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The kaon to pion ratio $K^+/\pi^+$ and the scaled variance $\omega^-$ for fluctuations of negatively charged particles are studied within the statistical hadron resonance gas (HRG) model and the Ultra relativistic Quantum Molecular Dynamics (UrQMD) transport model. The calculations are done for p+p, Be+Be, Ar+Sc, and Pb+Pb collisions at the CERN Super Proton Synchrotron energy range to reveal the system size dependence of hadron production. For the HRG calculations the canonical ensemble is imposed for all conserved charges. In the UrQMD simulations the centrality selection in nucleus-nucleus collisions is done by calculating the forward energy $E_{\rm F}$ deposited in the Projectile Spectator Detector, and the acceptance maps of the NA61/SHINE detectors are used. A comparison of the HRG and UrQMD results with the data of the NA61/SHINE Collaboration is done. To understand a difference of the event-by-event fluctuations in p+p and heavy ion collisions the centrality selection procedure in the sample of all inelastic p+p events is proposed and analyzed within the UrQMD simulations.

We analyze the newest data from the NA61/SHINE collaboration which, in addition to previous results on pions and kaons, include mean multiplicities of $p$, $\Lambda$, and $\phi$-mesons produced in inelastic proton-proton (p+p) interactions at $\sqrt{s_{NN}}=6.3-17.3$~GeV. The canonical ensemble formulation of the ideal hadron resonance gas (HRG) model is used with exact conservation of net baryon number $B=2$, electric charge $Q=2$, and strangeness $S=0$. The chemical freeze-out parameters in p+p interactions are obtained and compared to those in central nucleus-nucleus collisions. Several features of p+p interactions at the CERN SPS within a statistical model are studied: 1) the inclusion of the $\phi$-meson yields in thermal fits worsens significantly the fit quality; 2) the data show large event-by-event multiplicity fluctuations in inelastic p+p interactions which can not be explained by a single fireball described by a statistical model; 3) the fits within the canonical ensemble formulation of HRG do not give any improvement over the fits within the grand canonical ensemble formulation, i.e., there are no indications for existence of a single statistical system in p+p inelastic interactions in the considered energy range., Comment: 5 figures, 4 tables

Detailed information on decay channel probabilities is absent for many high mass resonances, which are typically included in thermal models. In these cases, the sum over all known decay branching probabilities is smaller than 1. Due to this systematic uncertainty of the model, the exact charge conservation may appear to be violated. We estimate the corresponding number of missing charge states in the canonical ensemble formulation of the hadron resonance gas for p+p reactions at the SPS energy $E_{\rm lab}=158$ GeV: $\Delta B \simeq 0.16$ for baryon charge, $\Delta Q\simeq 0.12$ for electric charge, and $\Delta S=-0.01$ for strangeness. The value of the considered effect is 5-8%, which seems to be important enough to include it as a systematic error in the calculations within a hadron gas., Comment: Proceedings of YSTAR2016 conference. Excited Hyperons in QCD Thermodynamics at Freeze-Out. November 16-17, 2016. Thomas Jefferson National Accelerator Facility, Newport News, VA

We show that the new data on mean multiplicities measured in p+p and A+A collisions together with the updated list of resonances lead to the significant changes of the obtained freeze-out lines. The new A+A line gives much smaller temperatures at high collision energies and agrees with the values obtained at the LHC. The newly obtained p+p line is much closer to the A+A line than previously expected, and even touches it in the region where the $K^+/\pi^+$ horn appears in the data. It indicates that the temperatures that will be obtained in the beam energy and system size scan by the NA61/SHINE Collaboration might be very close. However, our analysis shows that the chemical potentials could be very different for the same energies in A+A and p+p. It adds more puzzles to the set of surprising coincidences at the energies close to the possible onset of deconfinement., Comment: Contribution to the conference Strangeness in Quark Matter 2016 (SQM2016)

New results of the NA61/SHINE Collaboration at the CERN SPS on mean hadron multiplicities in proton-proton (p+p) interactions are analyzed within the transport models and the hadron resonance gas (HRG) statistical model. The chemical freeze-out parameters in p+p interactions and central Pb+Pb (or Au+Au) collisions are found and compared with each other in the range of the center of mass energy of the nucleon pair $\sqrt{s_{NN}}=3.2-17.3$ GeV. The canonical ensemble formulation of the HRG model is used to describe mean hadron multiplicities in p+p interactions and the grand canonical ensemble in central Pb+Pb and Au+Au collisions. The chemical freeze-out temperatures in p+p interactions are found to be larger than the corresponding temperatures in central nucleus-nucleus collisions., Comment: Some misprints are corrected in the text, as well as in Fig.1(f)

We illustrate the role of repulsive interactions in a hadron-resonance gas at freeze-out and in a gas of quark-gluon bags. Taking into account non-zero size of particles in hadron gas leads to a significant decrease and shift of the net-baryon density maximum. The transition point from baryon to meson dominated matter depends on the difference between baryon and meson radii. We also show that depending on the properties of the quark-gluon bags one may obtain any type of the phase transition from hadron gas to quark-gluon plasma: the first or second order, as well as four types of the crossover., Comment: Talk at the 32th Max-Born-Symposium and HECOLS workshop on "Three Days of Phase Transitions in Compact Stars, Heavy-Ion Collisions and Supernovae", Institute for Theoretical Physics, University of Wroc\l{}aw, Wroc\l{}aw, Poland, February 17--19, 2014

An influence of the repulsive interactions on matter properties is considered within the excluded volume van der Waals hadron-resonance gas model. Quantitative results are presented for matter at the chemical freeze-out in central nucleus-nucleus collisions at relativistic energies. In particular, it is shown that repulsive interactions connected to non-zero size of created particles lead to a significant decrease of collision energy at which the net-baryon density has a maximum. A position of the transition point from baryon to meson dominated matter depends on the difference between baryon and meson hard-core radiuses.

The recently proposed two families of strongly intensive measures of fluctuations and correlations are studied within Hadron-String-Dynamics (HSD) transport approach to nucleus-nucleus collisions. We consider the measures $\Delta^{K\pi}$ and $\Sigma^{K\pi}$ for kaon and pion multiplicities in Au+Au collisions in a wide range of collision energies and centralities. These strongly intensive measures appear to cancel the participant number fluctuations. This allows to enlarge the centrality window in the analysis of event-by-event fluctuations up to at least of 10% most central collisions. We also present a comparison of the HSD results with the data of NA49 and STAR collaborations. The HSD describes $\Sigma^{K\pi}$ reasonably well. However, the HSD results depend monotonously on collision energy and do not reproduce the bump-deep structure of $\Delta^{K\pi}$ observed from the NA49 data in the region of the center of mass energy of nucleon pair $\sqrt{s_{NN}}= 8\div 12$ GeV. This fact deserves further studies. The origin of this `structure' is not connected with simple geometrical or limited acceptance effects, as these effects are taken into account in the HSD simulations.

Single particle momentum spectra are calculated within three micro-canonical statistical ensembles, namely, with conserved system energy, system momentum, as well as system energy and momentum. Deviations from the exponential spectrum of the grand canonical ensemble are quantified and discussed. For mean particle multiplicity and temperature, typical for p+p interactions at the LHC energies, the effect of the conservation laws extends to transverse momenta as low as about 3 GeV/c. The results may help to interpret spectra measured in nuclear collisions at high energies, in particular, their system size dependence., Comment: 11 pages, 3 figures

The statistical system of pions with zero total isospin is studied. The suppression effects for the average yields due to isospin conservation are the same for $\pi^0$, $\pi^+$ and $\pi^-$. However, a behavior of the corresponding particle number fluctuations are different. For neutral pions there is the enhancement of the fluctuations, whereas for charged pions the isospin conservation suppresses fluctuations. The correlations between the numbers of charged and neutral pions are observed for finite systems. This causes a maximum of the total pion number fluctuations for small systems. The thermodynamic limit values for the scaled variances of neutral and charged pions are calculated. The enhancements of the fluctuations due to Bose statistics are found and discussed.

The modified versions of the bag model equation of state (EoS) are considered. They are constructed to satisfy the main qualitative features observed for the quark-gluon plasma EoS in the lattice QCD calculations. A quantitative comparison with the lattice results at high temperatures T are done in the SU(3) gluodynamics and in the full QCD with dynamical quarks. Our analysis advocates a negative value of the bag constant B., Comment: We have added the fit for entropy density, pressure over energy density and speed of sound. The recent lattice results from Wuppertal-Budapest collaboration for QCD with quarks are fitted

The semi-inclusive properties of the system of neutral and charged particles with net charge equal to zero are considered in the grand canonical, canonical and micro-canonical ensembles as well as in micro-canonical ensemble with scaling volume fluctuations. Distributions of neutral particle multiplicity and charged particle momentum are calculated as a function of the number of charged particles. Different statistical ensembles lead to qualitatively different dependencies. They are being compared with the corresponding experimental data on multi-hadron production in $p+p$ interactions at high energies., Comment: Two subsections are added: "Average multiplicities, fluctuations and correlations" and "Quantum statistics"

Volume fluctuations are introduced in a statistical modelling of relativistic particle collisions. The micro-canonical ensemble is used, and the volume fluctuations are assumed to have the specific scaling properties. This leads to the KNO scaling of the particle multiplicity distributions as measured in p+p interactions. A striking prediction of the model is a power law form of the single particle momentum spectrum at high momenta. Moreover, the mean multiplicity of heavy particles also decreases as a function of the particle mass according to a power law. Finally, it is shown that the dependence of the momentum spectrum on the particle mass and momentum reduces to the dependence on the particle energy. These results resemble the properties of particle production in collisions of high energy particles.

We consider the Bose-Einstein condensation (BEC) in a relativistic pion gas. The thermodynamic limit when the system volume $V$ goes to infinity as well as the role of finite size effects are studied. At $V\to \infty$ the scaled variance for particle number fluctuations, $\omega=< \Delta N^2>/< N>$, converges to finite values in the normal phase above the BEC temperature, $T>T_C$. It diverges as $\omega \propto V^{1/3}$ at the BEC line $T=T_C$, and $\omega \propto V$ at $T

Multiplicity distributions of hadrons produced in central nucleus-nucleus collisions are studied within the hadron-resonance gas model in the large volume limit. In the canonical ensemble conservation of three charges (baryon number, electric charge, and strangeness) is enforced. In addition, in the micro-canonical ensemble energy conservation is included. An analytical method is used to account for resonance decays. Multiplicity distributions and scaled variances for negatively charged hadrons are presented along the chemical freeze-out line of central Pb+Pb (Au+Au) collisions from SIS to LHC energies. Predictions obtained within different statistical ensembles are compared with preliminary NA49 experimental results on central Pb+Pb collisions in the SPS energy range. The measured fluctuations are significantly narrower than a Poisson reference distribution, and clearly favor expectations for the micro-canonical ensemble., Comment: 6 pages, 3 figures

The aim of this paper is to introduce a new technique for calculation of observables, in particular multiplicity distributions, in various statistical ensembles at finite volume. The method is based on Fourier analysis of the grand canonical partition function. Taylor expansion of the generating function is used to separate contributions to the partition function in their power in volume. We employ Laplace's asymptotic expansion to show that any equilibrium distribution of multiplicity, charge, energy, etc. tends to a multivariate normal distribution in the thermodynamic limit. Gram-Charlier expansion allows additionally for calculation of finite volume corrections. Analytical formulas are presented for inclusion of resonance decay and finite acceptance effects directly into the system partition function. This paper consolidates and extends previously published results of current investigation into properties of statistical ensembles., Comment: 53 pages, 7 figures

The multiplicity distributions of hadrons produced in central nucleus-nucleus collisions are studied within the hadron-resonance gas model in the large volume limit. The microscopic correlator method is used to enforce conservation of three charges - baryon number, electric charge, and strangeness - in the canonical ensemble. In addition, in the micro-canonical ensemble energy conservation is included. An analytical method is used to account for resonance decays. The multiplicity distributions and the scaled variances for negatively, positively, and all charged hadrons are calculated along the chemical freeze-out line of central Pb+Pb (Au+Au) collisions from SIS to LHC energies. Predictions obtained within different statistical ensembles are compared with the preliminary NA49 experimental results on central Pb+Pb collisions in the SPS energy range. The measured fluctuations are significantly narrower than the Poisson ones and clearly favor expectations for the micro-canonical ensemble. Thus this is a first observation of the recently predicted suppression of the multiplicity fluctuations in relativistic gases in the thermodynamical limit due to conservation laws., Comment: 27 pages, 9 figures, corrected references

We present microcanonical ensemble calculations of particle number fluctuations in the ideal pion gas approaching Bose-Einstein condensation. In the samples of events with a fixed number of all pions, $N_{\pi}$, one may observe a prominent signal. When $N_{\pi}$ increases the scaled variances for particle number fluctuations of both neutral and charged pions increase dramatically in the vicinity of the Bose-Einstein condensation line. As an example, the estimates are presented for $p+p$ collisions at the beam energy of 70 GeV., Comment: 4 pages, 2 figures

The charged hadron multiplicity fluctuations are considered in the canonical ensemble. The microscopic correlator method is extended to include three conserved charges: baryon number, electric charge and strangeness. The analytical formulae are presented that allow to include resonance decay contributions to correlations and fluctuations. We make the predictions for the scaled variances of negative, positive and all charged hadrons in the most central Pb+Pb (Au+Au) collisions for different collision energies from SIS and AGS to SPS and RHIC., Comment: 19 pages, 4 figures

Particle multiplicities and ratios in the microcanonical ensemble of relativistic gases near production thresholds are studied. It is shown that the ratio of heavy to light particle multiplicity may be enhanced in comparison to its thermodynamic limit., Comment: 30 pages, 13 figures

Particle number fluctuations are studied in relativistic Bose and Fermi gases. The calculations are done within both the grand canonical and canonical ensemble. The fluctuations in the canonical ensemble are found to be different from those in the grand canonical one. Effects of quantum statistics increase in the grand canonical ensemble for large chemical potential. This is, however, not the case in the canonical ensemble. In the limit of large charge density a strongest difference between the grand canonical and canonical ensemble results is observed., Comment: 13 pages, 6 figures

A new method is proposed for a treatment of ideal quantum gases in the microcanonical ensemble near the thermodynamic limit. The method allows rigorous asymptotic calculations of the average number of particles and particle number fluctuations in the microcanonical ensemble. It gives also the finite-volume corrections due to exact energy conservation for the total average number of particles and for higher moments of the particle number distribution in a system approaching the thermodynamic limit. A present consideration confirms our previous findings that the scaled variance for particle number fluctuations in the microcanonical ensemble is different from that in the grand canonical ensemble even in the thermodynamic limit., Comment: ReVTeX4, 12 pages, 2 figures

Even though the first momenta i.e. the ensemble average quantities in canonical ensemble (CE) give the grand canonical (GC) results in large multiplicity limit, the fluctuations involving second moments do not respect this asymptotic behaviour. Instead, the asymptotics are strikingly different, giving a new handle in study of statistical particle number fluctuations in relativistic nuclear reactions. Here we study the analytical large volume asymptotics to general case of multispecies hadron gas carrying fixed baryon number, strangeness and electric charge. By means of Monte Carlo simulations we have also studied the general multiplicity probability distributions taking into account the decay chains of resonance states., Comment: 4 pages, 2 figures. The report of the talk given in Strangeness in Quark Matter 2004, Cape Town. Submitted to J. Phys. G: Nucl. Part. Phys

The particle number and energy fluctuations in the system of charged particles are studied in the canonical ensemble for non-zero net values of the conserved charge. In the thermodynamic limit the fluctuations in the canonical ensemble are different from the fluctuations in the grand canonical one. The system with several species of particles is considered. We calculate the quantum statistics effects which can be taken into account for the canonical ensemble fluctuations in the infinite volume limit. The fluctuations of the particle numbers in the pion-nucleon gas are considered in the canonical ensemble as an example of the system with two conserved charges - baryonic number and electric charge., Comment: 15 pages, 6 figures

Particle number fluctuations are studied in the microcanonical ensemble. For the Boltzmann statistics we deduce exact analytical formulae for the microcanonical partition functions in the case of non-interacting massless neutral particles and charged particles with zero net charge. The particle number fluctuations are calculated and we find that in the microcanonical ensemble they are suppressed in comparison to the fluctuations in the canonical and grand canonical ensembles. This remains valid in the thermodynamic limit too, so that the well-known equivalence of all statistical ensembles refers to average quantities, but does not apply to fluctuations. In the thermodynamic limit we are able to calculate the particle number fluctuations in the system of massive bosons and fermions when the exact conservation laws of both the energy and charge are taken into account., Comment: REVTeX, 17 pages, 9 figures, v3: misprints a corrected

Fluctuations of charged particle number are studied in the canonical ensemble. In the infinite volume limit the fluctuations in the canonical ensemble are different from the fluctuations in the grand canonical one. Thus, the well-known equivalence of both ensembles for the average quantities does not extend for the fluctuations. In view of a possible relevance of the results for the analysis of fluctuations in nuclear collisions at high energies, a role of the limited kinematical acceptance is studied., Comment: 13 pages, 9 figures, LaTeX