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T. A. Trainor
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Event-by-event analysis of heavy-ion collision events is an important tool for the study of the QCD phase boundary and formation of a quark-gluon plasma. A universal feature of phase boundaries is the appearance of increased fluctuations of conserved measures as manifested by excess measure variance compared to a reference. In this paper I consider a particular aspect of EbyE analysis emphasizing global-variables variance comparisons and the central limit theorem. I find that the central limit theorem is, in a broader interpretation, a statement about the scale invariance of total variance for a measure distribution, which in turn relates to the scale-dependent symmetry properties of the distribution.. I further generalize this concept to the relationship between the scale dependence of a covariance matrix for all conserved measures defined on a dynamical system and a matrix of covariance integrals defined on two-point measure spaces, which points the way to a detailed description of the symmetry dynamics of a complex measure system. Finally, I relate this generalized description to several recently proposed or completed event-by-event analyses.
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The event-by-event fluctuations of suitably chosen observables in heavy ion collisions at SPS, RHIC and LHC can tell us about the thermodynamic properties of the hadronic system at freeze-out. By studying these fluctuations as a function of varying control parameters, it is possible to learn much about the phase diagram of QCD. As a timely example, we stress the methods by which present experiments at the CERN SPS can locate the second-order critical endpoint of the first-order transition between quark-gluon plasma and hadron matter. Those event-by-event signatures which are characteristic of freeze-out in the vicinity of the critical point will exhibit nonmonotonic dependence on control parameters. We focus on observables constructed from the multiplicity and transverse momenta of charged pions. We first consider how the event-by-event fluctuations of such observables are affected by Bose-Einstein correlations, by resonances which decay after freeze-out and by fluctuations in the transverse flow velocity. We compare our thermodynamic predictions for such noncritical event-by-event fluctuations with NA49 data, finding broad agreement. We then focus on effects due to thermal contact between the observed pions and a heat bath with a given (possibly singular) specific heat, and due to the direct coupling between the critical fluctuations of the sigma field and the observed pions. We also discuss the effect of the pions produced in the decay of sigma particles just above threshold after freeze-out on the inclusive pion spectrum and on multiplicity fluctuations. We estimate the size of these nonmonotonic effects which appear near the critical point, including restrictions imposed by finite size and finite time, and conclude that they should be easily observable.
Fluctuations of thermodynamic quantities are fundamental for the study of the QGP phase transition. The ALICE experiment is well suited for precise event-by-event measurements of various quantities. In this article, we review the capabilities of ALICE to study the fluctuations of several key observables such as the net charge, the temperature, and the particle ratios. Among the observables related to correlations, we review the balance functions and the long range correlations.
A Monte Carlo study of identified particle ratio fluctuations at LHC energies is carried out in the frame work of hij model using the fluctuation variable $ u_{dyn}$. The simulated events for Pb-Pb collisions at $sqrt{s}_{NN}$ = 2.76 and 5.02 TeV and Xe-Xe collisions at $sqrt{s}_{NN}$ = 5.44 TeV are analyzed. From this study, it is observed that the values of $[pi,K]$, $[p,K]$ and $[pi,p]$ follow the similar trends of energy dependence as observed in the most central collision data by NA49, STAR and ALICE experiments. It is also observed that $ u_{dyn}$ for all the three combinations of particles for semi-central and central collisions, the model predicted values of $ u_{dyn}[A,B]$ for Pb-Pb collisions at $sqrt{s}_{NN}$ = 2.76 TeV agree fairly well with those observed in ALICE experiment. For peripheral collisions, however, the model predicted values of $ u_{dyn}[pi,K]$ are somewhat smaller, whereas for $[p,K]$ and $[pi,p]$ it predicts larger values as compared to the corresponding experimental values. The possible reasons for the observed differences are discussed. The $ u_{dyn}$ values scaled with charged particle density when plotted against $langle$N$_{part}$$rangle$, exhibit a flat behaviour, as expected from the independent particle emission sources. For $[p,K]$ and $[pi,p]$ combinations, a departure from the flat trend is, however, observed in central collisions in the case of low p$_{T}$ window when effect of jet quenching or resonances are considered. Furthermore, the study of $ u_{dyn}[A,B]$ dependence on particle density for various collision systems (including proton-proton collisions) suggests that at LHC energies $ u_{dyn}$ values for a given particle pair is simply a function of charged particle density, irrespective of system size, beam energy and collision centrality.
Within a dynamical quark recombination model we explore various proposed event-by-event observables sensitive to the microscopic structure of the QCD-matter created at RHIC energies. Charge fluctuations, charge transfer fluctuations and baryon-strangeness correlations are computed from a sample of central Au+Au events at the highest RHIC energy available ($sqrt{s_{NN}}$=200 GeV). We find that for all explored observables, the calculations yield the values predicted for a quark-gluon plasma only at early times of the evolution, whereas the final state approaches the values expected for a hadronic gas. We argue that the recombination-like hadronization process itself is responsible for the disappearance of the predicted deconfinement signatures. This might explain why no fluctuation signatures for the transition between quark and hadronic matter was ever observed in the experimental data up to now. However, it might also be interpreted as a clear indication for a recombination like hadronization process at RHIC.
Some of the more powerful results of mathematical statistics are becoming of increasing importance in statistical mechanics. Here the use of the central limit theorem in conjunction with the canonical ensemble is shown to lead to an interesting and important new insight into results associated with the canonical ensemble. This theoretical work is illustrated numerically and it is shown how this numerical work can form the basis of an undergraduate laboratory experiment which should help to implant ideas of statistical mechanics in students minds.
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