We investigate the role of the Knudsen number (Kn) as a scaling parameter governing the emergence of collective behavior in relativistic heavy-ion collisions. Using the Relativistic Boltzmann Transport approach, we explore different initial conditions for both conformal (massless) and non-conformal (massive) systems with a constant specific shear viscosity <italic>&#951;</italic>/ <italic>s</italic>. Observables such as the time evolution of anisotropic flow coefficients collapse onto universal curves for fixed classes of Knudsen number, when using a scaled time variable accounting for the system size and the speed of sound <italic>c<inf>s</inf></italic>. More differential quantities, such as <italic>v<inf>n</inf></italic>( <italic>p<inf>T</inf></italic>/&#10216; <italic>E<inf>T</inf></italic>&#10217;), show a larger sensitivity to <italic>c<inf>s</inf></italic>. We also study events with fluctuating initial profiles from the TRENTo model, simulating collision systems from O-O to Pb-Pb at RHIC and LHC energies. Universal scaling at a given Kn value also holds in these event-by-event simulations, suggesting that the Knudsen number provides a unified criterion for classifying collectivity across different systems, including small systems where thermalisation may not be fully realised.

Italy

Vincenzo Nugara

Germany

Nicolas Borghini

Vincenzo Greco

Salvatore Plumari

Knudsen number and universal behavior of collective flows in conformal and non-conformal systems

We determine the average state and the uncorrelated modes that characterize the event-by-event fluctuations of the initial state in two typical centrality classes of Pb–Pb collisions at 5.02 TeV. We find that modes in a narrow central bin are similar to those in events at fixed vanishing impact parameter, while those in a mid-peripheral centrality class are affected by the impact-parameter variation. We study how each fluctuation mode affects observables both in the initial state and in the final state of the collisions, at the end of a state-of-the-art boost-invariant hybrid evolution with KøMPøST + + iSS + SMASH, and show that implementing a hadronic transport cascade in such a mode-by-mode analysis with reasonable statistical noise is costly but feasible.

Renata Krupczak

United States

Hendrik Roch

Mode-by-mode evolution of Pb–Pb collisions at 5.02 TeV in a hybrid model

We determine the uncorrelated modes that characterize the fluctuations in a semi-realistic model for the initial state of high-energy nuclear collisions, consisting of hot spots whose positions are distributed independently. Varying the number of hot spots, their size, and the weights with which they contribute to the initial state, we find that the parameter that has the largest influence on the relative importance of the fluctuation modes is the source size, with more extended hot spots leading to a more marked predominance of the principal modes.

Alicia Schütte

Statistical analysis of the fluctuations of an initial-state model with independently distributed hot spots

State-of-the-art simulations of high-energy nuclear collisions rely on hybrid setups, involving in particular a pre-equilibrium stage to let the system evolve from a far-from-equilibrium initial condition towards a near-equilibrated state after which fluid dynamics can be applied meaningfully. A known issue is the mismatch between the equation of state in the fluid-dynamical evolution and the effective one in the previous stage, which leads to discontinuities at the interface between the two models. Here we introduce a new matching prescription at this interface, based on the entropy, and we compare it with the standard one relying on local energy conservation. We study the behavior of various quantities at the switching time between the models and investigate a number of final-state hadronic observables. For the latter, we show that they are not modified significantly by the choice of matching prescription, provided an appropriate normalization is chosen for the initial state. In turn, our approach reduces sizeably the ratio of bulk over thermodynamic pressure at the beginning of the fluid-dynamical stage.

Comparing matching prescriptions between pre-equilibrium and hydrodynamic models in high-energy nuclear collisions

We develop a general decomposition of an ensemble of initial density profiles in terms of an average state and a basis of modes that represent the event-by-event fluctuations of the initial state. The basis is determined such that the probability distributions of the amplitudes of different modes are uncorrelated. Based on this decomposition, we quantify the different types and probabilities of event-by-event fluctuations in Glauber and Saturation models and investigate how the various modes affect different characteristics of the initial state. We perform simulations of the dynamical evolution with KøMPøST and music to investigate the impact of the modes on final-state observables and their correlations.

Marc Borrell

Nina Feld

Sören Schlichting

Clemens Werthmann

Statistical analysis of initial-state and final-state response in heavy-ion collisions

To assess how anisotropic transverse flow is created in a system out of equilibrium, we compare several kinetic-theoretical models in the few-rescatterings regime. We compare the flow harmonics     $$v_n$$  <math> <msub> <mi>v</mi> <mi>n</mi> </msub> </math>   from three types of transport simulations, with either     $$2\rightarrow 2$$  <math> <mrow> <mn>2</mn> <mo>→</mo> <mn>2</mn> </mrow> </math>   or     $$2\rightarrow 0$$  <math> <mrow> <mn>2</mn> <mo>→</mo> <mn>0</mn> </mrow> </math>   collision kernels and in the former case allowing the particles to rescatter several times or not, and from analytical calculations neglecting the gain term of the Boltzmann equation. We find that the even flow harmonics are similar in all approaches, while the odd ones differ significantly. This suggests that while even     $$v_n$$  <math> <msub> <mi>v</mi> <mi>n</mi> </msub> </math>   harmonics may to a large extent be due to the anisotropic escape probability of particles, this is not the predominant mechanism underlying the odd     $$v_n$$  <math> <msub> <mi>v</mi> <mi>n</mi> </msub> </math>   coefficients.

Benedikt Bachmann

On differences between even and odd anisotropic-flow harmonics in non-equilibrated systems

We investigate the early time development of the anisotropic transverse flow and spatial eccentricities of a fireball with various particle-based transport approaches using a fixed initial condition. In numerical simulations ranging from the quasi-collisionless case to the hydrodynamic regime, we find that the onset of  $$v_n$$  <math> <msub> <mi>v</mi> <mi>n</mi> </msub> </math>   and of related measures of anisotropic flow can be described with a simple power-law ansatz, with an exponent that depends on the amount of rescatterings in the system. In the few-rescatterings regime we perform semi-analytical calculations, based on a systematic expansion in powers of time and the cross section, which can reproduce the numerical findings.

Early time behavior of spatial and momentum anisotropies in kinetic theory across different Knudsen numbers

We derive a general formula for the early time dependence of a phase space distribution evolving according to the kinetic Boltzmann equation. Assuming that the early evolution of the system created in high-energy nuclear collisions can be described by kinetic theory, we calculate the scaling behaviors for the onset of various characteristics of the transverse dynamics. In particular, we show that the scaling behavior of the anisotropic flow coefficients  $$v_n$$  <math> <msub> <mi>v</mi> <mi>n</mi> </msub> </math>   at early times does not depend on the details of the collision kernel or the system composition, while at the same time it differs from the prediction of fluid dynamics.

Early time behavior of spatial and momentum anisotropies in a kinetic approach to nuclear collisions

We calculate the elliptic flow of bottomonia produced in Pb $$\,+\,$$  <math> <mrow> <mspace width="0.166667em"></mspace> <mo>+</mo> <mspace width="0.166667em"></mspace> </mrow> </math>  Pb collisions at  $$\sqrt{s_{\mathrm{NN}}}=5.02$$  <math> <mrow> <msqrt> <msub> <mi>s</mi> <mi>NN</mi> </msub> </msqrt> <mo>=</mo> <mn>5.02</mn> </mrow> </math>   TeV. We consider temperature-dependent decay widths for the anisotropic escape of various bottomonium states and observe that the transverse momentum dependence of bottomonia elliptic flow provides a tomographic information about the QGP fireball at different stages of its evolution. For the space-time evolution of the fireball, we employ simulation results from the 3 + 1 D quasiparticle anisotropic hydrodynamic model. We find that our results for transverse momentum dependence of bottomonia elliptic flow are in reasonable agreement with experimental results from the ALICE and CMS collaborations.

India

Partha Bhaduri

Saudi Arabia

Mubarak Alqahtani

Amaresh Jaiswal

Michael Strickland

Fireball tomography from bottomonia elliptic flow in relativistic heavy-ion collisions

We investigate the fluctuations of anisotropic transverse flow due to the finite number of scatterings in a two-dimensional system of massless particles. Using a set of initial geometries from a Monte Carlo Glauber model, we study how flow coefficients fluctuate about their mean value at the corresponding eccentricity, for several values of the scattering cross section. We also show how the distributions of the second and third event planes of anisotropic flow about the corresponding participant plane in the initial geometry evolve as a function of the mean number of scatterings in the system.