In high energy heavy ion collisions of RHIC and LHC, a strongly interacting
quark gluon plasma (sQGP) is created. This medium undergoes a hydrodynamic
evolution, before it freezes out to form a hadronic matter. The initial state
of the sQGP is determined by the initial distribution of the participating
nucleons and their interactions. Due to the finite number of nucleons, the
initial distribution fluctuates on an event-by-event basis. The transverse
plane anisotropy of the initial state can be translated into a series of
anisotropy coefficients or eccentricities: second, third, fourth-order
anisotropy etc. These anisotropies then evolve in time, and result in
measurable momentum-space anisotropies, to be measured with respect to their
respective symmetry planes. In this paper we investigate the time evolution of
the anisotropies. With a numerical hydrodynamic code, we analyze how the speed
of sound and viscosity influence this evolution.Comment: 10 pages, 6 figures. To appear in the Gribov-85 Memorial Workshop's
  proceedings volume. Supported by OTKA NK 10143

Bagoly, Attila

Csanad, Mate

International Journal of Modern Physics A

English

arXiv

A. BAGOLY

M. CSANÁD

Crossref

TIME EVOLUTION OF THE ANISOTROPIES OF THE HYDRODYNAMICALLY EXPANDING SQGP

BAGOLY, A

CSANAD, M

ELTE Digital Institutional Repository (EDIT)

October 11, 2018 13:27 WSPC - Proceedings Trim Size: 9in x 6in bagolycsanad_numhidro1Time evolution of the anisotropies of the hydrodynamicallyexpanding sQGPA. BAGOLY, M. CSANÁD∗Department of Atomic Physics, Eötvös University,Budapest, Pázmány P. s. 1/a, H-1117, Hungary∗E-mail: csanad@elte.huIn high energy heavy ion collisions of RHIC and LHC, a strongly interactingquark gluon plasma (sQGP) is created. This medium undergoes a hydrody-namic evolution, before it freezes out to form a hadronic matter. The initialstate of the sQGP is determined by the initial distribution of the participat-ing nucleons and their interactions. Due to the finite number of nucleons, theinitial distribution fluctuates on an event-by-event basis. The transverse planeanisotropy of the initial state can be translated into a series of anisotropycoefficients or eccentricities: second, third, fourth-order anisotropy etc. Theseanisotropies then evolve in time, and result in measurable momentum-spaceanisotropies, to be measured with respect to their respective symmetry planes.In this paper we investigate the time evolution of the anisotropies. With a nu-merical hydrodynamic code, we analyze how the speed of sound and viscosityinfluence this evolution.1. Hydrodynamics in ultra-relativistic heavy ion collisionsThe medium created in ultra-relativistic heavy ion collisions is success-fully modeled by perfect fluid hydrodynamics, i.e. the observables of softhadrons, photons and leptons can described by hydro models. Exact solu-tions are particularly interesting, since one gains an analytic insight on theconnection between the initial and final state of the hydro evolution.These solutions fulfill a simple set of laws, governing many systems onmany scales. Hydrodynamics only assumes the local conservation of energyand momentum, and, depending on the circumstances, of entropy or somearXiv:1507.05005v2  [nucl-th]  13 Feb 2016October 11, 2018 13:27 WSPC - Proceedings Trim Size: 9in x 6in bagolycsanad_numhidro2conserved charges. The equations of non-relativistic hydrodynamics are:∂ρ∂t+ ∇ρv = 0 (1)ρ(∂v∂t+ (v∇)v)= −∇p+ µ∆v +(ζ +µ3)∇(∇v) (2)∂ε∂t+ ∇εv = −p∇v + ∇(σv) (3)where ρ is matter density, v is the velocity field, ε is the (internal) energydensity, p is pressure, σ is the viscous stress tensor of the homogeneousand isotropic medium, µ is the dynamic viscosity coefficient, λ the bulkviscosity coefficient, while t represents time and ∇ = (∂x, ∂y, ∂z) the spa-tial derivative vector. If we however would like to utilize hydrodynamicsin a relativistically expanding medium, we have to use the equations ofrelativistic hydrodynamics. In case of perfect hydro, these take the form ofTµν =(ε+ p)uµuνc2− pgµν , ∂µTµν = 0. (4)To close this set of equations, we need to specify a relation between energydensity ε and pressure p. This we obtain using the below Equation of State:ε = κ(T )p, (5)where κ = c−2s is the inverse square of speed of sound. It’s temperaturedependence can be taken from lattice QCD calculations, but often κ issimply kept constant.2. Initial conditions and anisotropiesThere are many solutions that fulfill the above equations, we however needto start from an initial state that corresponds to the energy distributiondeposited by the nucleons participating in the collision. In nucleus-nucleuscollisions the initially formed medium can be spatially approximated by anellipsoid, which can be described by the contour levels of a scale variable ss =x2X2+y2Y 2+z2Z2(6)where X,Y, Z are the axes of the ellipsoid described by the equation s =1. Multiple exact solutions of hydrodynamics were found1,2 that resemblethis symmetry, i.e. where the thermodynamical quantities are constant ons =const. curves. It is important to see however, that nuclei contain a finitenumber of nucleons, and thus the participants of a nucleus-nucleus collisionform an event-by-event fluctuating shape (see Fig. 1).October 11, 2018 13:27 WSPC - Proceedings Trim Size: 9in x 6in bagolycsanad_numhidro3Figure 1. Simulated participant nucleon distribution (figure from Ref.3)Thus a simple ellipsoidal symmetry is not sufficient to describe the hy-drodynamically expanding medium, not to mention that it does not de-scribe higher order flow coefficients either. To overcome this difficulty, onecan assume higher order anisotropies in the scaling variable, by redefiningit ass =r2R2(1 + ε2 cos(2φ)) +z2Z2(7)with r and φ being the polar coordinates in the transverse plane, while Rand ε2 are in a direct relation to X and Y . This can then be generalized tos =r2R2(1 +∑n=2,3,...εn cosnϕ)+z2Z2(8)in the cosnϕ expression a Ψn n-th order event plane can also be introduced.Example distributions generated with this scaling variable are shown inFig. 2. Solutions utilizing scaling variables like the above s were presentedin Ref.4 However, these solutions, as well as the ones in Ref.1 are validonly in case of a 3D Hubble-flow, which corresponds to the lack of pressuregradients. This also means that the anisotropies will remain constant intime, which may not be the case throughout the whole evolution. In ouranalysis, we thus apply numerical calculations to include pressure gradientsin the initial conditions, and analyze the time evolution of the anisotropies.Thus we will start the hydrodynamic evolution from the above describedset of initial conditions. We would like to extract the anisotropy coefficientsat later stages of the evolution, for which we need a definition. We in-troduced the anisotropies of density, energy density and velocity fields asfollows:εn = 〈cos(nϕ)〉ρ/ε/w, (9)October 11, 2018 13:27 WSPC - Proceedings Trim Size: 9in x 6in bagolycsanad_numhidro4Figure 2. Initial distribution with the outlined s scaling variable, for different εn values.where w = exp (−v2x − v2y) is defined to calculate the asymmetry of velocityfield. To understand the relation of the asymmetry parameters introducedby 9 to the scale variable anisotripy coefficients, εn’s, we checked theirrelation. As a first order approximation (in εn), we obtainedεn = −εn/2 (10)however, in a more accurate approximation the result is:ε1 ≈(ε2 + ε4)ε32 +∑n ε2n=ε3(ε2 + ε4)2+O(ε4n) (11)ε2 ≈−ε2 + ε2ε42 +∑n ε2n= −ε22+ε2ε42+14ε2∑nε2n +O(ε4n) (12)ε3 ≈−ε32 +∑n ε2n= −ε32+14ε3∑nε2n +O(ε4n) (13)ε4 ≈−ε4 + 12ε222 +∑n ε2n= −ε42+ε224− 14ε4∑nε2n +O(ε4n) (14)this means that even if there is no ε1 coefficient, we will still get an ε1.It is also important to note that ε2 6= 0 and εn 6=2 = 0 will result in anonzero ε4 = ε22/4. See an example with multiple nonzero εn coefficients intable 1, where we give an example how a set of εn values result in a differentset of εn values. The actual results can be accurately approximated witheqs. (12)-(14).3. Numerical methodLet us now discuss how we evolve the previously defined initial conditionsin time. In order to do that, we first transform the equations of (1+2 di-mensional) hydrodynamics to a convective form as usual:∂tQ + ∂xF(Q) + ∂yG(Q) = 0 (15)October 11, 2018 13:27 WSPC - Proceedings Trim Size: 9in x 6in bagolycsanad_numhidro5Table 1. Example for the εn − εn relation.n εnεndirect calculation approximation first order estimate1 0 0.00278 0.00298 02 0.1 -0.04927 -0.04869 -0.053 0.05 -0.02533 -0.02484 -0.0254 0.02 -0.00769 -0.00745 -0.01where Q is calculated from the hydrodynamical fields, while F and G arefunctions of these, transformed from the original hydro equations. The finitevolume method discussed below and introduced in Ref.5 works for manydifferent systems of partial differential equations (PDE’s), one just needsto determine the given functions to transform the PDE’s to the above form.This can be done in a straightforward manner, see for example Refs.5–8The basis of finite volume methods is to discretize the Q vector, bydefining it on a space-time grid, and obtaining the discrete values by av-eraging in each cell. Then, we have to evaluate fluxes F and G betweengrid points (these are then called intercell fluxes). This is difficult, since thefluxes depend on the values of Q, but these are defined on the grid points.Our method, outlined in Ref.5 and used for exampe in Ref,7 is based on anaccurate multi-stage (MUSTA) predictor-corrector estimation of the inter-cell fluxes. We discuss this method briefly in 1+1 dimensions (and with asingle-component Q only) – it is straightforward to generalize it to multiplespatial dimensions via operator splitting (we used the Lie type of splittingmethod here). In case of viscous hydrodynamics, we also used operatorsplitting to separate ideal and viscous fluxes.To estimate the intercell fluxes between cells i and i+ 1, we start fromthe Qni and Qni+1 values, where n represents the current time-step. Let uscall these values Q(0)i,i+1, indicating that these represent a “zeroth order”estimate of the intercell values. We will then make a series of estimates inthe given cell, the lth estimate being indicated with Q(l)i,i+1. The flux in thecell centers is then F (l)i,i+1 ≡ F(Q(l)i,i+1). We define an intermediate Q valueand flux F as follows:Q(l)i+ 12=12[Q(l)i +Q(l)i+1]− 12∆t∆x[F(l)i+1 − F(l)i], F(l)M ≡ F(Q(l)i+ 12)(16)Then our intercell flux estimate isF(l)i+ 12=14[F(l)i+1 + 2F(l)M + F(l)i −∆x∆t(Q(l)i+1 −Q(l)i)](17)October 11, 2018 13:27 WSPC - Proceedings Trim Size: 9in x 6in bagolycsanad_numhidro6This is the so called FORCE flux approximation. The essence of our theapplied MUSTA algorithm is that now we make a new prediction, Q(l+1)iand with that we repeat the steps and we get a better flux approximation,and with that we can make better prediction, and again better flux approx-imation. To make a new prediction simply we use the equations we wantto solve:Q(l+1)i = Q(l)i −∆t∆x[F(l)i+ 12− F (l)i](18)Q(l+1)i+1 = Q(l)i+1 −∆t∆x[F(l)i+1 − F(l)i+ 12](19)We may then stop at a given l value, by setting Fni+ 12= F(l)i+ 12. With this,we simply calculate the next time-step asQn+1i = Qni −∆x∆t(Fni+ 12− Fni− 12). (20)where Fni− 12can also be calculated by the procedure outlined above, if westart from cells i− 1 and i.In order to complete the method, let us mention that we used the initialconditions as described above, we used non-reflective boundary conditions,and the timestep was determined adaptively via the Courant-Friedrichs-Lewy condition. We tested our method with the usual Sod shock tube andknown hydrodynamic solutions as well.4. ResultsWith the above outlined method, we solved the equations of nonrelativis-tic hydrodynamics with viscosity, as well as the equations of relativisticperfect hydrodynamics. Our goal was to investigate how various speed ofsound values and various viscosity values affect the time evolution of theasymmetries.Let us first discuss the effects of viscosity (in nonrelativistic hydro). Asshown in Fig. 3 viscosity makes the disappearance of pressure anisotropiesslower, and the same is in the number density. In other words, in caseof nonzero viscosity the pressure asymmetries remain in the system for alonger time. This may be due to the fact that viscosity makes the flowslower, as illustrated in the first two plots of Fig. 4. As shown in the bot-tom plot of Fig. 3 and the third and fourth plots in Fig. 4, the effect ofviscosity on the flow field is however the opposite: viscosity makes the flowanisotropies disappear faster. The reason of this may be that the viscousforce contains the second spatial derivatives, thus the fluid elements withOctober 11, 2018 13:27 WSPC - Proceedings Trim Size: 9in x 6in bagolycsanad_numhidro7larger asymmetries feel a stronger viscous force, which means that viscosity“washes out” the anisotropies.We also investigated the effects of a change in the speed of sound. Asshown in Fig. 5, the reduction of speed of sound makes the anisotropiesdisappear slower. The reason for this is that pressure waves travel with thespeed of sound, so the equalization of pressure is slower in case of smallerspeed of sound values. This is confirmed by relativistic calculations as well.As shown in the top panels of Fig. 6, in case of a larger κ (i.e. a smallerc2s value) the anisotropies disappear slower, and also the cooling is slower.This also means that of course the system will freeze out later. The timeevolution of the flow field is shown in the bottom panels of Fig. 6.Figure 3. Time evolution of pressure anisotropies is shown in the left panel, whileanisotropies in the flow field are shown in the right panel.5. SummaryIn this paper we analyzed the time evolution of the asymmetries of theQCD matter created in high energy heavy ion collisions. We utilized nu-merical hydrodynamics instead of exact solutions to investigate situationsnot describable by the presently known hydrodynamic solutions. We how-ever did not start from the most realistic initial conditions, but startedinstead from an initial condition that is very similar to one described byknown analytic solutions – except in pressure, where we used a pressureprofile similar to the density profile given in these models. We analyzedboth non-relativistic and relativistic hydrodynamics, and arrived at similarconclusions. It turns out that a stiffer equation of state, i.e. a larger speed ofsound makes the pressure anisotropies disappear faster. The appearance ofviscosity also makes the flow anisotropies disappear faster, however, pres-October 11, 2018 13:27 WSPC - Proceedings Trim Size: 9in x 6in bagolycsanad_numhidro8Figure 4. Time evolution of the energy density (first two rows) and of the flow field(last two rows). The first and third rows show the viscosity free case, the second andfourth rows show the viscous case with µ = 10MeV · fm.sure anisotropies remain in the system longer, due to the slower flow of theviscous system.AcknowledgmentsThe authors would like to thank Julia Nyíri for her kind invitation to thisseries, and the organizers for keeping Vladimir Gribov’s work and memoryalive. M. Cs. would also like to thank Tamás Csörgő for inspiring and use-October 11, 2018 13:27 WSPC - Proceedings Trim Size: 9in x 6in bagolycsanad_numhidro9Figure 5. Time evolution of pressure anisotropies is shown in the left panel, whileanisotropies in the flow field are shown in the right panel.ful discussions. This manuscript was supported by the OTKA grant NK101438.Bibliography1. T. Csörgő, L. P. Csernai, Y. Hama and T. Kodama, Heavy Ion Phys. A21, 73(2004).2. T. Csörgő, S. V. Akkelin, Y. Hama, B. Lukács and Y. M. Sinyukov, Phys. Rev.C67, p. 034904 (2003).3. C. Loizides, J. Nagle and P. Steinberg (2014).4. M. Csanád and A. Szabó, Phys.Rev. C90, p. 054911 (2014).5. E. F. Toro and V. A. Titarev, J. Comp. Phys. 216, 403 (2006).6. H. Nishikawa and K. Kitamura, J. Comp. Phys. 227, 2560 (2008).7. M. Takamoto and S.-i. Inutsuka, J.Comput.Phys. 230, 7002 (2011).8. I. Karpenko, P. Huovinen and M. Bleicher, Comput.Phys.Commun. 185, 3016(2014).October 11, 2018 13:27 WSPC - Proceedings Trim Size: 9in x 6in bagolycsanad_numhidro10Figure 6. The time evolution of velocity field. The top row shows κ = 2 case, thebottom row the κ = 4 case. Time evolution of pressure anisotropies is shown in the leftpanel, while anisotropies in the flow field are shown in the right panel.

 In high energy heavy ion collisions of RHIC and LHC, a strongly interacting quark gluon plasma (sQGP) is created. This medium undergoes a hydrodynamic evolution, before it freezes out to form a hadronic matter. The initial state of the sQGP is determined by the initial distribution of the participating nucleons and their interactions. Due to the finite number of nucleons, the initial distribution fluctuates on an event-by-event basis. The transverse plane anisotropy of the initial state can be translated into a series of anisotropy coefficients or eccentricities: second, third, fourth-order anisotropy etc. These anisotropies then evolve in time, and result in measurable momentum-space anisotropies, to be measured with respect to their respective symmetry planes. In this paper we investigate the time evolution of the anisotropies. With a numerical hydrodynamic code, we analyze how the speed of sound and viscosity influence this evolution. </jats:p

A. Bagoly

M. Csanád

Time evolution of the anisotropies of the hydrodynamically expanding SQGP

Time evolution of the anisotropies of the hydrodynamically expanding
  sQGP

Time evolution of the anisotropies of the hydrodynamically expanding sQGP

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