reading external files. Parameter s_factor is designed to be used to normalize the energy density read-in from the external file Initial_Distribution_input_filename. Ideally it is not a parameter that should need to be used and it should be set to 1.0. Glauber initial conditions A comprehensive presentation of the Glauber model can be found in e.g. There are two main inputs to the Glauber model: the distribution of nucleon inside the nuclei, and the inelastic nucleonnucleon cross-section. In very simplified terms, if the nuclei colliding in heavy ion collisions are taken as bag of nucleons, the Glauber model describes the collision of two nuclei as the incoherent collision of the all the nucleons inside the nuclei. This is illustrated in the following figure. Fig. 1.5: [Ref: Hannu Holopainen, PhD thesis ] Every dark circle represents a nucleon that underwent a nucleon-nucleon collisions. They are refered to as participant nucleons. Light circles are nucleons that did not collide with any nucleons from the other nuclei. Nucleon-nucleon collisions are called binary collisions. The Glauber model can be used as initial condition for the energy density ε(x) in the plane transverse to the collision axis (x-y plane). Energy can be deposited either at the position of the binary nucleon collisions, or at the position of the participant nucleons, or a combination of the two. An input parameter in MUSIC controls binary_collision_scaling_fraction the fraction of energy density initialized with binary collision and participants nucleons. The picture above is technically a Monte Carlo implementation of Glauber, refered to as Monte Carlo Glauber or MC Glauber. There is another version of Glauber, optical Glauber, which is roughly equivalent to averaging a large number of MC Glauber initial conditions, producing a smooth energy density distribution. Both MC Glauber and optical Glauber are available in MUSIC Before hydrodynamics: initial conditions, energy deposition & pre-equilibrium dynamics 25