In this paper we present properties of relativistic and non-relativistic
perfect hydrodynamical models. In particular we show illustrations of the fact
that different initial conditions and equations of state can lead to the same
hadronic final state. This means that alone from the hadronic observables one
cannot determine either of the above, one needs for example penetrating probes
that inherit their properties from each timeslice of the evolution of the
fireball.Comment: Presented at the IV Workshop on Particle Correlations and Femtoscopy.
  6 pages, 3 figure

Csanad, Mate

English

arXiv

In this paper we present properties of relativistic and non-relativistic perfect hydrodynamical models. In particular we show illustrations of the fact that different initial conditions and equations of state can lead to the same hadronic final state. This means that alone from the hadronic observables one cannot determine either of the above, one needs for example penetrating probes that inherit their properties from each timeslice of the evolution of the fireball

Csanád, M

ELTE Digital Institutional Repository (EDIT)

arXiv:0903.1278v1  [nucl-th]  6 Mar 2009Initial conditions, equations of state and final state inhydrodynamics ∗Máté CsanádEötvös Loránd University, H - 1117 Budapest, Pázmány P. s. 1/A, HungaryIn this paper we present properties of relativistic and non-relativisticperfect hydrodynamical models. In particular we show illustrations of thefact that different initial conditions and equations of state can lead to thesame hadronic final state. This means that alone from the hadronic ob-servables one cannot determine either of the above, one needs for examplepenetrating probes that inherit their properties from each timeslice of theevolution of the fireball.PACS numbers: 24.10.Nz 25.75.Ag 25.75.Ld1. Perfect fluid hydrodynamicsPerfect fluid hydrodynamics is based on local conservation of entropy ordensity, energy and momentum, expressed by so-called conservation equa-tions. The fluid is perfect if the energy-momentum tensor is diagonal inthe local rest frame, i.e. there are no shear stress, viscosity or heat conduc-tion effects. The conservation equations are closed by the equation of state,which gives the relationship between energy density ǫ, pressure p. Typicallyǫ = κp, where the proportionality “constant” κ may depend on temperatureT , which is connected to the density n and pressure p via p = nT in manysolutions of the exactly solvable class, especially those discussed in following.In addition a bag constant B can be introduced with a modified equationof state of ǫ−B = κ(p +B), where B = 0 in the hadronic phase, non-zeroin a deconfined phase. In this paper only the ǫ = κp case is discussed.Solutions describing non-relativistic flows that are applicable to relativis-tic heavy ion collisions are e.g. described in refs. [1, 2, 3], we will investigatea family of such solutions.Solving the relativistic equations is much harder. There are only afew exact solutions for these equations. One (and historically the first)∗ Presented at the IV Workshop on Particle Correlations and Femtoscopy(1)2 csanad˙wpcf08 printed on April 5, 2019is the Landau-Khalatnikov implicit solution solution discovered more than50 years ago [4, 5, 6]. This is a 1+1 dimensional solution, and has real-istic properties: it describes a 1+1 dimensional expansion, does not lackacceleration and predicts an approximately Gaussian rapidity distribution.Another renowned solution of relativistic hydrodynamics is the Hwa-Bjorken solution [7, 8, 9], which is a simple, 1+1 dimensional, explicit andexact, but accelerationless solution. This solution is boost-invariant in itsoriginal form, hence fails to describe the data [10, 11]. However, the solutionallowed Bjorken to obtain a simple estimate of the initial energy densityreached in high energy reactions from final state hadronic observables.Important are solutions [12, 13] which are explicit and describe a rela-tivistic acceleration, i.e. combine the properties of the above solutions. Wewill investigate such a family of solutions.2. Investigated solutionsWe will investigate here an exact relativistic and an exact non-relativisticsolution. We will use these solutions to extract information on the depen-dence of the final state on initial state, the parameters of the exact solutionand the equation of state used.A non-relativistic hydro solutionThe below discussed non-relativistic solution describes a 3+1 dimen-sional ellipsoidally symmetric expansion [1, 14], with an arbitrary κ ∈ R.The velocity field v isv =(ẊXrx,ẎYry,ŻZrz), (1)where X, Y and Z are time-dependent principal axes of the expanding ellip-soid, Ẋ , Ẏ and Ż are their expansion rate versus time, and r = (rx, ry, rz)are the spatial coordinates. As for the thermodynamical quantities, one hasn(r, t) = n0X0Y0Z0XY Zexp(−r2x2X2−r2y2Y 2−r2z2Z2), (2)T (r, t) = T0(X0Y0Z0XY Z)1/κ, (3)where X0 = X(t0), Y0 = Y (t0), Z0 = Z(t0) are the principal axes at agiven (arbitrarily chosen) time t0, and n0 = n(0, t0), T0 = T (0, t0). Thiscsanad˙wpcf08 printed on April 5, 2019 3represents a solution if the ordinary differential equationsẌX = Ÿ Y = Z̈Z = α(X0Y0Z0XY Z)1/κ(4)are fulfilled. Here α is an “acceleration parameter”, in the above mentionedsolution α = T0/m, but note, that with a generalization in the expressionfor n(r, t) α can be made arbitrary.A relativistic hydro solutionNext being discussed is a family of analytic, explicit and simple so-lutions, which do not lack acceleration, and yield finite, realistic rapiditydistributions [12]. In these solutions the velocity field is given byuµ = γ(1,v) (5)with v being the d dimensional spherically symmetric velocity. The lengthof this vector is denoted by v, and this length can be expressed in the Rindlercoordinates η and τ byv = tanh(λη) (6)where λ is a kind of “acceleration parameter”, because relativistic accelera-tion uµ∂µuν vanishes if λ = 1, but is non-vanishing for a positive λ differentfrom 1. The pressure p and the temperature T are given byp = p0(τ0τ)λdκ+1κ, (7)T = T0(τ0τ)λd 1κ. (8)The value of the constants λ (“acceleration parameter”), d (number of spa-tial dimensions) and κ (adiabatic index of the fluid) are constrained, anddifferent possible set of values yield different solutions. For example if λ = 1,then d ∈ R and κ ∈ R are arbitrary (this is corresponds to the Bjorken so-lution). In this case there is no acceleration. If however λ = 2, i.e. thereis substantial acceleration, d ∈ R is arbitrary but κ = d must be fulfilled.Another important case is, that if d = 1 and κ = 1, then λ ∈ R is arbitrary.This last case has a remarkably general velocity field: the λ accelerationparameter can be arbitrary. On the other hand, this solution works only ford = 1 and κ = 1, which is obviously a drawback. For a discussion of greaterdetail Refs. [12, 15] are recommended.4 csanad˙wpcf08 printed on April 5, 20193. Dependence on the initial conditions and equation of stateTo calculate observables from the above solutions, freeze-out criteriahave to be utilized. For example the simple condition of T (r, t) = Tfreeze−outcan be used, or a condition where the freeze-out hypersurface is pseudo-orthogonal to the velocity field. This is an important aspect of hydrody-namics, that a freeze-out condition has to be chosen, but here it does notneed to be discussed: we fix the hydrodynamical final state and investigateonly those hydrodynamical evolutions that lead exactly to the same finalstate (i.e. to the same hadronic observables).First, let us see relativistic solutions of section 2. In Figs. 1 and 2 thespatial temperature distribution is shown at different times, for different ini-tial conditions and different λ and/or κ values. All the plotted solutions goto the same final temperature at mid-rapidity (η = 0), although their initialcondition, acceleration parameter and/or equation of state is different.Then let us see non-relativistic solutions of section 2. In Fig. 3 timeevolution of the principal axes of the expanding ellipsoid, their expansionrates and the temperature is shown, for different equations of state anddifferent initial conditions. Here parameters were chosen so that the finalstates of all cases are the same: T0 = 200 MeV, X = 11 fm, Y = 12 fm,Z = 13 fm, Ẋ0 = 0.6, Ẏ0 = 0.5, Ż0 = 0.7.ConclusionsWe have seen that the same hydrodynamical final state can be achievedwith different solutions, equations of state or initial conditions A very im-portant consequence is also, that the hydrodynamic scaling laws observedin relativistic heavy ion collisions [16] do not depend on the initial con-ditions or the equation of state separately - one needs to restrict one inorder to extract information on the other. These examples show that in or-der to extract the initial conditions or the equation of state from hadronic(final-state) observables, one needs further restrictions or experimental con-straints. With penetrating probes one has access to the earlier times of theevolution thus can restrict results on initial conditions or equation of state.AcknowledgementsThe author wishes to thank Tamás Csörgő and Márton Nagy for theirvaluable discussions, as well as the organizers of the IV Workshop on Par-ticle Correlations and Femtoscopy for their kind hospitality. This researchwas supported by the OTKA grants NK73143 and T049466, as well as theexchange program of the Hungarian Academy of Sciences and the PolishAcademy of Arts and Sciences.csanad˙wpcf08 printed on April 5, 2019 50τr/-10 -5 0 5 100T/T012345678910Temperature distribution from d=1 relativistic solutions = 1.0κ = 0.0, λ = 1.0κ = 0.5, λ = 1.0κ = 1.0, λ = 1.0κ = 1.2, λ = 1.0κ = 2.0, λ = 2.00τt/0τr/ 0τr/-10 -5 0 5 100T/T012345678910Temperature distribution from d=1 relativistic solutions = 1.0κ = 0.0, λ = 1.0κ = 0.5, λ = 1.0κ = 1.0, λ = 1.0κ = 1.2, λ = 1.0κ = 2.0, λ = 4.00τt/0τr/0τr/-10 -5 0 5 100T/T012345678910Temperature distribution from d=1 relativistic solutions = 1.0κ = 0.0, λ = 1.0κ = 0.5, λ = 1.0κ = 1.0, λ = 1.0κ = 1.2, λ = 1.0κ = 2.0, λ = 6.00τt/0τr/ 0τr/-10 -5 0 5 100T/T012345678910Temperature distribution from d=1 relativistic solutions = 1.0κ = 0.0, λ = 1.0κ = 0.5, λ = 1.0κ = 1.0, λ = 1.0κ = 1.2, λ = 1.0κ = 2.0, λ = 10.00τt/0τr/Fig. 1. These plots show the temperature distribution of a d = 1 exact relativisticsolutions from section 2, for different λ parameters and different initial temperaturedistributions. All approximate the same final temperature distribution (T ≈ T0).REFERENCES[1] T. Csörgő et al., Phys. Rev. C67, 034904 (2003).[2] T. Csörgő et al., Phys. Lett. B565, 107 (2003).[3] Y. M. Sinyukov and I. A. Karpenko, Acta Phys. Hung. A25, 141 (2006).[4] L. D. Landau, Izv. Akad. Nauk SSSR Ser. Fiz. 17, 51 (1953).[5] I. M. Khalatnikov, Zhur. Eksp. Teor. Fiz. 27, 529 (1954).[6] S. Z. Belenkij and L. D. Landau, Nuovo Cim. Suppl. 3S10, 15 (1956).[7] R. C. Hwa, Phys. Rev. D10, 2260 (1974).[8] C. B. Chiu, E. C. G. Sudarshan, and K.-H. Wang, Phys. Rev.D12, 902 (1975).[9] J. D. Bjorken, Phys. Rev. D27, 140 (1983).[10] B. B. Back et al., Phys. Rev. Lett. 87, 102303 (2001).[11] I. G. Bearden et al., Phys. Rev. Lett. 88, 202301 (2002).[12] T. Csörgő, M. I. Nagy, and M. Csanád, Phys. Lett. B663, 306 (2008).[13] A. Bialas, R. A. Janik, and R. B. Peschanski, Phys. Rev. C76, 054901 (2007).[14] T. Csörgő, Acta Phys. Polon. B37, 483 (2006).[15] M. I. Nagy, T. Csörgő, and M. Csanád, Phys. Rev. C77, 024908 (2008).[16] M. Csanád et al., Eur. Phys. J. A38, 363 (2008).6 csanad˙wpcf08 printed on April 5, 20190τr/-10 -5 0 5 100T/T012345678910Temperature distribution from d=1 relativistic solutions = 1.0κ = 1.0, λ = 2.0κ = 1.0, λ = 3.0κ = 1.0, λ = 4.0κ = 1.0, λ = 5.0κ = 1.0, λ = 2.00τt/0τr/ 0τr/-10 -5 0 5 100T/T012345678910Temperature distribution from d=1 relativistic solutions = 1.0κ = 1.0, λ = 2.0κ = 1.0, λ = 3.0κ = 1.0, λ = 4.0κ = 1.0, λ = 5.0κ = 1.0, λ = 4.00τt/0τr/0τr/-10 -5 0 5 100T/T012345678910Temperature distribution from d=1 relativistic solutions = 1.0κ = 1.0, λ = 2.0κ = 1.0, λ = 3.0κ = 1.0, λ = 4.0κ = 1.0, λ = 5.0κ = 1.0, λ = 6.00τt/0τr/ 0τr/-10 -5 0 5 100T/T012345678910Temperature distribution from d=1 relativistic solutions = 1.0κ = 1.0, λ = 2.0κ = 1.0, λ = 3.0κ = 1.0, λ = 4.0κ = 1.0, λ = 5.0κ = 1.0, λ = 10.00τt/0τr/Fig. 2. Temperature distribution is shown for a d = 1 exact relativistic solutionfrom section 2 with different initial temperatures and equations of state (κ). Allapproximate the same final temperature distribution. 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5  0X,Y,Z [fm]t [fm/c]X(t), κ=1.0Y(t), κ=1.0Z(t), κ=1.0X(t), κ=1.5Y(t), κ=1.5Z(t), κ=1.5X(t), κ=3.0Y(t), κ=3.0Z(t), κ=3.0X(t), κ=10.Y(t), κ=10.Z(t), κ=10.-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5  0Xv,Yv,Zv [c]t [fm/c]Xv(t), κ=1.0Yv(t), κ=1.0Zv(t), κ=1.0Xv(t), κ=1.5Yv(t), κ=1.5Zv(t), κ=1.5Xv(t), κ=3.0Yv(t), κ=3.0Zv(t), κ=3.0Xv(t), κ=10.Yv(t), κ=10.Zv(t), κ=10. 0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5  0T [GeV]t [fm/c]T(t), κ=1.0T(t), κ=1.5T(t), κ=3.0T(t), κ=10.Fig. 3. Time evolution of the principal axes of the expanding ellipsoid (X,Y, Z),their expansion rates (Ẋ, Ẏ , Ż, denoted by Xv etc.), as well as that of the temper-ature (T ) is plotted, for different equations of state (κ). It is evident that differentequations of states combined with different initial conditions can lead to the samefinal state. The solution used here is the exact non-relativistic solution of section 2.

Initial conditions, equations of state and final state in hydrodynamics

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