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The iEBE-VISHNU code package for relativistic heavy-ion collisions

Chun Shen, Zhi Qiu, Huichao Song, Jonah Bernhard, Steffen Bass, Ulrich Heinz

arXiv:1409.8164v3nucl-thhep-phnucl-ex

TL;DR

Relativistic heavy-ion collisions require detailed event-by-event modeling because their evolution is inaccessible to direct external probing. The paper presents iEBE-VISHNU, a hybrid package combining viscous hydrodynamics, particle sampling, and hadronic cascading, and documents its implementation and tests. The package reproduces the measured p+Pb multiplicity distribution at √sNN = 5.02 TeV while supporting large-scale simulations and rare-probe interfaces.

  • Problem

    The tiny, short-lived QGP cannot be measured with external probes, motivating detailed theoretical simulations of relativistic heavy-ion collision dynamics.

  • Method

    iEBE-VISHNU links initial-condition generation, (2+1)-dimensional viscous hydrodynamics, Cooper-Frye particle sampling, and hadronic cascade evolution for event-by-event simulations.

  • Results

    The multiplicity-fluctuation model correctly reproduces the measured multiplicity distribution for p+Pb collisions at √sNN = 5.02 TeV.

  • Takeaways & Limitations

    The package provides a general framework for large-scale model-data comparisons and calculations of rare probes coupled to the evolving medium.

  • Takeaways & Limitations

    The document does not compare iEBE-VISHNU hadron and photon spectra or anisotropy coefficients with experimental data.

Abstract

from arXiv · show

The iEBE-VISHNU code package performs event-by-event simulations for relativistic heavy-ion collisions using a hybrid approach based on (2+1)-dimensional viscous hydrodynamics coupled to a hadronic cascade model. We present the detailed model implementation, accompanied by some numerical code tests for the package. iEBE-VISHNU forms the core of a general theoretical framework for model-data comparisons through large scale Monte-Carlo simulations. A numerical interface between the hydrodynamically evolving medium and thermal photon radiation is also discussed. This interface is more generally designed for calculations of all kinds of rare probes that are coupled to the temperature and flow velocity evolution of the bulk medium, such as jet energy loss and heavy quark diffusion.

PROGRAM SUMMARY

iEBE-VISHNU is a computational package for event-by-event relativistic heavy-ion collision simulations, implemented with multiple programming languages and tested across common computing environments. Its running-time summary covers individual components for several collision types at √sNN = 5.02 TeV.

  • Package scope: iEBE-VISHNU provides event-by-event relativistic heavy-ion collision simulations using Fortran, C++, Python, Bash, and SQLite.The package is tested on GNU/Linux and Mac OS X systems and can run on laptops, desktops, or clusters.
  • Scientific scope: The package targets relativistic viscous hydrodynamics, quark-gluon plasma, and Monte-Carlo simulation applications.
  • Runtime characterization: Table 1 reports typical running times in seconds for individual package components across different collision types at √sNN = 5.02 TeV.

1. Introduction

Relativistic heavy-ion collisions create an extremely small, short-lived QGP whose evolving stages require different theoretical descriptions. Because strongly interacting particles retain indirect evolution information while electromagnetic probes interact weakly and can be produced early, simulations connect the evolving fireball to measurable observables.

  • 1. Introduction: RHIC and LHC collisions create a QGP in systems with volume approximately 10^-42 m3 and lifetime approximately 5×10^-23 s.These scales make external probing of QGP properties impossible.
  • 1. Introduction: The collision evolution comprises multiple stages governed by different underlying physics, including early gluon dynamics and later hadronic evolution.The fireball evolves from saturated small-x gluons through expansion, cooling, hadronization, and freeze-out.
  • 1. Introduction: As the fireball expands and cools, decreasing hadronic collision rates lead first to chemical freeze-out and later to kinetic freeze-out.After kinetic freeze-out, particles free-stream to detectors.
  • 1. Introduction: Photons and dileptons have mean free paths much longer than the system size and therefore undergo negligible final-state interactions after production.Their weak electromagnetic interaction makes them penetrating probes of the collision medium.
  • 1. Introduction: Theoretical models and corresponding codes are assigned to the different fireball stages, as illustrated in Figure 1.

2. General Framework

The iEBE-VISHNU framework links stage-specific simulation codes into a hybrid event-by-event workflow. It supports sequential event processing, parallel multi-job execution, database aggregation, and downstream observable analysis.

  • 2. General Framework: A Python shell script links superMC, VISHNew, iSS, and UrQMD into a large-scale event-by-event simulation framework.These components generate initial conditions, evolve viscous hydrodynamics, sample particles, and simulate hadronic cascades.
  • 2.2. Large scale event-by-event simulations: For N computing cores, Nev events are divided into N parallel jobs with M = Nev/N events per job, while events within each job run sequentially.
  • 2.1. Work flow for a single sequential simulation: Each single job begins with M fluctuating initial conditions from superMC, followed by VISHNew evolution, iSS particle sampling, and UrQMD rescattering.The cascade continues until particles stop interacting and unstable particles decay.
  • 2. General Framework: The hybrid VISHNU algorithm combines (2+1)-dimensional hydrodynamic evolution for the QGP stage with a microscopic hadronic cascade.
  • 2.2. Large scale event-by-event simulations: Multiple-job scripts configure and submit simulations locally, through qsub, or with adaptations for other supercomputing facilities and the Open Science Grid.After completion, job databases are combined for later analysis.
  • 2.2. Large scale event-by-event simulations: The UHG Python utility queries the combined database, computes experimental observables, and performs statistical analyses.

3. Initial condition generator SuperMC

SuperMC generates fluctuating nuclear initial conditions with MC-Glauber or MCKLN models, samples finite-size nucleons, and deposits energy or entropy through wounded nucleons and binary collisions. Collision-by-collision multiplicity fluctuations affect p+Pb and Pb+Pb multiplicity distributions and increase initial eccentricities.

  • Collision geometry: SuperMC generates fluctuating initial conditions using MC-Glauber or MCKLN models, with nucleon positions sampled by a Monte Carlo procedure.The nucleons follow Woods-Saxon distributions, with finite nucleon size included in the density modeling.
  • Collision geometry: Finite nucleon size requires modified Woods-Saxon sampling so the folded nuclear density reproduces the experimentally measured Woods-Saxon distribution.The package supports cylindrical and Gaussian nucleon density parameterizations.
  • Energy deposition: Binary collisions deposit energy or entropy around pairwise collision points, while wounded nucleons deposit it either around their centers or around their binary-collision partners.Gaussian profiles are used for Gaussian nucleons, with the collision point defined from the participating nucleon positions.
  • Energy deposition: The partner-based wounded-nucleon deposition scheme increases initial eccentricity at large impact parameters, while its difference from center-based deposition is negligible in central collisions.The partner-based scheme distributes density over a more compact transverse area.
  • Multiplicity fluctuations: 20-40%: collision-by-collision multiplicity fluctuations increase eccentricity coefficients across harmonic orders, with larger increases at higher order n and effects at all centralities.The same fluctuations also increase the variance of εn.

4. (2+1)-d viscous hydrodynamics VISHNew

VISHNew is a (2+1)-dimensional boost-invariant viscous hydrodynamics module that solves second-order viscous equations with lattice-based equations of state. The package combines numerical inversion and regulation procedures with validation tests showing good agreement with analytic or semi-analytic solutions, while parameter studies delimit stable and physically reliable regulation choices.

  • 4.1. Solving the hydrodynamic equations: VISHNew improves the VISH2+1 algorithm for efficiency and stability and solves second-order Israel-Stewart viscous hydrodynamic equations.The equations are coupled to an equation of state; supported choices include s95p-v0-PCE, s95p-v1, and s95p-v1-PCE.
  • 4.1. Solving the hydrodynamic equations: The hydrodynamic solver reconstructs local energy density and fluid velocity from evolved energy-stress components before obtaining pressure from the equation of state.In (2+1) dimensions it uses the vector M^μ = (T^ττ − π^ττ, T^τx − π^τx, T^τy − π^τy) for this inversion.
  • 4.1. Solving the hydrodynamic equations: Newton’s root-finding method efficiently obtains a positive energy-density solution after boundary conditions and bulk-pressure regulation ensure a valid solution.The implementation exploits the approximate proportionality between pressure and energy density for numerical stability and efficiency.
  • 4.2. Numerical tests: The viscous hydrodynamic simulations agree very well with semi-analytical results for local temperature, flow velocity, and shear stress tensor evolution.An ideal-mode test using Gubser’s solution also reports excellent agreement between simulation and analytic evolution.
  • 4.3. Stabilizing: The regulation suppresses all shear-stress components by a common factor only where the validity criterion is violated or nearly violated, while leaving other grid points nearly unchanged.This treatment stabilizes the code against large viscous terms with negligible effects on the physics and negligible extra numerical cost.
  • 4.4. Hydrodynamic evolution with regulation: Final hadronic observables are insensitive to ξ0 from 0.1 to 10, whereas ξ0 = 0.01 over-suppresses viscous effects and ξ0 > 10 causes numerical instability.For ξ0 = 0.01, spectra become steeper and anisotropic flow is damped; the preferred calculation uses ξ0 = 0.1.

5. Cooper-Frye freezeout using iS and particle sampler iSS

This section introduces iS as a fast Cooper–Frye momentum-distribution calculator and iSS as a particle event generator for hydrodynamic outputs. The implementation emphasizes modularity, efficiency, maintainability, and reuse for event-by-event calculations.

  • iS calculates continuous invariant momentum distributions for specified hadron species at user-provided discrete momenta.Its name stands for “iSpectra.”
  • Figure 14 maps inverse Reynolds and Knudsen numbers while marking the kinetic freeze surface at Tdec = 120 MeV.The contour plot uses η/s = 0.20 for 20-30% LHC collisions.
  • iSS uses iS distributions as relative emission probabilities to generate complete events of emitted hadrons.These sampled particles are intended to resemble experimentally generated collision events.
  • The framework is divided into documented functionality modules to support readability, extendability, maintainability, and re-usability.The code uses comments, informative variable names, and carefully chosen modules.
  • Both iS and iSS target intensive event-by-event calculations, with algorithmic optimization used to improve efficiency.The package is designed so that CPU cycles matter in repeated simulations.

5.1. Cooper-Frye freeze-out

The Cooper–Frye freeze-out formulation computes particle emission from a hydrodynamic freeze-out hypersurface. It combines equilibrium and viscous corrections, then integrates the emission function to obtain momentum spectra and anisotropic flow.

  • The Cooper–Frye formula gives the particle emission function from each surface element on the freeze-out hypersurface.Emission depends on the surface element and particle momentum.
  • The emission distribution consists of the local equilibrium term f0 plus the viscous correction δf.The correction represents deviation from local thermal equilibrium caused by viscous effects.
  • For boost-invariant systems, the hypersurface element is expressed using ηs and transverse derivatives of the freeze-out surface.The surface element is specified explicitly in terms of τ, x, y, and ηs.
  • Integrating the emission function over the freeze-out surface produces the particle momentum distribution.The formulation also defines the momentum-dependent viscous correction through χ(p · u/T) = (p · u/T)^α with 1 ≤ α ≤ 2.
  • The code computes azimuthally averaged pT spectra and anisotropic flow coefficients from the resulting momentum distribution.The corresponding quantities are introduced through separate expressions for the spectrum and flow coefficients.
  • Gaussian quadrature and additional loop, variable, and function optimizations make iS approximately 7 times faster than AZSpectra.These optimizations target the numerical integral in the momentum-distribution calculation.

5.2. Methodology for particle sampling

iSS offers numerical and semi-analytic strategies for sampling particle positions and momenta from the Cooper–Frye emission function. The methods trade memory and repeated-sampling efficiency against speed and sensitivity to negative contributions.

  • 5.2. Methodology for particle sampling: iSS treats the Cooper–Frye emission function as the probability for emitting a particle from a fluid cell with specified momentum.The generated positions and momenta are passed to the subsequent microscopic hadron cascade.
  • 5.2.1. Purely numerical approach: The purely numerical approach computes the required quantities numerically and supports two inverse-CDF sampling orderings.It can sample spatial variables first or momentum variables first, with the complementary variables sampled from the Cooper–Frye probability.
  • 5.2.1. Purely numerical approach: The spatial-first method builds an inverse CDF over freeze-out positions after integrating over transverse momentum.The momentum-first method instead constructs the inverse CDF over (p⊥, φp) and samples spatial coordinates with a direct PDF.
  • 5.2.2. Semi-analytic approach: The semi-analytic approach calculates particle yields analytically, builds an inverse CDF for spatial positions, and samples momenta with a direct PDF.The total rapidity density is obtained by summing cell yields, while momentum variables are sampled from the local Cooper–Frye probability.
  • 5.2.2. Semi-analytic approach: Under boost invariance, rapidity and space-time rapidity enter through y − ηs, after which y is sampled uniformly over a user-specified range.The remaining variables include pT, φp, and y − ηs.
  • 5.2.2. Semi-analytic approach: Direct-PDF sampling requires estimating the maximum of the probability distribution, including cases determined by particle statistics and the constraint E ≥ m.The extrema involve a transcendental equation whose solutions can be tabulated; fermion and boson cases differ.
  • 5.2.2. Semi-analytic approach: The semi-analytic treatment assumes f0 < 1 for light mesons, but this can fail in some cells at low freeze-out temperature and large chemical potential.The authors report that violations are uncommon and provide a more rigorous inequality when needed.
  • 5.2.3. Comparison of sampling approaches: The semi-analytic approach is extremely fast for small numbers of repeated samplings, whereas the purely numerical approach suits many repeated samplings of one hydrodynamic event.The latter is useful for rare multi-strange hadrons requiring high statistical precision.

5.3. Code verification

The sampler tests compare generated particle distributions with direct Cooper–Frye results for spatial and momentum observables. The purely numerical method agrees well, while the semi-analytic and momentum-first variants show method-specific deviations.

  • The verification tests compare sampled spatial and momentum distributions for pions, kaons, and protons against Cooper–Frye emission results.The tests include spectra and anisotropy coefficients v2, v3, and v4.
  • Purely numerical method I: The purely numerical method I reproduces the spatial distributions very accurately and agrees very well for spectra and v2, v3, and v4.The comparison uses repeated samplings of a single fluctuating hydrodynamic event.
  • Semi-analytic approach: The semi-analytic method reproduces transverse spatial distributions well but shows slight differences in ηs and τ distributions.Its higher-order momentum anisotropies show noticeable disagreement linked to the method’s removal of negative contributions.
  • Purely numerical method II: The purely numerical method II reproduces momentum distributions very well but yields noticeable spatial deviations from Cooper–Frye results.The spatial deviations are attributed to removing negative contributions.

6. OSCAR to URQMD: osc2u

The osc2u converter transforms particle-list output from iSS into synchronized initial conditions for UrQMD hybrid simulations. It reformats the data and propagates particles to a common computational frame.

  • iSS produces hadron ensembles containing production coordinates and times, momenta, particle identities, masses, and quantum numbers.
  • The ensemble is written in the standard OSCAR1997A particle-list format before conversion for UrQMD.
  • osc2u converts OSCAR1997A output into the initial-condition format required by UrQMD.
  • To satisfy UrQMD’s common-frame requirement, osc2u propagates hadrons backward from their production times to t = 0 and assigns formation times.

7. Hadronic Rescattering: UrQMD

UrQMD models the post-particlization evolution of hadronic matter with microscopic transport, binary collisions, resonance processes, and decays. Its interaction treatment combines experimental inputs with parameterized or model-based cross sections and detailed balance.

  • The model describes an expanding hadron gas initialized in local equilibrium at the switching temperature, without imposing equilibrium afterward.
  • UrQMD evolves hadrons by solving a Boltzmann equation through binary collisions and 2−N-body decays.
  • Binary collisions occur when particles’ minimum center separation in the center-of-mass frame satisfies the collision criterion, using free cross sections for the relevant channel.
  • UrQMD includes 53 baryon species and 24 meson species, together with corresponding antiparticles and isospin-projected states.
  • Cross sections use tabulated or parameterized experimental results when available, and otherwise rely on One-Boson-Exchange or modified additive-quark models.
  • Detailed balance governs resonance absorption, decays, and scattering, with modifications for reactions involving finite-width resonances.

8. Interface for thermal photon emission

The package provides an interface that exports hydrodynamic evolution data and combines it with thermal emission rates to calculate photon observables. Tensor decomposition makes viscous corrections more efficient and extends the workflow to other rare probes.

  • A dedicated iEBE branch supports electromagnetic-probe calculations and supplies the workflow illustrated in Fig. 21.
  • The interface outputs local temperature, flow velocity, and shear-stress histories, storing them in HDF5 binary format to reduce storage demands and improve I/O efficiency.
  • Thermal-photon spectra are obtained by integrating emission over the radiating medium’s space-time volume, including equilibrium and shear-viscous contributions.
  • Tensor decomposition evaluates the viscous correction using laboratory-frame shear stress and local-rest-frame emission factors, reducing repeated Lorentz boosts and rotations.
  • The generic workflow is designed for rare probes coupled to bulk temperature and flow evolution, including jet energy loss and heavy-quark diffusion.

9. Conclusions

The paper documents iEBE-VISHNU’s event-by-event simulation components, including fluctuating initial conditions, viscous hydrodynamics, particle sampling, and numerical tests. Experimental comparisons are outside this document’s scope.

  • The package implements event-by-event numerical simulations of relativistic heavy-ion collisions and adds multiplicity fluctuations to the MC-Glauber model using KNO scaling.
  • The KNO-based multiplicity model reproduces the measured multiplicity distribution for p+Pb collisions at √sNN = 5.02 TeV.
  • VISHNew improves numerical stability for fluctuating initial conditions, studies regulation-parameter sensitivities, and tests against semi-analytical Gubser-flow solutions.
  • iSS converts fluid cells into particles with the Cooper–Frye formula and documents performance tests for three thermal-particle sampling methods.
  • The document does not compare calculated hadron and photon spectra or anisotropy coefficients with experimental data.
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