Similar final states from different initial states using new exact solutions of relativistic hydrodynamics
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Similar final states from different initial states usingnew exact solutions of relativistic hydrodynamics
M. Csan´ad , a , M. I. Nagy , b , and T. Cs¨org˝o , c ELTE, E¨otv¨os Lor´and University, H - 1117 Budapest, P´azm´any P. s. 1/A, Hungary MTA KFKI RMKI, H-1525 Budapest 114, POBox 49, Hungary
Abstract.
We present exact, analytic and simple solutions of relativistic perfectfluid hydrodynamics. The solutions allow us to calculate the rapidity distributionof the particles produced at the freeze-out, and fit them to the measured rapiditydistribution data. We also give an advanced estimation of the energy densityreached in heavy ion collisions, and an improved estimation of the life-time of thereaction.
Since the birth of hydrodynamics, there were many interesting exact solutions found to thecomplicated non-linear coupled differential equations of them. In contemporary research, suchas the description of collective properties of high-energy elementary particle and heavy ionreactions, one has to deal with not only hydrodynamics but relativistic hydrodynamics. Thisis an even more complicated topic, and likely this is the reason why only a few exact solutionsexist for relativistic hydrodynamics, in contrast to the nonrelativistic case.Exact solutions are important in at least two ways: first, they can be used to test numericalcodes reliably. Second, they provide an invaluable insight into the details of the evolution ofthe matter created in high-energy reactions. After recalling some presently known solutions,such as the Landau-Khalatnikov solution and the Hwa-Bjorken solution, we present a classof accelerating exact and explicit solutions of relativistic hydrodynamics. These solutions areadvantageous compared to the previously mentioned ones. Then we show how the new solutionscan be applied to describe the evolution of the matter created in high energy reactions. Ourtreatment uses simple formulas and takes the presence of acceleration into account.This article is based on refs. [1,2,3]. These results are extended here with more detailedsimulations that indicate that even for a fixed equation of state, different initial conditionscould lead to the same freeze-out distributions at mid-rapidity. However, the width of therapidity distribution can be utilized to select from among these time evolution scenarios.
In this section we recapitulate the equations of relativistic hydrodynamics. We will use thefollowing notations: x µ = ( t, r ) is the coordinate four-vector, r = ( r x , r y , r z ) is the coordinatethree-vector. The metric tensor is g µν = diag (1 , − , − , − u µ ,the normalization is u µ u µ = 1. The three-velocity v is defined as u µ = γ (1 , v ), with γ = a e-mail: [email protected] b e-mail: [email protected] c e-mail: [email protected] Will be inserted by the editor (cid:0) − v (cid:1) − / . We denote the so-called comoving derivative by dd t , that is, dd t = ∂∂t + v ∇ . Thethermodynamical quantities are the following: ε is the energy density, p is the pressure, w = ε + p is the enthalpy density, T is the temperature, σ is the entropy density. When there are some(conserved or non-conserved) charges present, we denote them by n i , and the correspondingchemical potentials by µ i .The equations of relativistic hydrodynamics are obtained in the simplest way by Landau’sheuristic argumentation: for perfect fluid (without viscosity and heat conductivity) the energy-momentum tensor is T µν = diag ( ε, p, p, p ) (1)in the local rest frame, so we have T µν = wu µ u ν − pg µν (2)in arbitrary frame. Projecting the energy-momentum conservation law ∂ ν T µν = 0 orthogo-nal and parallel to u µ , we obtain the relativistic Euler equation and the energy conservationequation as wu ν ∂ ν u µ = ( g µρ − u µ u ρ ) ∂ ρ p, (3) w∂ µ u µ = − u µ ∂ µ ε. (4)The charge conservation equation (for one charge) is ∂ µ ( nu µ ) = 0 , (5)but if there are many different particles with the same charge, then this has to be supplementedby the chemical potentials. For instance, in case of baryonic or electric charge, particles andantiparticles carry opposite charges, and they chemical potentials are the same but of oppositesign, so we have X i µ i ∂ µ ( n i u µ ) = 0 . (6) In this section we summarize some already known exact results: the renowned Landau-Khalat-nikov solution, the Hwa-Bjorken solution and other recent multi-dimensional solutions.
Landau invented relativistic hydrodynamics in the early 1950s and proposed the applicationof it to describe high-energy particle reactions (in that time mostly cosmic ray events). In acollaboration with I. M. Khalatnikov, they found the first exact solution [4,5], which is a 1+1dimensional implicit solution, with ε = 3 p equation of state (EoS). The Landau-Khalatnikovsolution has nice realistic features: it is an accelerating one, the initial condition describes afinite piece of matter at rest, then it starts to expand. Another important feature is that therapidity distribution of the particles is approximately Gaussian. However, the solution itself isvery complicated, since the independent variables — the time and spatial coordinate — aregiven in terms of horrendous integral formulas involving Bessel functions integrated over thetemperature and the fluid rapidity “variables”. Thus we don’t quote the whole result here. ill be inserted by the editor 3Case λ d κ φ λ a.) 1 ∈ R ∈ R ∈ R d ∈ R κ +1 κ d.) ∈ R d − κ +1 κ e.) ∈ R Table 1.
The new family of solutions.
The Hwa-Bjorken solution [6,7] is a 1+1 dimensional, expanding accelerationless solution. It issimple and explicit, this is its main advantage compared to the Landau-Khalatnikov solution.It is written down in the easiest way in Rindler-coordinates τ and η , which are defined by t = τ cosh η , r = τ sinh η (7)with r being the spatial coordinate. The Hwa-Bjorken solution is given by the velocity andentropy profiles v = tanh η = rt , σ = σ τ τ , (8)where the subscript refers to the initial condition. It works for ǫ + B = κ ( p + B ), for arbitraryvalues of κ and B . This boost-invariant solution results in a constant rapidity distribution of theparticles. This is a rough prediction clearly not valid at the present experimental situations.However, Bjorken has given a simple estimate of the initial energy density of the reactionbased on this solution, this is why it became so renowned. In the next section we present suchsolutions that are simple and explicit (as the Hwa-Bjorken solution) and are accelerating, andyield realistic rapidity distributions (as the Landau-Khalatnikov solution). The Landau-Khalatnikov solution contains acceleration, and yields a finite, realistic, Gaussian-like rapidity distribution. On the other hand, the Hwa-Bjorken solution is very simple and easyto handle, this is why it became so important, although the accelerationless, boost-invariantflow profile and the constant rapidity distribution obviously does not agree with the results ofreal high-energy experiments.We found such new, analytic, explicit and simple solutions, which do not lack acceleration,and yield finite, realistic rapidity distributions [1,2]. We found solutions for 1+1 dimensions,and also spherical flows in arbitrary number of spatial dimensions ( d will denote their number).For presenting the solutions we use Rindler-coordinates, which are defined by eq. (7) for t > r ,that is ,,inside the lightcone”. (From now on t is the time and r is the radial coordinate in the1+d dimensional case, and the single spatial coordinate in the 1+1 dimensional case.)For a discussion of the mathematical derivations in greater detail, apart from Refs. [1,2],we recommend Ref. [3]. Here we present only the main results. The velocity and the pressureis given by v = tanh( λη ) , p = p (cid:16) τ τ (cid:17) λd κ +1 κ φ λ ( d − η . (9)The value of the constants λ , φ λ , d and κ are constrained: different set of values yield differentsolutions. Table 1 shows the possible cases, every row of the table being a solution. In thefollowing we discuss them separately. – Case a) isn’t a new solution, it is just the well-known Hwa-Bjorken solution in 1+1 di-mensions, and the only recently discovered Buda-Lund type of solutions [12,13] in 1+ddimensions. We quoted it for the sake of completeness, since this solution is also a memberof the class. Apart from this case, all the other solutions possess non-vanishing acceleration.
Will be inserted by the editor – Case b) was found first, with other methods. It describes spherical expansion in d dimensions.The EoS is constrained in a way that κ must be equal to d . (For instance, in case of three-dimensional expansion κ = 3, which is the EoS of an ultra-relativistic ideal photon gas.)It is interesting to calculate the equation of trajectories R ( t ) from the velocity field inMinkowskian coordinates: v = 2 trr + t ⇒ R ( t ) = 1 a ( p a t ) + 1) (10)with a = r | r − t | . So the fluid elements have constant a acceleration in the local rest frame. – Case c ) and d ) were found first by T. S. Bir´o in 1+3 dimensions [14], we generalized themto arbitrary number of spatial dimensions. These solutions are also accelerating, and since φ λ = 0, if d = 1, the pressure field explicitly depends on η , so the solution is finite in η . – Case e ) has a remarkably general velocity field: the λ parameter can be arbitrary. Wecall λ acceleration parameter, because it somehow governs the acceleration of the flow.On the other hand, this solution works only for d = 1 and κ = 1, which is obviouslya drawback. Nevertheless, this solution can be considered as a ,,smooth extrapolation”between the previous cases. In the next section we shall use this solution to calculate therapidity distribution, and see that the width of it is controlled by the value of λ . Hencein principle, the value of the parameter λ of the hydrodynamical solution can be obtainedfrom measurements.For a more thorough review of these solutions we recommend ref. [3].For illustration, in Fig. 1 we plot the spatial temperature distribution at various pointsof the evolution of a collision for different λ values. All the plotted solutions go to the samefinal temperature at mid-rapidity ( η = 0), although their initial temperature and accelerationparameter λ are different. We shall see in the next section, that solutions with different accel-eration parameter λ have different rapidity distributions, and the acceleration parameter canbe determined from the measurement of the widths of the rapidity distribution. We calculated the rapidity distribution, d n d y of the particles numerically and (approximately)analytically as well. We used the solutions discussed in the previous section as case e. ), that is,where the λ parameter is arbitrary. We assumed that at a certain freeze-out massive particlesappear. (In the figures of this paper we used pions with m = 140 MeV, but other particles canbe used as well – and their contribution can be added to the one of pions). We have chosenthe freeze-out condition as follows: the temperature in η = 0 should reach a given T f value(subscript f means freeze-out) and the freeze-out hypersurface should be pseudo-orthogonal tothe four-velocity field u µ ( x ). The equation of this hypersurface is (cid:16) τ f τ (cid:17) λ − cosh (( λ − η ) = 1 . (11)The details of the calculation of d n d y is found in refs. [1,3]. We only quote the result here: usinga saddle-point integration in η , for λ > / m/T f ≫ ν σ ( s ) = 1 we gotd n d y ≈ d n d y (cid:12)(cid:12)(cid:12) y =0 cosh − α − (cid:16) yα (cid:17) e − mTf [ cosh α ( yα ) − ] , (12)where α = λ − λ − . Some typical cases are plotted in Fig. 2. This figure is just an illustration, theparameter values on this figure are not realistic ones. ill be inserted by the editor 5 t r/-10 -5 0 5 10 T / T Temperature distribution based on the relativistic hydro solutions = 0.0 l = 0.5 l = 1.0 l = 1.2 l = 2.0 l = 2.0 t t/ t r/ Temperature distribution based on the relativistic hydro solutions t r/-10 -5 0 5 10 T / T Temperature distribution based on the relativistic hydro solutions = 0.0 l = 0.5 l = 1.0 l = 1.2 l = 2.0 l = 4.0 t t/ t r/ Temperature distribution based on the relativistic hydro solutions t r/-10 -5 0 5 10 T / T Temperature distribution based on the relativistic hydro solutions = 0.0 l = 0.5 l = 1.0 l = 1.2 l = 2.0 l = 6.0 t t/ t r/ Temperature distribution based on the relativistic hydro solutions t r/-10 -5 0 5 10 T / T Temperature distribution based on the relativistic hydro solutions = 0.0 l = 0.5 l = 1.0 l = 1.2 l = 2.0 l = 10.0 t t/ t r/ Temperature distribution based on the relativistic hydro solutions
Fig. 1. (Color online) These plots show the temperature distribution of different one dimensionalsolutions (case e. of Table 1, with different λ parameters) with different initial temperature. All theplotted solutions go to the same final temperature at mid-rapidity ( η = 0). The λ = 1 solutioncorresponds to the Hwa-Bjorken solution. All the λ > λ = 0 case, corresponding to astatic, infinite, homogeneous medium with a constant temperature, and its domain extends beyond thelight-cone, while for λ > r < t only. -10 -8 -6 -4 -2 0 2 4 6 8 100.20.40.60.811.21.41.61.82 y=0 dydn/dydn y =1.001 l =1.035 l =1.1 l =1.5 l =200MeV, m=140MeV f T -6 -4 -2 0 2 4 60.20.40.60.811.21.41.61.82 y=0 dydn/dydn y p Kp =1.2 l =140MeV, f T Fig. 2. (Color online) Normalized rapidity distributions from the new solutions in 1+1 dimensions –case e.) of Table 1 – for various λ , T f and m values. Thick lines show the result of numerical integration,thin lines the analytic approximation from eq. (12). For λ > T f it can be used withinabout 10 % error. Now as we have an analytic approximation for the rapidity distribution, we are able to fitit to real experimental data and extract the λ acceleration parameter from them. Since ourtreatment includes acceleration effects, the new solutions provide a more realistic picture ofthe initial period of high energy reactions. In this section we show how this can be used forimproving the famous energy density estimation made by Bjorken. It is clear that initial energy Will be inserted by the editor
Fig. 3. (Color online) This figure shows that if there is no acceleration, η = η f , but for the acceleratingcase, a correction factor has to be applied. density is a quantity of crucial importance when one wants to interpret the conclusions drawnfrom high energy experiments.We follow Bjorken’s method [7] and modify it when acceleration effects become important.Let us focus on a thin transverse piece of the produced matter at mid-rapidity, as seen on by Fig.2 of ref. [7]. The radius R of this slab is estimated by the radius of the colliding hadrons or nuclei: R = 1 . A / fm. The volume is dV = ( R π ) τ d η , where τ is the proper time of observationand d η is the space-time rapidity extent of the slab. The energy content dE = h m t i dn , where h m t i is the average transverse mass at mid-rapidity, so similarly to Bjorken, the initial energydensity is ε = h m t i ( R π ) τ dndη . (13)Here τ is the proper-time of thermalization, estimated by Bjorken as τ ≈ η = η f = y , however, for accelerating solutionsone has to apply a correction of ∂y∂η f ∂η f ∂η = (2 λ −
1) ( τ f /τ ) λ − (14)(see Fig. 3). These two factors contain the acceleration effects on the energy density estimation,see ref. [3] for details.So the initial energy density ε can be accessed by an advanced estimation ε c as ε c ε Bj = (2 λ − (cid:18) τ f τ (cid:19) λ − , ε Bj = h m t i ( R π ) τ dndy . (15)Here ε Bj is the Bjorken estimation, which is recovered if d n d y is flat (i.e. λ = 1), but if λ > ε is under-estimated by the Bjorken formula. Fig. 4 shows our fits to BRAHMS dn/dy data [15]. From these fits wehave found λ = 1 . ± . ε Bj = 5 GeV/fm as given in ref. [16], and τ f /τ = 8 ± ε c = (2 . ± . ε Bj = 10 . ± . . ill be inserted by the editor 7 -4 -3 -2 -1 0 1 2 3 4100150200250300 ydydn yield - p BRAHMS 0-5% central t / f t Bj ˛ c ˛ fi E rr o r b a r /dy - p from BRAHMS 0-5% central dn Bj ˛ / c ˛ Fig. 4. (Color online) Left panel: dn/dy data of negative pions, as measured by the BRAHMS col-laboration [15] in central (0-5%) Au+Au collisions at √ s NN = 200 GeV, fitted with eq. (12) (1+3dimensional case). The fit range was − < y <
3, to exclude target and projectile rapidity region, CL= 0.6 %. Right panel: ε c /ε Bj ratio as a function of τ f /τ . As another application, let us focus on the life-time of a high energy reaction. For a Hwa-Bjorkentype of accelerationless, coasting longitudinal flow, Sinyukov and Makhlin [17] determined thelongitudinal length of homogeneity as R long = r T f m t τ Bj . (16)Here m t is the transverse mass and τ Bj is the (Bjorken) freeze-out time. This result provides ameans to determine the life-time of the reaction, if one simply identifies it with τ Bj . However,if the flow is accelerating, the case is a little bit more complicated: the estimated origin of thetrajectories is shifted back in proper-time, so the life-time of the reaction is under-estimatedby τ Bj . From our solutions we have (for a broad but finite rapidity distribution, so that thesaddle-point approximation can be used): R long = r T f m t τ c λ ⇒ τ c = λτ Bj . (17)Thus the new estimation of the life-time, τ c contains a λ multiplication factor. BRAHMSrapidity distributions in Fig. 4 yield λ = 1 . ± .
01, so they imply a 18 ± We presented a class of simple solutions of relativistic perfect fluid hydrodynamics. Becausethe new solutions are accelerating, but still explicit and exact, they can be utilized in the de-scription of high energy reactions in a more reliable way than either the Hwa-Bjorken of theLandau-Khalatnikov solution. We have shown, that even for a fixed equation of state, differ-ent initial conditions could lead to the same freeze-out distributions at mid-rapidity. However,these solutions of different acceleration parameter λ have different width of their rapidity distri-butions, and the acceleration parameter can thus be determined from the measurement of thewidths of the rapidity distribution. We have calculated the rapidity distribution of the producedparticles, and showed two possible applications: and advanced estimation of the initial energydensity of the high energy reaction and an advanced life-time estimation. We fitted the analyticapproximation of the rapidity distribution to experimental data (measured by the BRAHMScollaboration at RHIC) and determined the λ acceleration parameter of the flow. From it we Will be inserted by the editor concluded that the energy density estimations based on Bjorken’s method has to be correctedby a factor of 2. We also obtained a correction to the life-time measurements: a ∼
20% increaseis due to the presence of acceleration in the initial evolution of the matter. Thus the effectsof work, done even by the fluid cell at mid-rapidity, can be determined from the measurementof the widht of the rapidity distribution and this information is important for an advancedestimation of initial energy densities and the life-time of the rapidity distribution.
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