The effects of a finite system volume on thermodynamic quantities, such as the pressure, energy density, specific heat, speed of sound, conserved charge susceptibilities, and correlations, in hot and dense strongly interacting matter are studied within the parity-doublet chiral mean field model. Such an investigation is motivated by relativistic heavy-ion collisions, which create a blob of hot QCD matter of a finite volume, consisting of strongly interacting hadrons and potentially deconfined quarks and gluons. The effect of the finite volume of the system is incorporated by introducing lower momentum cutoffs in the momentum integrals appearing in the model, the numerical value of the momentum cutoff being related to the de Broglie wavelength of the given particle species. It is found that some of these quantities show a significant volume dependence, in particular, those sensitive to pion degrees of freedom, and the crossover transition is generally observed to become smoother in finite volume. These findings are relevant for the effective equation of state used in fluid dynamical simulations of heavy-ion collisions and efforts to extract the freeze-out properties of heavy-ion collisions with susceptibilities involving electric charge and strangeness.

India

Somenath Pal

Germany

Anton Motornenko

United States

Volodymyr Vovchenko

Abhijit Bhattacharyya

Jan Steinheimer

Horst Stoecker

Effect of finite volume on thermodynamics of quark-hadron matter

Bayesian methods are used to constrain the density dependence of the QCD equation of state (EOS) for dense nuclear matter using the data of mean transverse kinetic energy and elliptic flow of protons from heavy ion collisions (HICs), in the beam energy range <math><mrow><msqrt><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>NN</mi></mrow></msub></mrow></msqrt><mo>=</mo><mn>2</mn><mi>–</mi><mn>10</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math>. The analysis yields tight constraints on the density dependent EOS up to 4 times the nuclear saturation density. The extracted EOS yields good agreement with other observables measured in HIC experiments and constraints from astrophysical observations both of which were not used in the inference. The sensitivity of inference to the choice of observables is also discussed.

Manjunath Omana Kuttan

Kai Zhou

QCD Equation of State of Dense Nuclear Matter from a Bayesian Analysis of Heavy-Ion Collision Data

The centrality determination and the estimated fluctuations of number of participant nucleons  $$N_{part}$$  <math> <msub> <mi>N</mi> <mrow> <mi>part</mi> </mrow> </msub> </math>   in Au–Au collisions at 1.23 AGeV beam kinetic energy suffers from severe model dependencies. Comparing the Glauber Monte Carlo (MC) and UrQMD transport models, it is shown that  $$N_{part}$$  <math> <msub> <mi>N</mi> <mrow> <mi>part</mi> </mrow> </msub> </math>   is a strongly model dependant quantity. In addition, for any given centrality class, Glauber MC and UrQMD predicts drastically different  $$N_{part}$$  <math> <msub> <mi>N</mi> <mrow> <mi>part</mi> </mrow> </msub> </math>   distributions. The impact parameter b and the number of charged particles  $$N_{ch}$$  <math> <msub> <mi>N</mi> <mrow> <mi>ch</mi> </mrow> </msub> </math>   on the other hand are much more correlated and give an almost model independent centrality estimator. It is suggested that the total baryon number balance, from integrated rapidity distributions, can be used instead of  $$N_{part}$$  <math> <msub> <mi>N</mi> <mrow> <mi>part</mi> </mrow> </msub> </math>   in experiments. Preliminary HADES data show significant differences to both, UrQMD simulations and STAR data in this respect.

Marcus Bleicher

Model dependence of the number of participant nucleons and observable consequences in heavy-ion collisions

Molecular dynamics simulations are performed for a finite nonrelativistic system of particles with Lennard-Jones potential. We study the effect of liquid-gas mixed phase on particle number fluctuations in coordinate subspace. A metastable region of the mixed phase, the so-called nucleation region, is analyzed in terms of a noninteracting cluster model. Large fluctuations due to spinodal decomposition are observed. They arise due to the interplay between the size of the acceptance region and that of the liquid phase. These effects are studied with a simple geometric model. The model results for the scaled variance of particle number distribution are compared with those obtained from the direct molecular dynamic simulations.

Ukraine

Volodymyr A. Kuznietsov

Oleh Savchuk

Roman V. Poberezhnyuk

Mark I. Gorenstein

Molecular dynamics analysis of particle number fluctuations in the mixed phase of a first-order phase transition

As an alternative but unified and more fundamental description for quantum physics, Feynman path integrals generalize the classical action principle to a probabilistic perspective, under which the physical observables’ estimation translates into a weighted sum over all possible paths. The underlying difficulty is to tackle the whole path manifold from finite samples that can effectively represent the Feynman propagator dictated probability distribution. Modern generative models in machine learning can handle learning and representing probability distribution with high computational efficiency. In this study, we propose a Fourier-flow generative model to simulate the Feynman propagator and generate paths for quantum systems. As a demonstration, we validate the path generator on the harmonic and anharmonic oscillators. The latter is a double-well system without analytic solutions. To preserve the periodic condition for the system, the Fourier transformation is introduced into the flow model to approach a Matsubara representation. With this novel development, the ground-state wave function and low-lying energy levels are estimated accurately. Our method offers a new avenue to investigate quantum systems with machine learning assisted Feynman path integral solving.

China

Shile Chen

Shiqi Zheng

Baoyi Chen

Lingxiao Wang

Fourier-flow model generating Feynman paths

It is shown that the initial compression in central heavy ion collisions at beam energies of  $$E_{\mathrm {lab}}=1-10A$$  <math> <mrow> <msub> <mi>E</mi> <mi>lab</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>-</mo> <mn>10</mn> <mi>A</mi> </mrow> </math>   GeV depends dominantly on the underlying equation of state and only marginally on the model used for the dynamical description. To do so, a procedure to incorporate any equation of state in the UrQMD transport model is introduced. In particular we compare the baryon density, temperature and pressure evolution as well as produced entropy in a relativistic ideal hydrodynamics approach and the UrQMD transport model, where the same equation of state is used in both approaches. Not only is the compression similar if the same equation of state is used in either dynamical model, but it also strongly depends on the actual equation of state. These results indicate that the equation of state can be studied with observables which are sensitive to the initial compression phase and maximum compression achieved in heavy ion collisions at these beam energies.

Japan

Yasushi Nara

A chiral mean-field equation-of-state in UrQMD: effects on the heavy ion compression stage

We point out that the variance of net-baryon distribution normalized by the Skellam distribution baseline, <math><msub><mrow><mi>κ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>[</mo><mi>B</mi><mo>−</mo><mover><mrow><mi>B</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>]</mo><mo>/</mo><mo>〈</mo><mi>B</mi><mo>+</mo><mover><mrow><mi>B</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>〉</mo></math>, is sensitive to the possible modification of (anti)baryon yields due to <math><mi>B</mi><mover><mrow><mi>B</mi></mrow><mrow><mo>¯</mo></mrow></mover></math> annihilation in the hadronic phase. The corresponding measurements can thus place stringent limits on the magnitude of the <math><mi>B</mi><mover><mrow><mi>B</mi></mrow><mrow><mo>¯</mo></mrow></mover></math> annihilation and its inverse reaction. We perform Monte Carlo simulations of the hadronic phase in Pb-Pb collisions at the LHC via the recently developed subensemble sampler + UrQMD afterburner and show that the effect survives in net-proton fluctuations, which are directly accessible experimentally. The available experimental data of the ALICE Collaboration on net-proton fluctuations disfavors a notable suppression of (anti)baryon yields in <math><mi>B</mi><mover><mrow><mi>B</mi></mrow><mrow><mo>¯</mo></mrow></mover></math> annihilations predicted by the present version of UrQMD if only global baryon conservation is incorporated. On the other hand, the annihilations improve the data description when local baryon conservation is imposed. The two effects can be disentangled by measuring <math><msub><mrow><mi>κ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>[</mo><mi>B</mi><mo>+</mo><mover><mrow><mi>B</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>]</mo><mo>/</mo><mo>〈</mo><mi>B</mi><mo>+</mo><mover><mrow><mi>B</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>〉</mo></math>, which at the LHC is notably suppressed by annihilations but virtually unaffected by baryon number conservation.

Volker Koch

Constraining baryon annihilation in the hadronic phase of heavy-ion collisions via event-by-event fluctuations

The thermal fit to preliminary HADES data of Au+Au collisions at <math><msqrt><mrow><msub><mrow><mi>s</mi></mrow><mrow><msub><mrow></mrow><mrow><mi>N</mi><mi>N</mi></mrow></msub></mrow></msub></mrow></msqrt><mo>=</mo><mn>2.4</mn></math> GeV shows two degenerate solutions at <math><mi>T</mi><mo>≈</mo><mn>50</mn></math> MeV and <math><mi>T</mi><mo>≈</mo><mn>70</mn></math> MeV. The analysis of the same particle yields in a transport simulation of the UrQMD model yields the same features, i.e. two distinct temperatures for the chemical freeze-out. While both solutions yield the same number of hadrons after resonance decays, the feeddown contribution is very different for both cases. This highlights that two systems with different chemical composition can yield the same multiplicities after resonance decays. The nature of these two minima is further investigated by studying the time-dependent particle yields and extracted thermodynamic properties of the UrQMD model. It is confirmed, that the evolution of the high temperature solution resembles cooling and expansion of a hot and dense fireball. The low temperature solution displays an unphysical evolution: heating and compression of matter with a decrease of entropy. These results imply that the thermal model analysis of systems produced in low energy nuclear collisions is ambiguous but can be interpreted by taking also the time evolution and resonance contributions into account.

Reinhard Stock

Ambiguities in the hadro-chemical freeze-out of Au+Au collisions at SIS18 energies and how to resolve them

A novel method for identifying the nature of QCD transitions in heavy-ion collision experiments is introduced. PointNet based Deep Learning (DL) models are developed to classify the equation of state (EoS) that drives the hydrodynamic evolution of the system created in Au-Au collisions at 10 AGeV. The DL models were trained and evaluated in different hypothetical experimental situations. A decreased performance is observed when more realistic experimental effects (acceptance cuts and decreased resolutions) are taken into account. It is shown that the performance can be improved by combining multiple events to make predictions. The PointNet based models trained on the reconstructed tracks of charged particles from the CBM detector simulation discriminate a crossover transition from a first order phase transition with an accuracy of up to 99.8%. The models were subjected to several tests to evaluate the dependence of its performance on the centrality of the collisions and physical parameters of fluid dynamic simulations. The models are shown to work in a broad range of centralities (b=0–7 fm). However, the performance is found to improve for central collisions (b=0–3 fm). There is a drop in the performance when the model parameters lead to reduced duration of the fluid dynamic evolution or when less fraction of the medium undergoes the transition. These effects are due to the limitations of the underlying physics and the DL models are shown to be superior in its discrimination performance in comparison to conventional mean observables.

Manjunath Kuttan

Andreas Redelbach

An equation-of-state-meter for CBM using PointNet

Second-order susceptibilities <math><msubsup><mi>χ</mi><mrow><mi>i</mi><mi>j</mi></mrow><mn>11</mn></msubsup></math> of baryon, electric, and strangeness, <math><mi>B</mi></math>, <math><mi>Q</mi></math>, and <math><mi>S</mi></math>, charges are calculated in the chiral mean field (CMF) model and compared with available lattice QCD data. The susceptibilities are sensitive to the short-range repulsive interactions between different hadron species, especially to the hardcore repulsion of hyperons. Decreasing the hyperons size, as compared with the size of the nonstrange baryons, does improve significantly the agreement of the CMF model results with the lattice QCD data. The electric-charge-dependent susceptibilities are sensitive to the short-range repulsive volume of mesons. The comparison with lattice QCD data suggests that strange baryons, nonstrange mesons, and strange mesons have significantly smaller excluded volumes than nonstrange baryons. The CMF model with these modified hadron volumes allows for a mainly hadronic description of the QCD susceptibilities significantly above the chiral pseudocritical temperature. This improved CMF model, which is based on the lattice QCD data, has been used to study the properties of both cold QCD matter and neutron-star matter. The phase structure in both cases is essentially unchanged, i.e., a chiral first-order phase transition occurs at low temperatures <math><mrow><mo>(</mo><msub><mi>T</mi><mi>CP</mi></msub><mo>≈</mo><mn>17</mn><mspace width="4pt"></mspace><mi>MeV</mi><mo>)</mo></mrow></math>, and hyperons survive deconfinement to higher densities than nonstrange hadrons. The neutron-star maximal mass remains close to <math><mrow><mn>2.1</mn><msub><mi>M</mi><mo>⊙</mo></msub></mrow></math> and the mass-radius diagram is only modified slightly due to the appearance of hyperons and is in agreement with astrophysical observations.