Wojciech Florkowski

Description of the RHIC p ⊥ Spectra in a Thermal Model with Expansion

The assumption of simultaneous chemical and thermal freeze-outs of the hadron gas leads to a surprisingly accurate, albeit entirely conventional, explanation of the recently measured RHIC p(perpendicular) spectra. The original thermal spectra are supplied with secondaries from cascade decays of all resonances, and subsequently folded with a suitably parametrized expansion involving longitudinal and transverse flow. The predictions of this thermal approach, with various parametrizations for the expansion, are in a striking quantitative agreement with the data in the whole available range of 0 < or = p(perpendicular) < or = 3.5 GeV.

Description of the RHIC p(perpendicular) spectra in a thermal model with expansion.

The assumption of simultaneous chemical and thermal freeze-outs of the hadron gas leads to a surprisingly accurate, albeit entirely conventional, explanation of the recently measured RHIC p(perpendicular) spectra. The original thermal spectra are supplied with secondaries from cascade decays of all resonances, and subsequently folded with a suitably parametrized expansion involving longitudinal and transverse flow. The predictions of this thermal approach, with various parametrizations for the expansion, are in a striking quantitative agreement with the data in the whole available range of 0 < or = p(perpendicular) < or = 3.5 GeV.

THERMINATOR: THERMal heavy-IoN generATOR☆☆☆

THERMINATOR is a Monte Carlo event generator designed for studying of particle production in relativistic heavy-ion collisions performed at such experimental facilities as the SPS, RHIC, or LHC. The program implements thermal models of particle production with single freeze-out. It performs the following tasks: (1) generation of stable particles and unstable resonances at the chosen freeze-out hypersurface with the local phase-space density of particles given by the statistical distribution factors, (2) subsequent space–time evolution and decays of hadronic resonances in cascades, (3) calculation of the transverse-momentum spectra and numerous other observables related to the space–time evolution. The geometry of the freeze-out hypersurface and the collective velocity of expansion may be chosen from two successful models, the Cracow single-freeze-out model and the Blast-Wave model. All particles from the Particle Data Tables are used. The code is written in the object-oriented c++ language and complies to the standards of the ROOT environment. Program summary Program title:THERMINATOR Catalogue identifier:ADXL_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADXL_v1_0 Program obtainable from: CPC Program Library, Queens University of Belfast, N. Ireland RAM required to execute with typical data:50 Mbytes Number of processors used:1 Computer(s) for which the program has been designed: PC, Pentium III, IV, or Athlon, 512 MB RAM not hardware dependent (any computer with the c++ compiler and the ROOT environment [R. Brun, F. Rademakers, Nucl. Instrum. Methods A 389 (1997) 81, http://root.cern.ch] Operating system(s) for which the program has been designed:Linux: Mandrake 9.0, Debian 3.0, SuSE 9.0, Red Hat FEDORA 3, etc., Windows XP with Cygwin ver. 1.5.13-1 and gcc ver. 3.3.3 (cygwin special)—not system dependent External routines/libraries used: ROOT ver. 4.02.00 Programming language:c++ Size of the package: (324 KB directory 40 KB compressed distribution archive), without the ROOT libraries (see http://root.cern.ch for details on the ROOT [R. Brun, F. Rademakers, Nucl. Instrum. Methods A 389 (1997) 81, http://root.cern.ch] requirements). The output files created by the code need 1.1 GB for each 500 events. Distribution format: tar gzip file Number of lines in distributed program, including test data, etc.: 6534 Number of bytes in ditribution program, including test data, etc.:41 828 Nature of the physical problem: Statistical models have proved to be very useful in the description of soft physics in relativistic heavy-ion collisions [P. Braun-Munzinger, K. Redlich, J. Stachel, 2003, nucl-th/0304013. [2]]. In particular, with a few physical input parameters, such as the temperature, chemical potentials, and velocity of the collective flow, the models reproduce the observed particle abundances [P. Koch, J. Rafelski, South Afr. J. Phys. 9 (1986) 8; J. Cleymans, H. Satz, Z. Phys. C 57 (1993) 135, hep-ph/9207204; J. Sollfrank et al., Z. Phys. C 61 (1994) 659; P. Braun-Munzinger et al., Phys. Lett. B 344 (1995) 43, nucl-th/9410026; P. Braun-Munzinger et al., Phys. Lett. B 365 (1996) 1, nucl-th/9508020; J. Cleymans et al., Z. Phys. C 74 (1997) 319, nucl-th/9603004; F. Becattini, J. Phys. G 23 (1997) 1933, hep-ph/9708248; G.D. Yen, M.I. Gorenstein, Phys. Rev. C 59 (1999) 2788, nucl-th/9808012; P. Braun-Munzinger, I. Heppe, J. Stachel, Phys. Lett. B 465 (1999) 15, nucl-th/9903010; J. Cleymans, K. Redlich, Phys. Rev. C 60 (1999) 054908, nucl-th/9903063; F. Becattini et al., Phys. Rev. C 64 (2001) 024901, hep-ph/0002267; P. Braun-Munzinger et al., Phys. Lett. B 518 (2001) 41, hep-ph/0105229; W. Florkowski, W. Broniowski, M. Michalec, Acta Phys. Polon. B 33 (2002) 761, nucl-th/0106009], the transverse-momentum spectra [W. Broniowski, W. Florkowski, Phys. Rev. Lett. 87 (2001) 272302, nucl-th/0106050], balance functions [W. Florkowski, W. Broniowski, P. Bozek, J. Phys. G 30 (2004) S1321, nucl-th/0403038. [17]; P. Bozek, W. Broniowski, W. Florkowski, Acta Phys. Hung. A 22 (2005) 149, nucl-th/0310062. [18]], or the elliptic flow [W. Broniowski, A. Baran, W. Florkowski, AIP Conf. Proc. 660 (2003) 185, nucl-th/0212053. [19]; W. Florkowski, W. Broniowski, A. Baran, 2004, nucl-th/0412077. [20]] in both non-strange and strange sectors. The key element of the approach is the inclusion of the complete list of hadronic resonances, which at the rather high temperature at freeze-out, ∼165 MeV, contribute very significantly to the observed quantities. Their two- and three-body decays, taken from the tables, proceed in cascades, ultimately producing the stable particles observed in detectors. At the moment there exist several codes to compute the abundances of particles (the publicly available programs for this purpose are SHARE [G. Torrieri et al., 2004, nucl-th/0404083] and THERMUS [S. Wheaton, J. Cleymans, 2004, hep-ph/0407174]), which is a rather simple task, since the abundances are insensitive to the geometry of the fireball and its expansion. On the other hand, the calculation of the transverse-momentum spectra of particles is much more complicated due to the sensitivity to these phenomena. THERMINATOR deals with this problem, offering the full information on the space–time positions and momenta of the produced particles. As a result, the program allows to compute very efficiently the transverse-momentum spectra of identified particles and examine implications of the assumed expansion model. THERMINATOR allows easily for the departure from symmetries typically assumed in other approaches. This opens the possibility to study the dependence of physical quantities on rapidity and the azimuthal angle. The contribution of the resonances to various observables may be traced conveniently, and their role in the statistical approach may be verified. As a Monte Carlo event generator written in the object-oriented c++ language in the ROOT [R. Brun, F. Rademakers, Nucl. Instrum. Methods A 389 (1997) 81, http://root.cern.ch] environment, THERMINATOR can be straightforwardly interfaced to the standard software routinely used in the data analysis for relativistic heavy-ion colliders, such as SPS, RHIC, and, in the future, LHC. In this way the inclusion of experimental acceptance, kinematic cuts, or interfacing with other programs poses no difficulty. Method of solving the problem:THERMINATOR uses the particle data tables [Particle Data Group, K. Hagiwara et al., Phys. Rev. D 66 (2002) 010001] in the universal input form used by the SHARE [G. Torrieri et al., 2004, nucl-th/0404083] package. The user decides for the thermal parameters and the preferred expansion model. The optimum thermal parameters may be taken, e.g., as those obtained with the help of SHARE [G. Torrieri et al., 2004, nucl-th/0404083] or THERMUS [S. Wheaton, J. Cleymans, 2004, hep-ph/0407174]. At the moment there are two different expansion models implemented in the code: the model of Ref. [W. Broniowski, W. Florkowski, Phys. Rev. Lett. 87 (2001) 272302, nucl-th/0106050], based on the so-called Buda–Lund [T. Csorgo, B. Lorstad, Phys. Rev. C 54 (1996) 1390, hep-ph/9509213] parameterization, and the Blast-Wave model [E. Schnedermann, J. Sollfrank, U.W. Heinz, Phys. Rev. C 48 (1993) 2462, nucl-th/9307020; F. Retiere, M.A. Lisa, Phys. Rev. C 70 (2004) 044907, nucl-th/0312024]. The positions and velocities of the particles are randomly generated on the hypersurface according to the statistical (Bose–Einstein of Fermi–Dirac) distribution factors. All particles, stable and unstable, are included. The particles move along classical trajectories from their initial positions, with velocities composed of the thermal motion and the collective expansion of the system. Stable particles just stream freely, while the resonances decay after some (randomly generated) time, which is controlled by the particles lifetime. The decays are two-body or three-body, and their implementation involves simple kinematic formulas. The decays can proceed in cascades, down to the stage where only stable particles are present. All particles have tags indicating their parent. The secondary rescatterings are not considered in this approach. Full history of the event is stored in an output file, allowing for a detailed examination of the space–time evolutions and the calculation of the transverse-momentum spectra. Additional comment: The ongoing analyses of the SPS and the RHIC data as well as the future heavy-ion program at LHC will certainly benefit from THERMINATOR as a tool for generating events in a simple statistical model. The Monte Carlo code written in c++ and using the standard ROOT [R. Brun, F. Rademakers, Nucl. Instrum. Methods A 389 (1997) 81, http://root.cern.ch] environment can be easily adapted to purposes directly linked to experimental data analyses. The space–time tracking capability will allow, in the framework of the statistical approach, to better understand the physics of relativistic heavy-ion collisions. THERMINATOR calculates the particle spectra and other observables related to the space–time evolution of the system. It provides a c++ framework which may be easily developed for detailed analyses of more involved observables such as, e.g., correlation functions or HBT radii. Typical running time: The generation of 500 events from scratch takes about 1 hour 15 minutes on a PC with Athlon-Barthon 2.5 GHz under Red Hat Fedora 3. Each subsequent 500 events take about 1 hour. To store 500 events about 1.1 GB disk storage is needed, depending on the kinematic range. After converting the output to the ROOT TTree format, 900 MB may be freed.

Testing viscous and anisotropic hydrodynamics in an exactly solvable case

We exactly solve the one-dimensional boost-invariant Boltzmann equation in the relaxation time approximation for arbitrary shear viscosity. The results are compared with the predictions of viscous and anisotropic hydrodynamics. Studying different non-equilibrium cases and comparing the exact kinetic-theory results to the second-order viscous hydrodynamics results we find that recent formulations of second-order viscous hydrodynamics agree better with the exact solution than the standard Israel-Stewart approach. Additionally, we find that, given the appropriate connection between the kinetic and anisotropic hydrodynamics relaxation times, anisotropic hydrodynamics provides a very good approximation to the exact relaxation time approximation solution.

Anisotropic Hydrodynamics for Rapidly Expanding Systems

Abstract We exactly solve the relaxation-time approximation Boltzmann equation for a system which is transversely homogeneous and undergoing boost-invariant longitudinal expansion. We compare the resulting exact numerical solution with approximate solutions available in the anisotropic hydrodynamics and second order viscous hydrodynamics frameworks. In all cases studied, we find that the anisotropic hydrodynamics framework is a better approximation to the exact solution than traditional viscous hydrodynamical approaches.

Description of strange particle production in Au+Au collisions of s NN = 130 GeV in a single-freeze-out model

Strange particle ratios and pT-spectra are calculated in a thermal model with single freeze-out, previously used successfully to describe non-strange particle production at RHIC. The model and the recently released data for phi, Lambda, anti-Lambda, and K*(892) are in very satisfactory agreement, showing that the thermal approach can be used to describe the strangeness production at RHIC.

Highly-anisotropic hydrodynamics in 3+1 space-time dimensions

Recently formulated model of highly-anisotropic and strongly dissipative hydrodynamics is used in 3+1 dimensions to study behavior of matter produced in ultra-relativistic heavy-ion collisions. We search for possible effects of the initial high anisotropy of pressure on the final soft-hadronic observables. We find that by appropriate adjustment of the initial energy density and/or the initial pseudorapidity distributions, the effects of the initial anisotropy of pressure may be easily compensated and the final hadronic observables become insensitive to early dynamics. Our results indicate that the early thermalization assumption is not necessary to describe hadronic data, in particular, to reproduce the measured elliptic flow v_2. The complete thermalization of matter (local equilibration) may take place only at the times of about 1-2 fm/c, in agreement with the results of microscopic models.

Update of the Hagedorn mass spectrum

We present an update of the Hagedorn hypothesis of the exponential growth of the number of hadronic resonances with mass. We use the newest available experimental data for the nonstrange mesons and baryons, as well as fill in some missing states according to the observation that the high-lying states form chiral multiplets. The results show, especially for the case of the mesons, that the Hagedorn growth continues with the increasing mass, with the new states lining up along the exponential growth.

Non-boost-invariant motion of dissipative and highly anisotropic fluid

The recently formulated framework of anisotropic and dissipative hydrodynamics (ADHYDRO) is used to describe non-boost-invariant motion of the fluid created at the early stages of heavy-ion collisions. Very strong initial asymmetries of pressure are reduced by the entropy production processes. By the appropriate choice of the form of the entropy source we can describe isotropization times of about 1 fm, which agrees with the common expectations that already at such times the perfect-fluid hydrodynamics may be applied. Our previous results are generalized by including the realistic equation of state as the limit of the isotropization processes.

Geometric relation between centrality and the impact parameter in relativistic heavy-ion collisions

We show, under general assumptions which are well satisfied in relativistic heavy-ion collisions, that the geometric relation of centrality c to the impact parameter b, namely c ~ pi b^2/sigma_inel, holds to a very high accuracy for all but most peripheral collisions. More precisely, if c(N) is the centrality of events with the multiplicity higer than N, then b is the value of the impact parameter for which the average multiplicity is equal to N. The corrections to this geometric formula are of the order (Delta n(b)/ )^2, where Delta n(b) is the width of the multiplicity distribution at a given value of b, hence are very small. In other words, the centrality effectively measures the impact parameter.