CCompact Stars in the QCD Phase Diagram IV (CSQCD IV)September 26-30, 2014, Prerow, Germany
Hydrodynamization Physics from Holography
Jakub Jankowski
Institute of Physics, Jagiellonian University, ul. (cid:32)Lojasiewicza 11, 30-348 Krak´ow, Poland
The quest for a better understanding of the properties of nuclear matter under ex-treme conditions (such as those created in relativistic heavy ion collisions) has ledto a number of theoretical challenges. One of them is to explain the success of thehydrodynamic description of quark-gluon plasma (QGP) evolution at time scales oforder τ ∼ . − τ H T H ≤ η/s ) and vanishing bulk viscosity suggests that theplasma state may be viewed effectively as a strongly coupled, conformal fluid. This hasmotivated studies aimed at understanding how such a hydrodynamic description couldemerge in a case which is amenable to theoretical studies – the N = 4 supersymmetricYang-Mills theory (SYM). In this example one has the option to apply gauge/gravityduality [3] to model the approach to equilibrium. Such studies were initiated in [4],where the case of Bjorken flow (discussed in more detail below) was considered. Janikand Peschanski showed in that context that the hydrodynamic description does indeedemerge at late proper times. This work was followed by numerous articles devotedto the description of hydrodynamic states in the context of gauge/gravity duality. Inparticular, second order transport coefficients were computed [5, 6], and many newinsights into the meaning of relativistic hydrodynamics were gained [7, 8, 9, 10, 11].To describe far from equilibrium states of N = 4 SYM it is necessary to resortto numerical calculations. A decisive step opening this field of research was madeby Chesler and Yaffe [12] who devised a very effective numerical scheme for solvingEinstein equations in asymptotically AdS spaces based on characteristic evolution.This was soon applied to the case of Bjorken flow [13], which is a particularly attrac-tive setting, since it was an important model used to understand basic features ofQGP evolution (such as entropy production), and at the same time is simple enoughto implement easily in the context of gauge/gravity duality. The results describedhere [14] follow from a modification of this scheme. A different method was usedin [15, 16], where some basic features of the approach to hydrodynamics were stud-ied. Most importantly, it was found there that the system reaches the hydrodynamic1 a r X i v : . [ h e p - t h ] A p r egime quickly (in the sense described earlier). Another physically important conclu-sion from these papers (earlier noted in [13]) was that hydrodynamics works very wellalready at a time when pressure gradients are large. Thus the process of reaching thisstage of evolution is often referred to as hydrodynamization instead of thermalization.The numerical studies [15, 16] used a different numerical scheme from [12, 13],which was however limited by the fact that only a few (29) consistent initial stateswere known. The study [14] reported here adapted the approach of [13] in a way whichallows an arbitrary number of initial conditions to be analysed, making it possible tolook for generic features. In our work we looked at 600 initial conditions randomlygenerated on the gravity side of the duality. In the context of gauge/gravity dualitythere is a natural characteristic of the initial state called the initial entropy [15] (tobe defined precisely below). Observables such as hydrodynamization time depend inparticular on this quantity.In this note we will focus on two natural questions • Is hydrodynamization generically a fast process? • Are there any universal physical characteristics of hydrodynamization?The answer to the first question seems to be positive and confirms previous investiga-tions [15, 16]; regardless of the values of the initial entropy chosen the system reacheshydrodynamic description on time scales of the order of the inverse local tempera-ture. The second question is more subtle. In the sample of initial data analysed in ourstudy the hydrodynamization time appears not to be correlated with initial entropy(in contrast to [15], where such a correlation seemed to be present on the basis of asmaller sample of initial states). However, when looking at the energy density at thetime when hydrodynamic evolution starts, there appears to be a linear correlationwith initial entropy.One should also mention the work [17, 18] where the process of isotropization wasconsidered in a similar spirit. Recently a technically different (closer to the originalformulation of the problem [12]) but conceptually similar studies appeared in [19].
As discussed in the introduction, N = 4 SYM is amenable to quantitative studies inthe strongly coupled limit. In the absence of methods which could be used in QCDfor the study of strongly coupled, real time dynamics, this theory has become a fo-cus of much attention. This theory shares some features of QCD (especially at hightemperature), but is rather different from it. It contains, apart from gluons, 6 realmassless scalars and 4 Majorana massless fermions, all in the adjoint representationof the U ( N ) gauge group. The theory is known to be conformal even at the quantum2evel. As mentioned in the introduction, we work within the AdS/CFT correspon-dence [3] which becomes an effective computational tool in the ’t Hooft limit N → ∞ and λ = g N → ∞ , where quantum and stringy corrections on the gravity side canbe neglected.There are important similarities and differences between SYM and QCD which onehas to keep in mind. Among the similarities are the existence of the deconfined phase.Also, in the perturbative regime at T >
0, both theories have been shown to behavesimilarly, with the difference coming mostly from the different number of degreesof freedom [20]. The most crucial difference is that N = 4 SYM has a vanishingbeta function, which implies that it is not confining, has no finite temperature phasetransition and has an exactly conformal equations of state.The modeling of nuclear collisions is an extremely complex task. To reduce thecomplexity of the problem we adopt strong symmetry assumptions introduced byBjorken [21] for the description of matter following a heavy ion collision. The dy-namics of the system is assumed to be independent of boosts along the longitudinal(collision) axis and is independent of transverse coordinates. In proper time-rapiditycoordinates t = τ cosh y , z = τ sinh y this reduces to the statement that observablesdependent only on proper time τ . This approximation becomes exact in the limit ofan infinite energy collision of infinitely large nuclei.Two physical quantities of interest to us will be the energy momentum tensor andentropy. In the present circumstances first one takes the form T µν = Diag { (cid:15) ( τ ) , p L ( τ ) , p T ( τ ) , p T ( τ ) } , (1)where (cid:15) ( τ ) = p L ( τ ) + 2 p T ( τ ) as required by conformal symmetry. The energy densitydefines local effective temperature by the relation (cid:15) ( τ ) = 38 N π T ( τ ) . (2)This is the temperature of an equilibrium system with the same energy density. Itcan be shown that for late times the dynamics is governed by the equations of hydro-dynamics; up to third order the effective temperature follows [4, 22, 5, 23] T ( τ ) = Λ(Λ τ ) / (cid:110) − π (Λ τ ) / + − π (Λ τ ) / + (3)+ −
21 + 2 π + 51 log 2 −
24 log π (Λ τ ) (cid:111) . The energy scale Λ appearing in Eq. (3) depends on the initial conditions chosenand it is the only trace of initial state information contained in the hydrodynamicexpansion. It also sets the scale for the energy density at hydrodynamization.3he calculation strategy follows usual lore of holography [24]. States in the bound-ary theory correspond to asymptotically AdS geometries in the bulk. For examplean equilibrium, finite temperature, deconfined plasma state corresponds to a static(planar) black hole in the bulk. The Hawking temperature of this black object isinterpreted as the temperature in the dual N = 4 SYM. Extending this notion tothe out-of-equilibrium situation we assume that non-equilibrium plasma states corre-spond to geometries with non-static horizons. Such geometries are assumed to posesan event horizon, but there are reasons to believe [25, 26, 27] that the physical notions(such as entropy for instance) should be associated with apparent horizons .This conjecture allows us to extend the notion of entropy to non-equilibrium statesby the Bekenstein-Hawking relation, which with our normalization translates to S = a AH π , (4)where a AH is apparent horizon area [23]. By the area law theorems this quantity isnon-decreasing and agrees with thermodynamic definition for late times.In calculations performed here we took 600 different initial states described byrandomly generated initial geometries – with each of these we associate an initialentropy as defined above. We then evolved these geometries according to Einsteinequations well into the hydrodynamic regime. From the rules of the holographiccorrespondence we are able to read of the relevant physical observables i.e. energydensity (cid:15) ( τ ) and entropy S ( τ ). An important assumption on the initial conditionsis that (cid:15) (0) (cid:54) = 0, which allows us to normalize the initial effective temperature as T (0) = 1 /π . For more technical details of construction of initial geometries andsolving for the time evolution we refer to the original paper [14]. In order to present quantitative results on the hydrodynamization process we needto give it a precise definition. The approach to hydrodynamics can be observed bymonitoring the pressure anisotropy∆ ≡ p T − p L (cid:15) . (5)It is convenient to measure time in units of inverse local temperature, that is, touse w = τ T ( τ ) as a parameter of evolution. The pressure anisotropy can then beexpressed as ∆( w ) = 6 f ( w ) − f ( w ) = τw dwdτ . (6)4hich for dimensional reasons is independent of Λ. At large times (large w ) thisfunction attains a universal (hydrodynamic) form f H ( w ), determined by an infiniteset of transport coefficients, which up to the third order reads f H ( w ) = 23 + 19 πw + 1 − log 227 π w + 15 − π −
45 log 2 + 24 log π w . (7)The beginning of the hydrodynamic stage might now be defined as the value of w (hence proper time τ ) when the difference between the actual f ( w ) and the hydrody-namic form f H ( w ) is less than some arbitrary small number; for example | f H ( w ) f ( w ) − | < . . (8)While this definition involves some arbitrary choice, varying this criterion within rea-son leads to no appreciable change in the calculated value of the hydrodynamizationtime. w th Figure 1: Histogram of hydrodynamization times w th in units of effective hydrody-namization temperature.Figure 1 shows a histogram of hydrodynamization times obtained this way. Re-gardless the initial entropy, the hydrodynamization time is of the order if the inversehydrodynamization temperature. On the average w av = 0 .
57. It is instructive to com-pare it to the estimation from the RHIC data; T = 500 MeV and τ = 0 .
25 fm/c gives w RHIC = 0 .
63 which is very close to theoretical prediction. The pressure anisotropyat hydrodynamization is found to be quite high: ∆ ≈ . .0 0.2 0.4 0.6 0.80.00.20.40.60.81.01.2 S i Λ / T ( i ) Figure 2: Hydrodynamic energy scale Λ in units of initial effective temperature.Results for this scale are shown in figure 2. For an intermediate range of entropiesbelow S ∼ . . η/s .Our simulations suggest that, at least in some regions of initial state parameters,there might exist characteristic regularities, reflecting the nature of collective non-hydrodynamic degrees of freedom [28]. At this point an important question to whatextent is this correlation a consequence of the strong symmetry assumptions imposedand to what extent it reflects the true nature of the process. Acknowledgements
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