Traces of non-equilibrium dynamics in relativistic heavy-ion collisions

Yingru Xu,1, ∗ Pierre Moreau,2, 3, 4 Taesoo Song,2, 3, 5 Marlene

Nahrgang,1, 6 Steffen A. Bass,1 and Elena Bratkovskaya3, 4

1Department of Physics, Duke University, Durham, NC 27708, USA2Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe Universität, Frankfurt am Main, Germany

3Institute for Theoretical Physics, Johann Wolfgang Goethe Universität, Frankfurt am Main, Germany4GSI Helmholtzzentrum für Schwerionenforschung GmbH,

Planckstrasse 1, 64291 Darmstadt, Germany5Institut für Theoretische Physik, Universität Gießen,

Heinrich-Buff-Ring 16, 35392 Gießen, Germany6SUBATECH UMR 6457 (IMT Atlantique, Université de Nantes,

IN2P3/CNRS), 4 rue Alfred Kastler, 44307 Nantes, France

The impact of non-equilibrium effects on the dynamics of heavy-ion collisions is investigated bycomparing a non-equilibrium transport approach, the Parton-Hadron-String-Dynamics (PHSD), toa 2D+1 viscous hydrodynamical model, which is based on the assumption of local equilibrium andconservation laws. Starting the hydrodynamical model from the same non-equilibrium initial con-dition as in the PHSD, using an equivalent lQCD Equation-of-State (EoS), the same transportcoefficients, i.e. shear viscosity η and the bulk viscosity ζ in the hydrodynamical model, we comparethe time evolution of the system in terms of energy density, Fourier transformed energy density, spa-tial and momentum eccentricities and ellipticity in order to quantify the traces of non-equilibriumphenomena. In addition, we also investigate the role of initial pre-equilibrium flow on the hydro-dynamical evolution and demonstrate its importance for final state observables. We find that dueto non-equilibrium effects, the event-by-event transport calculations show large fluctuations in thecollective properties, while ensemble averaged observables are close to the hydrodynamical results.

I. INTRODUCTION

Relativistic heavy-ion collisions produce a hot,dense phase of strongly-interacting matter com-monly known as the quark-gluon plasma (QGP)which rapidly expands and freezes into discrete par-ticles [1–7]. Since the QGP is not directly observ-able – only final-state hadrons and electromagneticprobes are detected – present research relies on dy-namical models to establish the connection betweenobservable quantities and the physical properties ofinterest. The output of these dynamical model simu-lations are analogous to experimental measurementsas they provide final state particle distributions fromwhich a wide variety of observables related to thebulk properties of QCD matter can be obtained. Inaddition, dynamical models also provide informationon the full space-time evolution of the QCD medium.This information can be utilized for the study of rareprobes, such as jets, heavy quarks and electromag-netic radiation that are sensitive to the properties ofthe medium either via re-interaction with the QGPconstituents or its production.

There exist a variety of different dynamical mod-

∗ yx59@phy.duke.edu

els, based on different transport concepts that havebeen successful in describing current data measuredat the Relativistic Heavy-Ion Collider at BrookhavenNational Laboratory and at the Large Hadron Col-lider at CERN. These models may differ in their as-sumptions on the formation of the deconfined phaseof QCD matter, the nature of its dynamical evolu-tion, the mechanism of hadron formation and freeze-out and on many other details. A key question thatneeds to be addressed is how sensitive current ob-servables are to these differences and what kind ofstrategy to pursue to ascertain which of these modelfeatures reflect the actual physical nature of the hotand dense QCD system.

In this paper we perform a comparison of twoprominent models for the evolution of bulk QCDmatter. The first one is a non-equilibrium trans-port approach, the Parton-Hadron-String-Dynamics(PHSD) [8–10], and the second one a 2D+1 viscoushydrodynamical model, VISHNew [11, 12] which isbased on the assumption of local equilibrium andconservation laws.

Non-equilibrium effects are considered to bestrongest during the early phase of the heavy-ionreaction and thus may significantly impact the prop-erties of probes with early production times, such asheavy quarks (charm and bottom hadrons), electro-magnetic probes (direct photons and dileptons), and

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jets. Moreover, some bulk observables, such as cor-relation functions and higher-order anisotropy coeffi-cients, might also retain traces of non-equilibrium ef-fects [13–15]. In particular, the impact of the event-by event fluctuations on the collective observableshas been studied by Kodama et al. [14]. Based onthe comparison of the coarse-grained hydrodynam-ical evolution with the PHSD dynamics, they findthat in spite of large fluctuations on event-by-eventbasis in the PHSD, the ensemble averages are closeto the hydrodynamical limit. A similar behavior hasbeen pointed out before within the PHSD study inRef. [8] where a linear correlation of the elliptic flowv2 with the initial spatial eccentricity ε2 has been ob-tained for the model study of an expanding partonicfireball (cf. Fig. 7 in Ref. [8]). Such correlations ofv2 versus ε2 are expected in the ideal hydrodynam-ical case [16]. The large event-by-event fluctuationsof the charge distributions has been addressed alsoin another PHSD study [17].

In the present paper our focus will be on isolatingdifferences in the dynamical evolution of the systemthat can be attributed to non-equilibrium dynam-ics. The groundwork laid in this comparative studywill hopefully lead to the development of new ob-servables that have an enhanced sensitivity to thenon-equilibrium components of the evolution of bulkQCD matter and that will allow us to quantify howfar off equilibrium the system actually evolves.

II. DESCRIPTION OF THE MODELS

A. PHSD transport approach

The Parton-Hadron-String Dynamics (PHSD)transport approach [8–10, 18] is a microscopic co-variant dynamical model for strongly interactingsystems formulated on the basis of Kadanoff-Baymequations [19, 20] for Green’s functions in phase-space representation (in first order gradient expan-sion beyond the quasi-particle approximation). Theapproach consistently describes the full evolutionof a relativistic heavy-ion collision from the initialhard scatterings and string formation through thedynamical deconfinement phase transition to thestrongly-interacting quark-gluon plasma (sQGP) aswell as hadronization and the subsequent interac-tions in the expanding hadronic phase as in theHadron-String-Dynamics (HSD) transport approach[21]. The transport theoretical description of quarksand gluons in the PHSD is based on the Dynami-cal Quasi-Particle Model (DQPM) for partons that

is constructed to reproduce lattice QCD results forthe QGP in thermodynamic equilibrium [18, 22] onthe basis of effective propagators for quarks and glu-ons. The DQPM is thermodynamically consistentand the effective parton propagators incorporate fi-nite masses (scalar mean-fields) for gluons/quarksas well as a finite width that describes the mediumdependent reaction rate. For fixed thermodynamicparameters (T, µq) the partonic width’s Γi(T, µq) fixthe effective two-body interactions that are presentlyimplemented in the PHSD [23]. The PHSD dif-fers from conventional Boltzmann approaches in acouple of essential aspects: i) it incorporates dy-namical quasi-particles due to the finite width ofthe spectral functions (imaginary part of the prop-agators); ii) it involves scalar mean-fields that sub-stantially drive the collective flow in the partonicphase; iii) it is based on a realistic equation of statefrom lattice QCD and thus describes the speed ofsound cs(T ) reliably; iv) the hadronization is de-scribed by the fusion of off-shell partons to off-shellhadronic states (resonances or strings) and does notviolate the second law of thermodynamics; v) allconservation laws (energy-momentum, flavor cur-rents etc.) are fulfilled in the hadronization con-trary to coalescence models; vi) the effective par-tonic cross sections are not given by pQCD butare self-consistently determined within the DQPMand probed by transport coefficients (correlators) inthermodynamic equilibrium. The latter can be cal-culated within the DQPM or can be extracted fromthe PHSD by performing calculations in a finite boxwith periodic boundary conditions (shear- and bulkviscosity, electric conductivity, magnetic susceptibil-ity etc. [24, 25]). Both methods show a good agree-ment.

In the beginning of relativistic heavy-ion colli-sions color-neutral strings (described by the LUNDmodel [26]) are produced in highly energetic scat-terings of nucleons from the impinging nuclei. Thesestrings are dissolved into ’pre-hadrons’ with a forma-tion time of ∼ 0.8 fm/c in the rest frame of the cor-responding string, except for the ’leading hadrons’.Those are the fastest residues of the string ends,which can re-interact (practically instantly) withhadrons with a reduced cross sections in line withquark counting rules. If, however, the local energydensity is larger than the critical value for the phasetransition, which is taken to be ∼ 0.5 GeV/fm3, thepre-hadrons melt into (colored) effective quarks andantiquarks in their self-generated repulsive mean-field as defined by the DQPM [18, 22]. In the DQPMthe quarks, antiquarks and gluons are dressed quasi-particles and have temperature-dependent effective

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masses and widths which have been fitted to lat-tice thermal quantities such as energy density, pres-sure and entropy density. The nonzero width of thequasi-particles implies the off-shellness of partons,which is taken into account in the scattering andpropagation of partons in the QGP on the same foot-ing (i.e. propagators and couplings).

The transition from the partonic to hadronicdegrees-of-freedom (for light quarks/antiquarks) isdescribed by covariant transition rates for the fu-sion of quark-antiquark pairs to mesonic resonancesor three quarks (antiquarks) to baryonic states, i.e.by the dynamical hadronization. Note that due tothe off-shell nature of both partons and hadrons,the hadronization process described above obeys allconservation laws (i.e. four-momentum conservationand flavor current conservation) in each event, aswell as the detailed balance relations and the in-crease in the total entropy S. In the hadronic phasePHSD is equivalent to the hadron-strings dynam-ics (HSD) model [21] that has been employed in thepast from SchwerIonen-Synchrotron (SIS) to SPS en-ergies. On the other hand the PHSD approach hasbeen applied to p+p, p+A and relativistic heavy-ioncollisions from lower SPS to LHC energies and beensuccessful in describing a large number of experimen-tal data including single-particle spectra, collectiveflow as well as electromagnetic probes [9, 10, 27, 28].

B. 2D+1 viscous hydrodynamics

Relativistic hydrodynamical models calculate thespace-time evolution of the QGP medium via theconservation equations

∂µTµν = 0 (1)

for the energy-momentum tensor

Tµν = e uµuν −∆µν(P + Π) + πµν , (2)

provided a set of initial conditions for the fluidflow velocity uµ, energy density e, pressure P ,shear stress tensor πµν , and bulk viscous pressureΠ. For our analysis, we use VISH2+1 [11], whichis an extensively tested implementation of boost-invariant viscous hydrodynamics that has been up-dated to handle fluctuating event-by-event initialconditions [12]. We use the method from Ref. [29]for the calculation of the shear stress tensor πµν .

This particular implementation of viscous hydro-dynamics calculates the time evolution of the viscouscorrections through the second-order Israel-Stewart

equations [30, 31] in the 14-momentum approxima-tion, which yields a set of relaxation-type equa-tions [32]

τΠΠ̇ + Π = −ζθ − δΠΠΠθ + φ1Π2

+ λΠππµνσµν + φ3π

µνπµν , (3a)

τππ̇〈µν〉 + πµν = 2ησµν + 2π〈µα w

ν〉α − δπππµνθ+ φ7π

〈µα π

ν〉α − τπππ〈µα σν〉α

+ λπΠΠσµν + φ6Ππ

µν . (3b)

Here, η and ζ are the shear and bulk viscosities. Forthe remaining transport coefficients, we use analyticresults derived for a gas of classical particles in thelimit of small but finite masses [32].

The hydrodynamical equations of motion must beclosed by an equation of state (EoS), P = P (e).We use a modern QCD EoS based on continuumextrapolated lattice calculations at zero baryon den-sity published by the HotQCD collaboration [33] andblended into a hadron resonance gas EoS in the in-terval 110 ≤ T ≤ 130 MeV using a smooth stepinterpolation function [34]. While not identical, thisEoS is compatible with the one that the DQPMmodel (underlying the PHSD approach) is tuned toreproduce.

In order to start the hydrodynamical calculation,an initial condition needs to be specified. Initialcondition models provide the outcome of the col-lision’s pre-equilibrium evolution at the hydrody-namical thermalization time, at approximately 0.5fm/c. This pre-equilibrium stage is the least under-stood phase of a heavy-ion collision. While somehydrodynamical models explicitly incorporate pre-equilibrium dynamics [35] starting from a full initialstate calculation, others sidestep the uncertainty as-sociated with this early regime by generating para-metric initial conditions directly at the materializa-tion time [36–38].

For our study here, we shall initialize the hydro-dynamical calculation with an initial condition ex-tracted from PHSD that provides us with a commonstarting configuration for both models regarding ourcomparison of the dynamical evolution of the sys-tem.

III. NON-EQUILIBRIUM INITIALCONDITIONS

In this section we describe the construction ofthe initial condition for the hydrodynamical evolu-tion from the non-equilibrium PHSD evolution. One

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should note that PHSD starts its calculation ab ini-tio with two colliding nuclei and makes no equilib-rium assumptions regarding the nature of the hotand dense system during the course of its evolutionfrom initial nuclear overlap to final hadronic freeze-out. For the purpose of our comparison we have toselect the earliest possible time during the PHSDevolution where the system is in a state in which ahydrodynamical evolution is feasible (e.g. the vis-cous corrections are already small enough) and gen-erate an initial condition for the hydrodynamical cal-culation at that time (note that this criterion is lessstringent than assuming full momentum isotropiza-tion or local thermal equilibrium).

A. Evaluation of the energy-momentum tensorTµν in PHSD

The energy-momentum tensor Tµν(x) of an idealfluid (by removing viscous corrections in Eq. 2) isgiven by

Tµν = (e+ P )uµuν − Pgµν (4)

where e is the energy density, P the thermody-namic pressure expressed in the local rest frame(LRF) and the 4-velocity is uµ = γ (1, βx, βy, βz).Here β is the (3-)velocity of the considered fluid el-ement and the associated Lorentz factor is given by

γ = 1/√

1− β2.In order to calculate Tµν in PHSD which fully de-

scribes the medium in every space-time coordinate,the space-time is divided into cells of size ∆x = 1 fm,∆y = 1 fm (which is comparable to the size of ahadron) and ∆z ∝ 0.5 × t/γNN scaled by γNN toaccount for the expansion of the system. We notethat choosing a high resolution has been shown inRef. [27] to lead to very similar results. In each cell,we can obtain Tµν in the computational frame from:

Tµν(x) =∑i

∫ ∞0

d3pi(2π)3

fi(Ei)pµi p

νi

Ei. (5)

where fi(E) is the distribution function correspond-ing to the particle i, pµi the 4-momentum and Ei =p0i is the energy of the particle i. In the case of anideal fluid, if the matter is at rest (uµ = (1, 0, 0, 0))Tµν(x) should only have diagonal components andthe energy density in the cell can be identified tothe T 00 component. However, in heavy-ion collisionsthe matter is viscous, anisotropic and relativistic,thus the different components of the pressure arenot equal and it becomes more difficult to extract

the relevant information. This especially holds truefor the early reaction time at which the initial condi-tions for hydrodynamical model are taken. In orderto obtain the needed quantities (e,β) from Tµν forthe hydrodynamical evolution, we have to expressthem in the local rest frame (LRF) of each cells ofour space-time grid. In the general case, the energy-momentum tensor can always be diagonalized, i.epresented as

Tµν (xν)i = λi (xµ)i = λi g

µν (xν)i, (6)

with i = 0, 1, 2, 3, where its eigenvalues are λi andthe corresponding eigenvectors (xν)i. When i = 0,the local energy density e is identified to the eigen-value of Tµν (Landau matching) and the correspond-ing time-like eigenvector is defined as the 4-velocityuν (multiplying (4) by uν):

Tµνuν = euµ = (egµν)uν (7)

using the normalization condition uµuµ = 1. Inorder to solve this equation, we have to calculatethe determinant of the corresponding matrix whichis the 4th order characteristic polynomial associatedto the eigenvalues λ:

P (λ) =

∣∣∣∣∣∣∣T 00 − λ T 01 T 02 T 03T 10 T 11 + λ T 12 T 13

T 20 T 21 T 22 + λ T 23

T 30 T 31 T 32 T 33 + λ

∣∣∣∣∣∣∣ (8)Having the four solutions for this polynomial, wecan identify the energy density being the larger andpositive solution, and the 3 other solutions are (−Pi)the pressure components expressed in the LRF. Toobtain the 4-velocity of the cell, we use (7) whichgives us this set of equations:

(T 00 − e) + T 01X + T 02Y + T 03Z = 0T 10 + (T 11 + e)X + T 12Y + T 13Z = 0T 20 + T 21X + (T 22 + e)Y + T 23Z = 0T 30 + T 31X + T 32Y + (T 33 + e)Z = 0

(9)

Rearranging these equations, we can obtain thesolutions which are actually for the vector uν =γ (1, X, Y, Z) = γ (1,−βx,−βy,−βz). To obtainthe physical 4-velocity uµ, we have to multiply bygµνuν = u

µ.

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Figure 1. Initial conditions for hydrodynamics: the en-ergy density profiles from a single PHSD event (upperpanel) and averaged over 100 PHSD events (lower panel)taken at t = 0.6 fm/c for a peripheral (b = 6 fm) Au+Aucollision at

√sNN = 200 GeV.

B. PHSD initial conditions for hydrodynamics

By the Landau matching procedure describedabove, we can obtain the initial conditions such as

the local energy density e and initial flow ~β for thehydrodynamical evolution. In the PHSD simulationthe parallel ensemble algorithm is used for the testparticle method, which has an impact on the fluctu-ating initial conditions. For a larger number of par-allel ensembles (NUM), the energy density profile issmoother since it is calculated on mean-field levelby averaging over all ensembles. From a hydrody-namical point of view, gradients should not be toolarge and some smoothing of the initial conditionsis therefore required. Here, we choose NUM= 30,

which provides the same level of smoothing of theinitial energy density as in typical PHSD simula-tions. In fig. 1 we show the initial condition at timeτ = 0.6 fm/c extracted from a single PHSD event av-eraged over (NUM=30) parallel events (upper panel)and averaged over 100 parallel events (lower panel),the color maps represent the local energy densitywhile the arrows shows the initial flow at each of thecells. Even though the initial profiles are averagedover NUM= 30 parallel events, the distribution stillcaptures the feature of event-by-event initial statefluctuations.

In Fig. 2 we investigate the dependence of thePHSD initial conditions on the equilibration time τ0,at which the non-equilibrium evolution is switchedto a hydrodynamical evolution in local thermal equi-librium. As expected, for larger initial times τ0, thelocal initial flow increases and the local energy den-sity decreases.

IV. MEDIUM EVOLUTION:HYDRODYNAMICS VERSUS PHSD

In this section we compare the response of thehydrodynamical long-wavelength evolution to thePHSD initial conditions with the microscopic PHSDevolution itself. In order to avoid as many biases aspossible we apply the temperature-dependent shearviscosity as determined in PHSD simulations [24]and shown in the upper panel of Fig. 3: the blueand red symbols correspond to η/s obtained fromthe Kubo formalism and from the relaxation timeapproximation method, respectively. The black linein Fig. 3 shows the parametrization of the PHSDη/s(T ), which is used in the viscous hydrodynamicsfor the present study.We note that the parametrizedcurve is very similar to the recently determined tem-perature dependence of η/s via Bayesian analysis ofthe available experimental data [39].

While the effect of shear viscosity on the hydro-dynamical evolution has been studied extensivelyfor simulations of heavy-ion collisions, bulk viscos-ity has not been treated as carefully so far. Thisis because at higher temperatures the bulk viscosityshould be very small, and vanish in the conformallimit. Moreover, an enhanced bulk viscosity at thepseudo-critical temperature causes problems for theapplicability for hydrodynamics itself. Studies con-ducted for dynamical quasi-particle models, like theone used in PHSD, show that the magnitude andtemperature behavior of the bulk viscosity dependon details of the parametrization of the equationof state and properties of the underlying degrees-of-

5

−10 −5 0 5 10x [fm]

−10

−5

0

5

10

e [G

eV.fm

−3] τ0 = 0.2 fm/c

−10 −5 0 5 10x [fm]

τ0 = 0.4 fm/c

−10 −5 0 5 10x [fm]

τ0 = 0.6 fm/c

−10 −5 0 5 10x [fm]

τ0 = 0.8 fm/c

−10 −5 0 5 10x [fm]

−10

−5

0

5

10

β x0,

β y0

−10 −5 0 5 10x [fm]

−10 −5 0 5 10x [fm]

−10 −5 0 5 10x [fm]

0 2 4 6 8 10distance [fm]

0.00.10.20.30.4

initi

al fl

ow β

0

0 2 4 6 8 10distance [fm]

0 2 4 6 8 10distance [fm]

0 2 4 6 8 10distance [fm]

10−1

100

101

102

0.060.120.180.240.300.360.420.480.54

Figure 2. (color online) Initial conditions of a peripheral Au+Au collision (b=6fm) at√sNN = 200 GeV (calculated

within the PHSD): upper contour: the local energy density in the transverse plane; middle: quiver plot of initialflow vx, vy in the transverse plane (colormap indicate the magnitude of local initial flow); lower: initial flow withrespect to the distance of the cell from the center of energy density.

freedom [22, 24]. For the relaxation time approxima-tion in quasi-particle models slightly different valuesfor the bulk viscosity are obtained [40, 41]. Giventhese uncertainties for the values of the bulk vis-cosity, we decide to use the bulk viscosity that hasrecently been determined by the Bayesian analysisof experimental data in our hydrodynamical simula-tions [39]. In the low panel of Fig. 3 we compare theratio of bulk viscosity to entropy ζ/s that is adaptedin our hydrodynamical simulations and the one ex-tracted from PHSD simulations. It should be notedthat the maximum ζ/s that hydrodynamical modelcan handle is much smaller than the bulk viscosityfrom PHSD simulations, and its effect on the mo-mentum anisotropy will be discussed at the end ofthis section.

A. Pressure isotropization

In order to justify the choice of initial time τ0 =0.6 fm, we first take a look at the evolution of thedifferent pressure components in PHSD. In the pre-equilibrium stage deviations from thermal equilib-rium are very large. It has been argued that onecan relax this strict requirement and instead applyhydrodynamics once the pressure is isotropic, whichimplies that both transverse and longitudinal pres-sure are about equal.

As mentioned in the previous section, the devi-ations from equilibrium are strongest at the begin-ning of the heavy-ion collision. In this case the vis-cous corrections can have a large contribution to theenergy-momentum tensor, and the pressure compo-nents can differ substantially from the isotropic pres-sure given by the EoS. This situation is illustratedin Fig. 4 which shows the evolution of the trans-

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1 . 0 1 . 5 2 . 0 2 . 5 3 . 0

0 . 1

1

f i t h y d r o

P H S D : k i n e t i c t h e o r y K u b o f o r m a l i s m

h / s K S S = 1 / ( 4 p )

l a t t i c e Q C D

p Q C Dcros

sover

h/s

T / T c

1 . 0 1 . 5 2 . 0 2 . 5 3 . 00 . 0

0 . 5

1 . 0

crosso

ver

l a t t i c e Q C D

P H S D : w i t h m e a n - f i e l d e f f e c t s w / o m e a n - f i e l d e f f e c t s

h y d r o : d e f a u l t 2 ´ 4 ´ 8 ´

z/s

T / T cFigure 3. (Color online) η/s and ζ/s versus scaledtemperature T/TC : Top: The symbols indicate thePHSD results of η/s from Ref. [24], calculated us-ing different methods: the relaxation-time approxi-mation (red line+diamonds) and the Kubo formalism(blue line+dots); the black line corresponds to theparametrization of the PHSD results for η/s. The or-ange short dashed line demonstrates the Kovtun-Son-Starinets bound [42] (η/s)KSS = 1/(4π). For compar-ison, the results from the virial expansion approach(green line) [43] are shown as a function of temperature,too. The orange dashed line is the η/s of VISHNU hy-drodynamical model that has been recently determinedby Bayesian analysis; Bottom: ζ/s from PHSD simula-tion from Ref. [24] and the ζ/s that is adapted in ourhydrodynamical simulations.

verse and longitudinal pressures divided by the localenergy density e in different cells along the x-axisextracted from PHSD as a function of time for aperipheral Au+Au collision at

√sNN = 200 GeV.

These pressure components correspond to the eigen-

Figure 4. (Color online) Evolution of the ratio of thetransverse PT and longitudinal PL pressures over the cellenergy density e extracted from PHSD as a function oftime for different cells along the x-axis in a peripheral(b = 6 fm) Au+Au collision at

√sNN = 200 GeV. Note

that Tµν has been averaged over 100 PHSD events.

values of Tµν(x) where the latter have been aver-aged in this case over 100 PHSD events in order toget a smooth evolution. As seen from Fig. 4, atearly reaction times the deviation between the pres-sure components is large and the longitudinal pres-sure dominates. The transverse pressure starts fromzero but grows with time and approximately reachesthe isotropic pressure within a range of 0.3 to 1.fm/c. On the other hand, the longitudinal pressuredecreases to very low values and remains small forlarge times. One of the reasons for this behavior isthat we took only a few cells on the z-axis whichcorrespond to a pseudorapidty gap ∆η ≈ O(10−2).By taking into account more cells in the longitudi-nal direction, the longitudinal pressure increases butthe collective expansion cannot be removed properlyin this case (as it has already been studied in Sec-tion 7 of Ref. [13]). By looking at more peripheralcells (bottom panel of Fig. 4), we can see that thepressure components deviate more from the isotropicpressure given by the EoS compared to more centralcells (top panel of Fig. 4).

7

Figure 5. Evolution of the relative value between the transverse pressure extracted from PHSD PT (see also Fig.4 for the pressure components) and the pressure P given by the EoS for different times in the transverse plane ofa peripheral (b = 6 fm) Au+Au collision at

√sNN = 200 GeV. Note that T

µν has been averaged over 100 PHSDevents.

We illustrate in Fig. 5 the (non-)equilibrated re-gions in the PHSD simulation. We evaluated therelative value between the transverse pressure ex-tracted from PHSD PT (see Fig. 4 for the pressurecomponents) and the pressure P given by the EoSin the full transverse plane. One can see that thecentral region in grey is rather equilibrated for alltimes ((PT − P )/P is around 0). The peripheralcells have a higher pressure when the initial condi-tion for the hydrodynamical model is taken (t = 0.6fm/c), and then fluctuate around the isotropic pres-sure as depicted by the red and blue colors. We cantherefore conclude that by averaging over the PHSDevents, the medium reaches with time a transversepressure comparable to the isotropic one as given bythe lQCD EoS. This statement is of course not validfor a single PHSD event where the pressure compo-nents show a much more chaotic behavior and wherethe high fluctuations in density and velocity profilesindicate that the medium is in a non-equilibriumstate, as we will see in the next section.

B. Space-time evolution of energy density e

and velocity ~β

Starting with the same initial conditions (as dis-cussed in section III), the evolution of the QGPmedium is now simulated by two different models:the non-equilibrium dynamics model – PHSD, andhydrodynamics – (2+1)-dimensional VISHNU.

Fig. 6 shows the time evolution of the local energydensity e(x, y, z = 0) (from Tµν) (left) and the corre-sponding temperature T (right) as calculated usingthe lQCD EoS in the transverse plane from a single

PHSD event (NUM=30) at different times for a pe-ripheral (b = 6 fm) Au+Au collision at

√sNN = 200

GeV. As seen in figure 1 for t = 0.6 fm/c, the en-ergy density profile is far from being smooth. Notealso that the energy density decreases rapidly as themedium expands in the transverse and longitudinaldirection. By converting the energy density to thetemperature given by the lQCD EoS, we can seethat the variations are less pronounced in that case.Fig. 7 shows the same quantities for a single eventevolved through hydrodynamics. In particular forthe energy density at later times one can already ob-serve a significant smoothing compared to the PHSDevolution.

Fig. 8 shows the time evolution of the local energydensity e(x, y) in the transverse plane from a singlePHSD event (NUM=30) at different proper timesfor a peripheral Au+Au collision at

√sNN = 200

GeV, while Fig. 9 shows the same time evolution ofe(x, y) from a hydrodynamical evolution using thesame initial condition as the PHSD event above. Acomparison of the two medium evolutions shows dis-tinct differences: in PHSD the energy density retainsmany small hot spots during its evolution due to itsspatial non-uniformly. In hydrodynamics, the ini-tial hot spots of energy density quickly dissolve andthe medium becomes much smoother with increas-ing time. Moreover, as a result of the initial spatialanisotropy, the pressure gradient in x-direction islarger than that in y-direction, resulting in a slightlyfaster expansion in x-direction. We attribute thesedifferences directly to the non-equilibrium nature ofthe PHSD evolution.

In Fig. 10 and Fig. 11 we show the time evolu-

tion of the velocity ~β = (βx, βy, βz) in the transverse

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Figure 6. Local energy density e(x, y, z = 0) (left) and the corresponding temperature T (x, y, z = 0) given by theEoS (right) in the transverse plane from a single PHSD event (NUM=30) at different times for a peripheral (b=6fm)Au+Au collision at

√sNN = 200 GeV.

9

Figure 7. Local energy density e(x, y, z = 0) (left) and the corresponding temperature T (x, y, z = 0) given by the EoS(right) in the transverse plane from a single hydrodynamical event at different proper times for a peripheral (b=6fm)Au+Au collision at

√sNN = 200 GeV.

10

−10

0

10y

[fm]

t = 0.6 fm/c

PHSD

t = 1.6 fm/c

PHSD

t = 2.6 fm/c

PHSD

t = 3.6 fm/c

PHSD

−10 0 10x [fm]

−10

0

10

y [fm

]

t = 4.6 fm/c

PHSD−10 0 10

x [fm]

t = 5.6 fm/c

PHSD−10 0 10

x [fm]

t = 6.6 fm/c

PHSD−10 0 10

x [fm]

t = 7.6 fm/c

PHSD 10−1

100

101

Figure 8. (Color online) Contour plots of the local energy density e in the transverse plane from the PHSD simulationof one event, for a peripheral Au+Au collision (b = 6 fm) at

√sNN =200 GeV

−10

0

10

y [fm

]

τ0 = 0.6 fm/c

Hydro

τ = 1.6 fm/c

Hydro

τ = 2.6 fm/c

Hydro

τ = 3.6 fm/c

Hydro

−10 0 10x [fm]

−10

0

10

y [fm

]

τ = 4.6 fm/c

Hydro−10 0 10

x [fm]

τ = 5.6 fm/c

Hydro−10 0 10

x [fm]

τ = 6.6 fm/c

Hydro−10 0 10

x [fm]

τ = 7.06 fm/c

Hydro 10−1

100

101

Figure 9. (Color online) Contour plots of the local energy density e in the transverse plane from the hydrodynamicalsimulation starting from the same initial conditions as in Fig. 8 including initial flow.

plane for the same PHSD initial condition evolvedthrough PHSD and hydrodynamics. The longitudi-nal velocity βz shown in the PHSD event remains onaverage approximately 0 and much smaller than thetransverse flow since we only consider a narrow inter-val in the z-direction. At τ0 = 0.6 fm/c, transverseflow has already developed and the transverse veloc-ity can reach values of 0.5 at the edge of the profile.Even though the velocity increase with time in both

PHSD and hydrodynamical events, it is clearly seenthat the development of flow in a hydrodynamicalevent is much faster than in a PHSD event. In ad-dition, local fluctuations in a single event are morevisible in the PHSD event. Moreover, the velocityin x-direction is slightly larger than the one in y-direction in both events, as a result of the initial spa-tial anisotropy of the energy density, and that spatialanisotropy is converted into momentum anisotropy,

11

Figure 10. (Color online) Components of the 3-velocity (βx, βy, βz) in the transverse plane from a single PHSD event(NUM=30) at different proper time for a peripheral (b=6fm) Au+Au collision at

√sNN = 200 GeV. βz is scaled

differently from βx, βy for better orientation.

Figure 11. (Color online) Components of the velocity (βx, βy) in the transverse plane from a single hydrodynamicalevent (taking the same initial condition as the PHSD event above) at different proper time for a peripheral (b=6fm)Au+Au collision at

√sNN = 200 GeV.

12

which increases with time.

C. Fourier images of energy density

The inhomogeneity of a medium can be quanti-fied by the Fourier transform of the energy density,ẽ(kx, ky). For a discrete spatial grid with an energydistribution as e(x, y)m×n, the Fourier coefficientsare given by

ẽ(kx, ky) =1

m

1

n

m−1∑x=0

n−1∑y=0

e(x, y)e2πi(xkxm +

ykyn ) .

(10)

The zero mode ẽkx=0,ky=0 is the total sum of theenergy density, while higher order coefficients con-tain information about the correlations of the lo-cal energy density on different length scales. Fora medium with large wave-length structures thehigher-order coefficients should be suppressed andthe typical global shape of the event should domi-nate. Given that our simulations in both PHSD andhydrodynamics are performed for the same central-ity classes, we expect these structures to give sim-ilar Fourier coefficients for lower modes. However,if structures are dominated by smaller length scales,the higher Fourier modes are excited as well.

In Figs. 12 and 13 we present the Fourier trans-form ẽ(kx, ky) for a medium evolved by PHSD andhydrodynamics, respectively, for different stages ofthe evolution. For the hydrodynamical evolution ofmedium only the dominant lower Fourier modes sur-vive in the later stages and shorter wavelength irreg-ularities are washed out. The microscopic transportevolution of PHSD generates the same level of shortwavelength phenomena at all times of the evolution;only the overall dilution of the medium reduces thestrength.

This difference can be identified more easily inFig. 14, where we plot the distribution of the Fourier

coefficients 〈ẽ(√

k2x + k2y

)〉 for different evolution

times. For the lower order Fourier modes, which car-ries the information about the global event scale, themicroscopically evolving medium and the hydrody-namical medium are identical. We observe that thestrength of the shorter wavelength modes rapidly de-creases with respect to the zero mode at the begin-ning of the hydrodynamical evolution.

D. Time evolution of the spatial andmomentum anisotropy

Much interest is given to the medium’s response toinitial spatial anisotropies. For the hydrodynamicalmodels the spatial anisotropies lead to substantialcollective flow, measured by Fourier coefficients ofthe azimuthal particle spectra. Initial spatial gra-dients are transformed into momentum anisotropiesvia hydrodynamical pressure. While experimentallyonly the final state particle spectra are known, mod-els for the space-time evolution of the medium cangive insight into the evolution of the spatial and themomentum anisotropy. For hydrodynamical modelsthe latter is directly related to the elliptic flow v2.Similar statements apply to the transport modelswhere the initial spatial anisotropies are convertedto momentum anisotropies [8].

The spatial anisotropy of the matter distributionis quantified by the eccentricity coefficients �n de-fined as

�n exp(inΦn) = −∫rdrdφrn exp(inφ)e(r, φ)∫

rdrdφrne(r, φ)(11)

where e(r, φ) is the local energy density in the trans-verse plane.

The second-order coefficient �2 is also called ellip-ticity and to leading order the origin of the ellipticflow v2. It can be simplified to

�2 =

√{r2 cos(2φ)}2 + {r2 sin(nφ)}2

{r2}(12)

where {...} =∫dxdy(...)e(x, y) describes an event-

averaged quantity weighted by the local energy den-sity e(x, y) [44].

The importance of event-by-event fluctuations inthe initial state has been realized in particular forhigher-order flow harmonics but also as a contri-bution to the elliptic flow and has been exten-sively investigated both experimentally and theoret-ically [45–47]. As shown earlier, the PHSD modelnaturally produces initial state fluctuations due toits microscopic dynamics. We therefore apply event-by-event hydrodynamics and all subsequent quanti-ties are averaged over many events.

In Fig. 15 we show the time evolution of the el-lipticity 〈�2〉 for both medium descriptions. For thePHSD simulations we observe large oscillations in〈�2〉 at the beginning of the evolution due to theinitialization geometries and formation times. Aftersufficient overlap of the colliding nuclei at the initialtime τ0 the average 〈�2〉 is stabilized in PHSD. There

13

−10

0

10

k y

t = 0.6 fm/c

PHSD

t = 1.6 fm/c

PHSD

t = 2.6 fm/c

PHSD

t = 3.6 fm/c

PHSD

−10 0 10kx

−10

0

10

k y

t = 4.6 fm/c

PHSD−10 0 10

kx

t = 5.6 fm/c

PHSD−10 0 10

kx

t = 6.6 fm/c

PHSD−10 0 10

kx

t = 7.6 fm/c

PHSD 100

101

102

103

104

Figure 12. (color online) Contour plots of the Fourier transform of the energy density ẽ(x, y, z = 0) for the simulationobtained within PHSD as shown in Fig. 8.

−10

0

10

k y

τ0 = 0.6 fm/c

Hydro

τ = 1.6 fm/c

Hydro

τ = 2.6 fm/c

Hydro

τ = 3.6 fm/c

Hydro

−10 0 10kx

−10

0

10

k y

τ = 4.6 fm/c

Hydro−10 0 10

kx

τ = 5.6 fm/c

Hydro−10 0 10

kx

τ = 6.6 fm/c

Hydro−10 0 10

kx

τ = 7.06 fm/c

Hydro 100

101

102

103

104

Figure 13. (color online) Contour plots of the Fourier transform of the energy density ẽ(x, y, z = 0) for the simulationobtained within hydrodynamics as shown in Fig. 9.

are, however, still significant event-by-event fluctua-tions of this quantity at later times and strong vari-ations between individual events.

In contrast, in a single hydrodynamical event �pdeviates from the average, but remains a smoothfunction of time. Due to the faster expansion inx-direction the initial spatial anisotropy decreasesduring the evolution for both medium descriptions.

However, the spatial anisotropy decreases fasterwhen initial pre-equilibrium flow βi (extracted fromthe early PHSD evolution) is included in the hydro-dynamical evolution. In this case, the time evolutionof the event-by-event averaged spatial anisotropy isvery similar in PHSD and in hydrodynamics. Ini-tializing with the shear-stress tensor πµνi may haveslight effects on the spatial eccentricity but not large

14

10−410−2100102104106 τ = 0.6 fm/c

HydroPHSD

τ = 1.6 fm/c τ = 2.6 fm/c τ = 3.6 fm/c

0 4 8 12 1610−4

10−2100102104106 τ = 4.6 fm/c

0 4 8 12 16

τ = 5.6 fm/c

0 4 8 12 16

τ = 6.6 fm/c

0 4 8 12 16

τ = 7.6 fm/c

k = √k2x + k2y

FT m

agni

tude

Figure 14. (color online) Radial distribution of the Fourier modes of the energy density for different proper timein both PHSD and hydrodynamical events. The red lines correspond to the PHSD simulations and the black linecorresponds to the hydrodynamical simulations.

Figure 15. (Color online) Event-by-event averaged spa-tial eccentricity �2 of 100 PHSD events and 100 VISHNUevents with respect to proper time, for a peripheralAu+Au collision (b = 6 fm) at

√sNN = 200 GeV. The

green dots show the distribution of each of the 100 PHSDevents used in this analysis. The solid red line is the av-erage over all the green dots. The blue, yellow and blackline correspond to hydrodynamical evolution taking dif-ferent initial condition scenarios.

enough to be visible.A similar feature is also seen in the evolution of

the momentum ellipticity, which is directly relatedto the integrated elliptic flow v2 of light hadrons.

The total momentum ellipticity is determined fromthe energy-momentum tensor as [29, 48]:

�p =

∫dxdy(T xx − T yy)∫dxdy(T xx + T yy)

(13)

Here the energy-momentum tensor includes the vis-cous corrections from πµν and Π.

In the left panel of Fig. 16 we show the time evo-lution of the event-by-event averaged 〈�(p)〉 for thehydrodynamical medium description with and with-out pre-equilibrium flow in the initial conditions. In-cluding the initial flow leads to a finite momentumanisotropy at τ0 which subsequently increases as thepressure transforms the spatial anisotropy in collec-tive flow. Consequently, �p is larger than in the sce-nario without initial flow throughout the entire evo-lution of the medium and an enhanced elliptic flowcan be expected. Given the unresolved question ofbulk viscosity in heavy-ion collisions, we investigatethe effect of tuning the bulk viscosity from the stan-dard value discussed at the beginning of this sectionto four times of this value, which comes closer tothe bulk viscosity found in different quasi-particlecalculations [40, 41]. We see that for an enhancedbulk viscosity around Tc the momentum anisotropydevelops a bump at later times, which is more pro-nounced for larger bulk viscosity.

15

Figure 16. (Color online) Event-by-event averaged total momentum anisotropy of 100 PHSD events and 100 VISHNUevents with respect to proper time, for a peripheral Au+Au collision (b = 6 fm) at

√sNN = 200 GeV. Left: the total

momentum eccentricity of hydrodynamical evolution for different initial scenarios, as well as different bulk viscosityadapted in the hydrodynamical simulation. Right: comparison of the total momentum eccentricity from PHSDevents compared with the standard hydrodynamical events. The green dots show the distribution of each of the 100PHSD events used in this analysis. The solid red line is an average over the green dots. The black line correspondsto the standard hydrodynamical evolution taking the 100 initial conditions which are generated from PHSD events.

In the right panel of Fig. 16 the hydrodynamicalsimulation is compared to the results from PHSD,again for event-by-event averaged quantities and theevent-by-event fluctuations indicated by the spreadof the cloud. The PHSD momentum eccentricity isconstructed by Eq. (13) where Tµν is evaluated fromEq. (5). It can be observed that before τ0 the aver-aged momentum anisotropy in PHSD develops con-tinuously during the initial stage, before it reachesthe value which is provided in the initial conditionsfor hydrodynamics. Despite the seemingly large bulkviscosity, as discussed in the beginning of this sec-tion, the momentum anisotropy in PHSD does notshow any hint of a bump like in the hydrodynamicalcalculation. The response to intrinsic bulk viscosityin a microscopic transport model does not seem tobe as strong as in hydrodynamics.

V. SUMMARY

In this paper, we have compared two commonlyused descriptions of the evolution of a QGP mediumin heavy-ion collisions, the microscopic off-shelltransport approach PHSD and a macroscopic hy-drodynamical evolution. Both approaches give anexcellent agreement with numerous experimentaldata, despite the very different assumptions inher-ent in these models. In PHSD, quasi-particles aretreated in off-shell transport with thermal massesand widths which reproduce the lattice QCD equa-tion of state and are determined from parallel

event runs in the simulations. Hydrodynamics as-sumes local equilibrium to be reached in the initialstages of heavy-ion collisions and transports energy-momentum and charge densities according to thelattice QCD equation of state and transport coef-ficients such as the shear and bulk viscosity. Wehave tried to match the hydrodynamical evolutionas closely as possible to these quantities as obtainedwithin PHSD:

1. by construction the equation of state in PHSDis compatible with the lQCD equation of stateused in the hydrodynamical evolution

2. a new Landau-matching procedure was used todetermine initial conditions for hydrodynamicsfrom the PHSD simulation,

3. the hydrodynamical simulations utilize thesame η/s(T ) as obtained within PHSD and

4. different bulk viscosity parameterizations havebeen introduced in the hydrodynamical simu-lation that resemble to those obtained in (dy-namical) quasi-particle models, which are thebasis for PHSD simulations.

In general we find that the ensemble averages overPHSD events follow closely the hydrodynamical evo-lution. The major differences between the macro-scopic near-(local)-equilibrium and the microscopicoff-equilibrium dynamics can be summarized as:

1. A strong short-wavelength spatial irregularityin PHSD at all times during the evolution

16

versus a fast smoothing of initial irregulari-ties in the hydrodynamical evolution such thatonly global long-wavelength structures survive.These structures have been calculated on thelevel of the fluid velocity and energy densityand quantified in terms of the Fourier modes ofthe energy density. Due to the QCD equationof state the irregularities imprinted in the tem-perature are smaller than in the energy densityitself.

2. The hydrodynamical response to changingtransport coefficients, especially the bulk vis-cosity, has a strong impact on the time evolu-tion of the momentum anisotropy. In PHSDthese transport coefficients can be determinedbut remain intrinsically linked to the interac-tion cross sections. Although there are indica-tions for a substantial bulk viscosity in PHSD,it does not show the same sensitivity to themomentum space anisotropy as in hydrody-namical simulations.

3. Event-by-event fluctuations might be of simi-lar magnitude in quantities like the spatial andmomentum anisotropy but while they remainsmooth functions of time in hydrodynamicssignificant variations are observed within in asingle event in PHSD as a function of time.

After having gained an improved understanding

of the similarities and differences in the evolutionof bulk QCD matter between the non-equilibriumPHSD and the equilibrium hydrodynamic approach,we plan to utilize our insights in future projects re-garding the development of observables sensitive tonon-equilibrium effects and the impact these effectsmay have on hard probe observables.

ACKNOWLEDGEMENTS

We appreciate fruitful discussions with J. Aiche-lin, W. Cassing, P.-B. Gossiaux, T. Kodama. Thiswork in part was supported by the LOEWE centerHIC for FAIR as well as by BMBF and DAAD. Thecomputational resources have been provided by theLOEWE-CSC. SAB, MG and YX acknowledge sup-port by the U.S. Department of Energy under grantno. DE-FG02-05ER41367.

VI. APPENDIX

A. Fourier transform of energy density

For a discrete 2D Fourier transform, we have:

X̃(k, l) =

M−1∑m=0

N−1∑n=0

x(m,n)e−2πimkM e−2πi

nlN (14)

x(m,n) =1

MN

M−1∑m=0

N−1∑n=0

X̃(k, l)e2πi(mkM +

nlN ) (15)

=1

MN

M−1∑m=0

N−1∑n=0

X̃(k, l)

[cos

(2π(

mk

M+nl

N)

)+ i sin

(2π(

mk

M+nl

N)

)](16)

=1

MN

M−1∑m=0

N−1∑n=0

[X̃real(k, l) cos

(2π(

mk

M+nl

N)

)− X̃imag(k, l) sin

(2π(

mk

M+nl

N)

)](17)

X̃real and X̃imag are the real and imagine part of

Fourier transform coefficients X̃

X̃real(k, l) =

M−1∑m=0

N−1∑n=0

x(m,n) cos

(2π(

mk

M+nl

N)

)(18)

X̃real(k, l) = −M−1∑m=0

N−1∑n=0

x(m,n) sin

(2π(

mk

M+nl

N)

)(19)

Therefore, FT is defined as:

17

∣∣∣X̃(k, l)∣∣∣2 = ∣∣∣X̃real(k, l)∣∣∣2 + ∣∣∣X̃imag(k, l)∣∣∣2 (20)=

M−1∑m=0

N−1∑n=0

x2(m,n) (21)

+∑

m 6=m′,n6=n′x(m,n)x(m′, n′) (22)

cos

(2π(

(m−m′)kM

+(n− n′)l

N)

)(23)

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19

Traces of non-equilibrium dynamics in relativistic heavy-ion collisionsAbstractI IntroductionII Description of the modelsA PHSD transport approachB 2D+1 viscous hydrodynamics

III Non-equilibrium initial conditionsA Evaluation of the energy-momentum tensor T in PHSDB PHSD initial conditions for hydrodynamics

IV Medium evolution: hydrodynamics versus PHSDA Pressure isotropizationB Space-time evolution of energy density e and velocity C Fourier images of energy densityD Time evolution of the spatial and momentum anisotropy

V Summary AcknowledgementsVI AppendixA Fourier transform of energy density

References