^{*}

^{†}

^{‡}

^{3}.

We exploit the concept of hydrodynamic attractors to establish a macroscopic description of the early time out-of-equilibrium dynamics of high energy heavy-ion collisions. One direct consequence is a general relation between the initial state energy and the produced particle multiplicities measured in experiments. When combined with an

Understanding the equilibration of isolated quantum systems is a fundamental question that touches physical phenomena across vastly different energy scales, from microkelvin temperatures in cold atom experiments to trillion kelvin temperatures in the dense strong-interaction matter produced in ultrarelativistic nuclear collisions

In this Letter, we show that hydrodynamic attractors can be used to describe entropy production in relativistic nuclear collisions and to make robust estimates of initial-state energy before the onset of equilibration. We derive a simple formula, Eq.

We describe the early time dynamics (

Based on these insights, the conservation law in Eq.

Strikingly, the correspondence between initial-state energy density and charged hadron multiplicity can be quantified further using the theory of hydrodynamic attractors. By factoring out the late time Bjorken scaling from Eq.

Hydrodynamic attractor for preequilibrium evolution of the energy density obtained from QCD and Yang Mills (YM) kinetic theory

Based on Eq.

Equations

One important phenomenological consequence of the entropy production in the preequilibrium phase concerns the determination of initial conditions for hydrodynamic simulations of heavy ion collisions (see, e.g.,

Now, in order to illustrate the impact of the preequilibrium phase, we will study the effects on the centrality dependence of the charged particle multiplicity within a simple initial state model based on the color-glass condensate effective theory of high-energy QCD

Since the saturation scale locally characterizes the longitudinally integrated density of color charge inside the nucleus, it is generically proportional to the nuclear thickness

The effect of preequilibrium dynamics on the centrality dependence of the charged-particle multiplicity,

Since

Even though Eq.

So far, we illustrated the utility of Eq.

By inverting Eqs.

We emphasize that, unlike the usual Bjorken estimate based on the measured final-state energy

Estimate of the initial energy per unit space time rapidity

Based on this analysis, we find that, especially in central collisions at high energies, the initial state

Entropy production in high-energy heavy-ion collisions occurs predominantly during the earliest stages, when the system is significantly out of equilibrium; therefore, measurements of the charged particle multiplicities—reflecting the total amount of entropy produced in the collision—provide a highly sensitive probe of the preequilibrium dynamics. Based on the concept of hydrodynamic attractors, which give a macroscopic description of the early time dynamics of the QGP, we established, for the first time, a direct relation between the initial-state energy and the final-state entropy. This relation, Eq.

By combining the information from

We thank A. Andronic, J. Berges, N. Borghini, M. Heller, C. Klein-Bösing, A. Kurkela, T. Lappi, M. Martinez, J. Y. Ollitrault, K. Reygers, C. Schmidt, D. Teaney, and R. Venugopalan for stimulating discussions and P. Romatschke for also providing the AdS/CFT attractor curve. This work was supported in part by the German Research Foundation (DFG) through the Collaborative Research Centres CRC-TR 211: Strong-interaction matter under extreme conditions Project No. 315477589 (S. S.) and SFB 1225: Isolated quantum systems and universality in extreme conditions (ISOQUANT) (A. M.). We thank the Institute for Nuclear Theory at the University of Washington for its kind hospitality and stimulating research environment.

We note that explicit comparisons of dilute-dense and dense-dense calculations in Ref.