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Thermal-FIST: A package for heavy-ion collisions and hadronic equation of state

Volodymyr Vovchenko, Horst Stoecker

arXiv:1901.05249v4nucl-thhep-phnucl-ex

TL;DR

Thermal-FIST addresses the need for a general-purpose HRG toolkit covering particle production and hadronic equation-of-state analysis. It implements interacting and chemically non-equilibrated HRG models with canonical options and applications across thermal analyses, while its event generator has stated approximations.

  • Problem

    General-purpose HRG analysis requires treatment of interactions, chemical non-equilibrium, conserved-charge constraints, decay effects, and fluctuation observables across particle-production and equation-of-state applications.

  • Method

    Thermal-FIST is a modular C++ HRG package implementing pair-dependent excluded-volume and van der Waals interactions, chemical non-equilibrium, canonical analyses, and thermal-model applications.

  • Results

    Thermal-FIST supports published applications spanning chemical freeze-out, equation-of-state studies, canonical calculations, nuclear liquid-gas phenomena, and light-nucleus production.

  • Takeaways & Limitations

    The package provides a unified framework for hadronic equation-of-state analysis and statistical descriptions of hadron production in heavy-ion collisions.

  • Takeaways & Limitations

    The event generator uses the Boltzmann approximation, assumes uncorrelated freeze-out momenta and coordinates, and treats many-body decays only approximately.

Abstract

from arXiv · show

Thermal-FIST (Thermal, Fast and Interactive Statistical Toolkit) is a C++ package designed for a convenient general-purpose physics analysis within the family of hadron resonance gas (HRG) models. This mainly includes the statistical analysis of particle production in heavy-ion collisions and the phenomenology of hadronic equation of state. Notable features include fluctuations and correlations of conserved charges, effects of probabilistic decay, chemical non-equilibrium, and inclusion of van der Waals hadronic interactions. Calculations are possible within the grand canonical ensemble, the canonical ensemble, as well as in mixed-canonical ensembles combining the canonical treatment of certain conserved charges with the grand-canonical treatment of other conserved charges. The package contains a fast thermal event generator, which generates particle yields in accordance with the HRG chemistry, and particle momenta based on the Blast Wave model. A distinct feature of this package is the presence of the graphical user interface frontend -- QtThermalFIST -- which is designed for fast and convenient general-purpose HRG model applications.

1. Introduction

Thermal-FIST extends HRG-based thermal-statistical analysis for particle production and the QCD equation of state. It combines interactions, chemical non-equilibrium, decay effects, and fluctuation observables in one package.

  • Motivation: HRG models describe particle production in heavy-ion collisions and the low-temperature, confined phase of QCD.The ideal HRG model reproduces many lattice QCD observables at T ∼100−150 MeV and zero chemical potential.
  • Extended HRG physics: Thermal-FIST includes excluded-volume and quantum van der Waals interactions, including arbitrary attractive and repulsive parameters between hadron species.These extensions address deviations from the ideal-gas picture and can capture repulsive baryon-baryon effects in fluctuation observables.
  • Extended HRG physics: Thermal-FIST combines chemical non-equilibrium with excluded-volume or van der Waals interactions and exact charge conservation in canonical calculations.The package extends chemical non-equilibrium analyses beyond their previously restricted use with SHARE.
  • Fluctuations: The package includes probabilistic decays and hadronic interactions for analyses of multiplicity fluctuations.These effects are relevant to recent thermal-model applications of fluctuation measurements.
  • Applications: Published applications cover chemical freeze-out, interaction effects, canonical-ensemble calculations, nuclear liquid-gas phenomena, and light-nucleus production.These applications span proton-proton and nucleus-nucleus collisions, equation-of-state studies, and thermal production analyses.

2. Hadron resonance gas

The HRG framework represents hadrons as ideal or interacting components and supports conserved-charge constraints, chemical non-equilibrium, and multiple excluded-volume formulations. Thermal-FIST solves the resulting thermodynamic equations numerically.

  • Ideal HRG: In the ideal HRG grand canonical ensemble, pressure and related thermodynamic quantities are sums of ideal-gas contributions from included hadron species.Species contributions use Fermi, Bose, or Boltzmann statistics with masses, degeneracies, and chemical potentials.
  • Ideal HRG: Conserved charges B, Q, S, and C are controlled on average by independent chemical potentials, with µi = Bi µB + Si µS + Qi µQ + Ci µC.The charge assignments enter each hadron species’ chemical potential.
  • Chemical non-equilibrium: Chemical non-equilibrium is modeled with fugacity parameters γq, γS, and γC regulating light-, strange-, and charm-quark abundances.γi = 1 corresponds to chemical equilibrium for the relevant flavor sector.
  • Excluded-volume corrections: Excluded-volume models represent repulsive interactions by reducing the available volume, with options ranging from species-specific to pair-specific parameters.Thermal-FIST implements these options and solves the resulting pressure equations numerically using Broyden’s method.
  • Non-diagonal EV model: The non-diagonal EV model uses a parameter matrix ˜bij for each particle pair, which may be asymmetric and reduces to the diagonal model when ebij ≡ vi.For hard spheres, the pair parameters can be expressed through the species’ hard-core radii.

2.4. Quantum van der Waals model

The quantum van der Waals HRG model incorporates both repulsive and attractive interactions between hadron species. Thermal-FIST determines its thermodynamics by numerically solving coupled equations for shifted chemical potentials and densities.

  • Model definition: The QvdW-HRG model extends the multi-component HRG with short-range repulsive and intermediate- or long-range attractive interactions.Repulsion is represented by ˜bij, while attraction is modeled in the mean-field approximation through aij.
  • Thermodynamic calculation: Particle densities ni satisfy a linear system, while shifted chemical potentials µ∗i satisfy a coupled system of transcendental equations.The density functions depend on the shifted chemical potentials.
  • Thermodynamic calculation: At fixed T and µ, Thermal-FIST solves the shifted-potential equations numerically and substitutes the solution into the pressure expression.This procedure determines the QvdW-HRG pressure.
  • Physical applications: The classical hard-sphere interpretation of the interaction parameters is not necessarily valid at the nuclear scale.This scope caveat accompanies the hard-sphere parameterization.
  • Model relations: Setting aij ≡ 0 reduces the QvdW-HRG model to the non-diagonal EV-HRG model.The attractive interaction parameters therefore distinguish QvdW-HRG from its purely repulsive counterpart.
  • Physical applications: QvdW-HRG permits nuclear liquid-gas phase-transition and criticality features, which are relevant to fluctuation observables in heavy-ion collisions.The model thereby connects hadronic interactions with nuclear-matter critical behavior.

2.5. Finite resonance widths

Thermal-FIST models finite resonance widths by integrating over resonance-mass distributions, with relativistic and nonrelativistic Breit-Wigner options and configurable integration intervals.

  • Finite resonance widths are incorporated by adding an integration over resonance masses to the HRG expressions.The mass distribution is denoted ρ_i(m).
  • Thermal-FIST implements relativistic and nonrelativistic Breit-Wigner forms for resonance mass distributions.
  • Attractive van der Waals terms must be introduced without double counting interactions that produce resonances.
  • The normalization factor A_i is fixed so that the resonance mass distribution integrates to one.
  • Two integration choices are available: a truncated interval using m_min = m_i + 2Γ_i with energy-independent widths, or a full interval beginning at the threshold mass.The implementation provides both options, with the second defined over the full interval.

2.6. Feeddown from resonance decays

Thermal-FIST computes final hadron yields and their fluctuations by combining primordial production with probabilistic resonance-decay feeddown, while allowing decay treatment to match experimental stability conditions.

  • Resonance decays can provide dominant contributions to measured yields, reaching 70% of all final pions in some cases.Figure 1 illustrates feeddown contributions for π+ and protons in the Id-HRG model at T = 155 MeV and V = 4000 fm3.
  • Final mean multiplicities are calculated as sums of primordial multiplicities and contributions from resonance decays.The decay contribution includes direct decays and chains proceeding through lower-mass resonances.
  • Users can select strong, electromagnetic, weak, or manually defined decay-stability flags when calculating final yields.
  • Conserved-charge susceptibilities provide information about equation-of-state details and can be compared with lattice-QCD calculations or heavy-ion observables.
  • Hadron-number fluctuations are calculated from primordial distributions and decay-chain moments, using derivatives with respect to hadron chemical potentials for primordial cumulants.Higher-order correlations are obtained analogously through mixed derivatives.
  • Probabilistic resonance decays modify variances, correlations, and higher-order fluctuations of final hadron yields.The package evaluates second-order effects generally, while third- and fourth-order final-yield cumulants are treated for the ideal HRG model.

2.8. Canonical ensemble

The canonical ensemble in Thermal-FIST enforces exact conserved-charge values, supports full and selective canonical treatments, and is especially relevant for small systems or rarely produced flavors.

  • Exact charge conservation affects mean hadron yields in small systems and strongly influences hadron-number fluctuations even in the thermodynamic limit.The relevant scale is typically a conserved-charge carrier count of order unity or smaller.
  • Analytic canonical-ensemble calculations in Thermal-FIST are restricted to the ideal HRG model.
  • The full canonical formulation fixes baryon number, electric charge, strangeness, and charm through the canonical partition function and correlation volume.The canonical correlation volume is the region over which exact conservation is enforced.
  • The canonical implementation includes quantum statistics, evaluates partition functions numerically, and supports fluctuations and correlations of particle numbers.
  • Quantum statistics and fluctuations extend functionality beyond many open-source packages, while THERMUS is more restricted in its canonical treatment and statistics.
  • Selective canonical treatment keeps some charges canonical while treating others grand canonically, helping isolate the role of individual charge-conservation effects.
  • Canonical calculations work reliably and quickly when system volumes or conserved charges are sufficiently small.This covers typical cases where exact conservation significantly affects mean multiplicities.
  • Strangeness- or charm-canonical treatments are useful because strange and especially charm hadrons are less abundant than light-flavored hadrons.The charm-canonical setup assumes no multicharmed particles and zero net charm.

2.9. Thermal fits

Thermal fits use HRG calculations of hadron multiplicities to infer thermal parameters at chemical freeze-out, with configurable constraints and chemical-equilibrium assumptions.

  • Thermal fits compare experimental hadron multiplicities with HRG predictions under assumed thermal and partial chemical equilibrium at chemical freeze-out.
  • The fit statistic uses experimental and calculated multiplicities, degrees of freedom, and combined statistical and systematic uncertainties.
  • Thermal-FIST includes resonance feeddown in calculated total yields and uses MINUIT2 for χ2 minimization.
  • In the simplest grand-canonical equilibrium setup, the fitted parameters are T, μB, and V.At each T and μB, μQ and μS are determined to satisfy two conservation laws rather than fitted independently.
  • The package can constrain μB using a fixed entropy-per-baryon ratio, S/B.
  • In the canonical formulation, chemical potentials are replaced by fixed total baryon, electric-charge, strangeness, and charm numbers.
  • Chemical under- or over-saturation of light, strange, or charm quarks adds γq, γS, and γC as possible fit parameters.

3. Monte Carlo event generator

Thermal-FIST’s Thermal Event Generator samples hadronic multiplicities, momenta, and optional resonance decays for HRG models, including grand-canonical and canonical formulations. It supports blast-wave flow and Monte Carlo treatment of observables that are difficult to calculate analytically, but uses simplifying approximations.

  • Generator capabilities: The Thermal Event Generator samples hadronic microstates for HRG models in grand-canonical or canonical ensembles and includes blast-wave momenta and probabilistic resonance decays.It is intended for heavy-ion applications and observables involving radial flow, momentum cuts, higher-order fluctuations, correlations, and residual interactions.
  • Event construction: Each event samples primordial multiplicities from the selected HRG partition function, generates particle momenta, and optionally decays resonances until stable hadrons remain.Momenta follow spherical or longitudinal blast-wave distributions, with kinetic and chemical freeze-out temperatures allowed to differ.
  • Limitations: The event generator assumes Boltzmann statistics, uncorrelated freeze-out momenta and coordinates, and approximate isotropic kinematics for many-body decays.Two- and three-body decays are treated isotropically, whereas four-or-more-body decays are only approximate.
  • Multiplicity sampling: Canonical sampling enforces exact global charge conservation by combining multi-Poisson sampling with rejection sampling and a multistep acceleration procedure.For interacting models, eigenvolume restrictions and QvdW weights are incorporated through rejection or importance sampling.
  • Momentum generation: The spherical blast-wave option uses isotropic angles and rejection sampling for transformed momentum magnitudes, while the cylindrical option samples azimuth, transverse momentum, and rapidity.The spherical scenario is described as appropriate for intermediate energies and the cylindrical scenario as more appropriate for high energies.

4. Thermal-FIST structure and implementation

Thermal-FIST is organized as a C++ library whose modules represent thermodynamic functions, particle data, HRG models, and thermal fits. A Qt5 frontend exposes general-purpose HRG calculations, while the implementation provides ideal-gas functions, configurable numerical methods, feeddown, and model constraints.

  • Thermodynamic functions: Thermal-FIST provides ideal-gas thermodynamic functions including pressure, densities, entropy, scalar density, and the leading four particle-number susceptibilities.Maxwell-Boltzmann calculations use analytical Bessel-function expressions, while quantum-gas calculations use cluster expansions or numerical quadratures.
  • Numerical methods: Quantum-gas calculations support cluster expansion and numerical integration, but cluster expansion must not be used for Fermi-Dirac functions when µ > m because it formally diverges.For µ > m, the implementation rewrites the Fermi-Dirac integrals and evaluates them with 32-point Gauss-Legendre and Gauss-Laguerre quadratures.
  • Particle representation: ThermalParticle stores each species’ PDG code, mass, degeneracy, statistics, quantum numbers, quark content, decay channels, and width, while ThermalParticleSystem manages the particle list.Particle lists are usually loaded from external tabular files.
  • HRG model classes: ThermalModelBase holds shared particle-system and parameter structures, and derived model classes implement HRG variants with chemical-potential constraints, primordial densities, feeddown, and density retrieval.Constraints can impose fixed S/B and Q/B ratios or zero strangeness and charm.
  • Graphical frontend: QtThermalFIST is a cross-platform Qt5 GUI frontend for convenient general-purpose HRG model calculations.The GUI can analyze yields, fluctuations, and equation-of-state properties across grand-canonical and canonical formulations and several HRG variants.

5. Installation

Thermal-FIST can be obtained from GitHub and built cross-platform with CMake. The documented workflow creates a build directory, compiles the library and applications, and runs the GUI or example calculations.

  • Obtaining and configuring: The source code is available from the Thermal-FIST GitHub repository, and the package is described as platform-independent.CMake is the preferred configuration method.
  • Building: The documented Linux workflow clones the repository, creates a build directory, runs CMake, and builds with make.The resulting artifacts are placed under the build directory.
  • Running applications: The QtThermalFIST executable can be launched from the build tree, and sample macros run the paper’s test calculations.Example programs cover HRG temperature dependence, chi-square fits, chemical non-equilibrium, and Monte Carlo HRG calculations.
  • Automation: Automated analyses can be performed with C++ macros that link against the Thermal-FIST library.Sample macros in src/examples are provided as templates.

6. Test results

The tests compare Thermal-FIST with THERMUS-2.3 across thermodynamic calculations, thermal fits, chemical non-equilibrium, and fluctuation observables. The results show quantitative agreement where both codes implement the same models, while also demonstrating the effects of non-ideal interactions, probabilistic decays, and chemical non-equilibrium.

  • Thermodynamic properties: Thermal-FIST and THERMUS-2.3 give quantitatively consistent Id-HRG and EV-HRG results for scaled pressure and energy density.The comparison uses the THERMUS-2.3 hadron list, quantum statistics, and energy-independent Breit-Wigner resonance widths in both codes.
  • Thermodynamic properties: EV/vdW effects become visible at higher temperatures, where hadron densities are large.The comparison includes Id-HRG, constant-radius EV-HRG with r = 0.3 fm, and QvdW-HRG with baryonic interactions.
  • Excluded-volume and van der Waals effects: The Id-HRG fit has a single χ2 minimum near T ∼155 MeV, whereas non-ideal models produce two-minimum structures.The high-temperature second minima require careful physical interpretation because lattice QCD suggests the crossover may already be complete at lower temperatures.
  • Chemical non-equilibrium fits: Thermal-FIST reproduces previously published chemical non-equilibrium systematics, including significantly smaller reduced χ2 values than chemical-equilibrium fits.The extracted γq values are γq = 1.6 −1.7, consistent with earlier results, while the authors do not select a preferred physical scenario.
  • Analytic calculations versus Monte Carlo: Monte Carlo thermal-event-generator results agree with analytic calculations, validating the implemented probabilistic decay treatment.The simplified average-decay procedure gives a markedly different Mk/σK 2 result, making full probabilistic decays important for interpreting the corresponding data.
  • Analytic calculations versus Monte Carlo: Thermal-FIST correlator calculations can differ substantially from measurable correlator ratios, affecting interpretation of experimental data in the QCD equation-of-state context.The comparison concerns ratios involving χS 11, χBQ 11, χB 2, χBS 11, and χS 2.

7. Concluding remarks

Thermal-FIST supports analyses of both the hadronic QCD equation of state and statistical hadron production in heavy-ion collisions. Its graphical interface and modular structure support convenient applications and continued extension of HRG model variants.

  • Thermal-FIST analyzes the hadronic part of the QCD equation of state and statistical hadron production in heavy-ion collisions.
  • The graphical user interface is intended to facilitate interpretation of future hadron-yield data from ongoing and future heavy-ion experiments.
  • The package’s modular structure supports continuous development of HRG variants and implementation of new features.
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