Unified description of Bjorken and Landau 1+1 hydrodynamics
aa r X i v : . [ nu c l - t h ] J un Unified description of Bjorken and Landau1+1 hydrodynamics
A.Bialas a , R.A.Janik a , and R.Peschanski b ∗ a M.Smoluchowski Institute of Physics, Jagellonian University,Reymonta 4, 30-059 Krakow, Poland. b CEA/DSM/SPhT Saclay Unit´e de Recherche associ´ee au CNRSCEA-Saclay, F-91191 Gif/Yvette Cedex, France.
Abstract
We propose a generalization of the Bjorken in-out Ansatz for fluidtrajectories which, when applied to the (1+1) hydrodynamic equa-tions, generates a one-parameter family of analytic solutions interpo-lating between the boost-invariant Bjorken picture and the non boost-invariant one by Landau. This parameter characterises the proper-time scale when the fluid velocities approach the in-out Ansatz. Wediscuss the resulting rapidity distribution of entropy for various freeze-out conditions and compare it with the original Bjorken and Landauresults.
There is an accumulating evidence that hydrodynamics may be relevant forthe description of the medium created in high-energy heavy ion collisions[1]. Indeed, experimental measurements such as the elliptic flow [2] showsthe existence of a collective effect on the produced particles which can bedescribed in terms of a motion of the fluid. More precisely, numerical sim-ulations of the hydrodynamic equations describe quite well the distribution ∗ e-mails: [email protected] , [email protected] , [email protected]
1f low- p ⊥ particles [1], with an equation of state close to that of a “perfectfluid” with a rather low viscosity. This evidence is of course indirect, sinceit relies on assumptions about the initial and final stages of the evolution ofthe fluid. Thus some doubts can be cast either on the full thermalizationof the medium, or on the possibility of accounting for some viscosity of thefluid [3]. Also, hydrodynamics are not expected to work for leading particles i.e. near the kinematic light-cone.Given these objections, it is important to separate precisely the conse-quences of the hydrodynamic flow from those of the initial and final con-ditions. From that point of view, it seems tempting to discuss a simplifiedpicture which can be qualitatively understood in physical terms. One suchsimplification, which we are going to follow in this paper, is the idea thatthe evolution of the system before freeze-out is dominated by the longitudi-nal motion [4, 5] and thus, in fact, the hydrodynamic transverse motion canbe neglected or at least factorized out. Thus we shall consider the (1 + 1)dimensional system.On the theoretical grounds, there are also quite appealing features forapplying hydrodynamic concepts to high-energy heavy-ion reactions. Suchconcepts have been already introduced some time ago and find a plausiblerealization nowadays. The fact that a rather dense medium is created in thefirst stage of the collision allows one to admit that the individual partonic orhadronic degrees of freedom are not relevant during the early evolution of themedium and justifies its treatment as a fluid. Moreover, the high quantumoccupation numbers allow one to use a classical picture and to assume thatthe “pieces of fluid” may follow quasi-classical trajectories in space-time,expressed as an in-out cascade [6] with the straight-line trajectories startingat the origin, i.e. y = η (1)where y = 12 log (cid:18) E + pE − p (cid:19) ; η = 12 log (cid:18) t + zt − z (cid:19) (2)are respectively the rapidity and “space-time rapidity” of the piece of thefluid.Note, for further use, that (1) can be rewritten in the form2 y = log u + − log u − = log z + − log z − (3)where u ± = e ± y are the light-cone components of the fluid (four-)velocityand z ± = t ± z are the light-cone kinematical variables.2aking (1) as the starting point and using the perfect fluid hydrodynam-ics, Bjorken developped in his seminal paper [5] a suggestive (and very usefulin many applications) physical picture of the central rapidity region of highlyrelativistic collisions of heavy ions. In this picture the condition (1) leads toa boost-invariant geometry of the expanding fluid and thus to the centralplateau in the distribution of particles.It is now experimentally established [7], however, that the central rapidityregion of heavy ion collisions is better described by a Gaussian fit with a widthproportional to Y = log s , the total rapidity range of the secondaries. Thisfinding has renewing interest [8] for the pioneering hydrodynamic descriptionby Landau [4] where, indeed, a Gaussian-like distribution of the fluid wasobtained. For the same reason, new families of 1+1 relativistic hydrodynamicsolutions have been recently proposed [9, 10].In the present paper we propose to study a generalization of the formula(1) for the classical trajectory which (as we show in the following) interpolatesnaturally between the Landau and Bjorken pictures:2 y = log u + − log u − = log f + ( z + ) − log f − ( z − ) (4)where f ± ( z ± ) are a priori arbitrary functions. They have to be determinedfrom the hydrodynamic equations.The hydrodynamic equations are rewritten in terms of light- cone vari-ables in the next section. The consequences of the quasi-classical Ansatz(1) and of the generalized one (4) are dicussed in Section 3 where also thecorresponding solutions of the hydrodynamical equations are derived. Var-ious selections of the freeze-out conditions are discussed in Section 4. Ourconclusions and comments are listed in the last section. We consider the perfect fluid for which the energy-momentum tensor is T µν = ( ǫ + p ) u µ u ν − pη µν (5)where ǫ is the energy density, p is the pressure and u µ is the 4-velocity. Weassume that the energy density and pressure are related by the equation ofstate: ǫ = gp (6)3here 1 / √ g is the sound velocity in the liquid. Using u ± ≡ u ± u = e ± y (7)and introducing z ± = t ± z = z ± z = τ e ± η → ( ∂∂z ± ∂∂z ) = ∂∂z ± ≡ ∂ ± (8)where τ = √ z + z − is the proper time and η is the spatial rapidity of the fluid,the hydrodynamic equations ∂ µ T µν = 0 (9)take the form ∂ ± T + 12 ∂ + ( T ± T ) − ∂ − ( T ∓ T ) = 0 . (10)Using now (5) and the equation of state (6) we deduce from this g∂ + [log p ] = − (1+ g ) ∂ + y − g − e − y ∂ − yg∂ − [log p ] = (1+ g ) ∂ − y + g − e y ∂ + y . (11)These are two equations for two unknowns which describe the state of theliquid: the pressure p and the rapidity y . They should be expressed in termsof the positions z + , z − in the liquid. Other thermodynamic quantities can beobtained from the equation of state (6) and the standard thermodynamicalidentities: p + ǫ = T s ; dǫ = T ds (12)where we have assumed for simplicity vanishing chemical potential.The result is ǫ = gp = ǫ T g +1 ; s = s T g → s ∼ ǫ g/ ( g +1) . (13)Note that (11) implies the consistency condition ∂ + ∂ − y = g − g ) (cid:8) ∂ − ∂ − [ e − y ] − ∂ + ∂ + [ e +2 y ] (cid:9) . (14)4 Generalized in-out Ansatz
The simplest possibility to describe the expansion of the fluid was suggestedby Bjorken [5] who proposed to use the Ansatz (1) in the hydrodynamicalcontext. Introducing (1) into (11) we obtain g∂ + [log p ] = − g z + ; g∂ − [log p ] = − g + 12 z − (15)from which we deduce p = ǫ g − = p ( z + z − ) − ( g +1) / g = p τ − ( g +1) /g , (16)where p is a constant. Thus the system is boost-invariant: the pressure doesnot depend neither on η nor on y . So are ǫ , s and T , given by (13). The data on both nucleon-nucleon and nucleus-nucleus collisions (see, e.g. [7]) show that the produced system strongly violates boost invariance (exceptperhaps in a narrow region of small c.m. rapidities). It is thus necessary togo beyond (1). As already indicated in the Introduction, we propose to study-as a simplest generalization of (1)- the Ansatz (4). Introducing (4) into (14)we obtain f − ∂ − ∂ − ( f − ) = f + ∂ + ∂ + ( f + ) = A / A is a constant. Thus both f + and f − satisfy an identical equation: f f ′′ = A / . (18)Note that A = 0 implies f ′′ = 0 and thus we recover the condition (1).We conclude that A describes the deviation of the system from the uniformHubble-like expansion.Eq. (18) can be solved multiplying by f ′ and dividing by f :[( f ′ ) ] ′ = A [log f ] ′ → f ′ = A p log( f /H ) (19)where H is an arbitrary constant. 5q. (19) can be solved in the standard manner. We obtain z − z = h Z FF dF ′ p log( F ′ ) (20)where we have introduced the notation F = f /H ; h = H/A . (21)When (4) is introduced into (19) we obtain g∂ + [log p ] = − (1+ g ) F ′ + F + + g − F ′− F + g∂ − [log p ] = − (1+ g ) F ′− F − + g − F ′ + F − . (22)From this we deduce g log p = − (1+ g ) F + + g − F ′− Z dz + F + + ∆ − ( z − ) g log p = − (1+ g ) F − + g − F ′ + Z dz − F − + ∆ + ( z − ) . (23)The integrals on the R.H.S. can be evaluated using (19). Indeed Z dzF = Z dFF F ′ = h Z d log F √ log F = 2 h p log F . (24)We thus obtain g log p = − (1+ g ) F + F − ) + g − p log( F + ) log( F − ) . (25)where, for the two equations (23) to be consistent with each other, we hadto take ∆ ± ( z ± ) = − (1+ g ) log F ± . This finally gives p ( z + , z − ) = p exp (cid:26) − (1+ g ) g (cid:2) l + l − (cid:3) + g − g l + l − (cid:27) (26) y ( z + , z − ) = 12 (cid:0) l − l − (cid:1) (27)6
20 40 60 80 100 1201.41.210.80.60.40.20 t /hy ç – ç=1/2ç=1ç=2 ç=4ç=8 Figure 1: The ratio y/η plotted versus proper-time τ /h for several values ofspace-time rapidity η . The boost-invariant picture corresponds to y/η ≡ . The “full-stopping” initial condition is y/η = 0 at τ = 0 . The asymptotic-value is still 1 , but it is reached only at very large proper-times.where, by definition l ± ( z ± ) = p log( F ± ) . (28)This completes the solution of hydrodynamic equations constrained bythe generalized in-out cascade (4). One sees that the pressure depends onboth τ and η . Thus the system is not boost-invariant but boost invariance isrecovered in the limit h → z ± fixed (see Eq. (20)). On the other hand, itcan be remarked that the Landau asymptotic solution [4] can be recoveredin the limit h fixed, z ± → ∞ . Other thermodynamic parameters are obtained from (13), giving s ∼ ǫ g/ (1+ g ) = s exp (cid:26) − g (cid:2) l + l − (cid:3) + g − l + l − (cid:27) = s exp ( − gθ ) (29)7here we have denoted θ ≡ log( T /T ) = 1 + g g (cid:2) l + l − (cid:3) − g − g l + l − (30)with s and T denoting the entropy and temperature at the beginning ofthe hydrodynamic evolution.To illustrate the deviation of our solution (27) from the in-out Bjorken-Gottfried-Low Ansatz (1), we show in the Fig.1 the ratio y/η vs τ /h (theproper time measured in units of h ) for several (fixed) values of η . One seesthat in the limit τ /h → y vanishes. In this limit we have ( e.g. for the region y ≥ y ≈ ( τ / h ) sinh 2 η = ( t/ h ) − ( z/ h ) ≤
14 ( t/h ) . (31)Thus for a fixed (small) t/h the fluid starts at rest and acquires somevelocity at later times, as in the Landau “full stopping” solution. At largetimes, τ → ∞ , one obtains y ≈ η , i.e. the in-out Ansatz (1) is approximatelyrecovered. Thus our solution does indeed interpolate between the Landauand Bjorken hydrodynamics.A last comment is in order. In all cases, the solution of the flow is alsodefined outside the kinematical light-cone. Indeed, there is some flow ofenergy entering the light-cone from outside. It could be physically inter-preted as mimicking energy sources on the light-cone (“leading particle ef-fect”). However, the relevance of hydrodynamical models near the light-coneis questionable. The observable results of the model depend in an essential way on the as-sumed freeze-out surface. The point is that the densities s and ǫ which enterthe hydrodynamic equations are densities per unit volume in the rest frameof the fluid . But we are generally interested in the distribution of entropy dS/dy and/or of energy dE/dy densities per unit of rapidity, as these quan-tities are possible to measure. For given s and ǫ , dS/dy and dE/dy dependon the hypersurface at which the hydrodynamic evolution stops and the fluidchanges into particles (freeze-out surface). To fix attention, in the followingwe discuss the entropy density. 8 .1 General freeze-out surface The evaluation of the entropy density per unit of rapidity for a given freeze-out surface can be performed in two steps.First we evaluate the amount of entropy in an infinitesimal volume alongthe freeze-out surface: dS = su µ dσ µ (32)where u µ is the 4-velocity of the fluid and dσ µ is the 4-vector orthogonal tothe surface satisfying dσ µ dσ µ = dz µ dz µ = dz + dz − (33)Consider the (space-like) surfaceΦ( z + , z − ) = C (34)where C is a constant. ThenΦ + dz + + Φ − dz − = 0 ; Φ ± ≡ ∂ Φ ∂z ± . (35)It follows that the unit vector orthogonal to the surface is n + = Φ − √ Φ + Φ − ; n − = Φ + √ Φ + Φ − . (36)The infinitesimal length along the surface is dσ = p dz + dz − = dz − p Φ − / Φ + = dz + p Φ + / Φ − (37)Therefore u µ dσ µ = [ u + Φ + + u − Φ − ] dz − + = [ u + Φ + + u − Φ − ] dz + − . (38)In the second step we express the infinitesimal volume along the freeze-out surface in terms of the infinitesimal interval of rapidity. This can be doneusing the relation (20) which gives z ± as a function of F ± = exp (cid:0) l ± (cid:1) . Wehave dz ± = dz ± dF ± dF ± dl ± dl ± = h exp (cid:0) l ± (cid:1) l ± dl ± (39)9sing this relation and (20), the R.H.S. of (38) can be expessed in terms of l + and l − .This in turn can be expressed in terms of rapidity y using (4) and thecondition (34) which gives an additional relation between z + and z − and thusfollowing (20) also between l + and l − . In particular, using the differentialforms, we have dl ± = ± ∓ e l ∓ /l ∓ Φ + exp ( l ) /l + +Φ − exp ( l − ) /l − dy → dz ± = ± h Φ ∓ Φ + l − e − l − +Φ − l + e − l dy (40)and thus, finally u µ dσ µ = u + Φ + + u − Φ − Φ + l − e − l − + Φ − l + e − l dy = e ( l + l − ) Φ + e y + Φ − e − y l − Φ + e y + l + Φ − e − y . (41) In this section we take the surface at t = const, for a first example. In thenotation from the previous section we writeΦ( z + , z − ) = t = 12 ( z + + z − ) = const ; Φ + = Φ − = 12 (42)Using (41) we thus have dS = su µ dσ µ = s e ( l + l − ) l + + l − + ( l + − l − ) tanh y dy == s e − ( g − l + − l − ) / l + + l − + ( l + − l − ) tanh y dy . (43)If, following Landau, we approximate both l + and l − by large constants,then for finite y the difference ( l + − l − ) is small and we have dS ∼ e − ( g − l + − l − ) / . (44)For g = 3 this formula is identical to the asymptotic result of Landau[4]. This can be seen when displaying the distribution dS/dy ; In Fig.2, oneshows dS/dy with t = const. (formula (43)) compared with the Landauapproximation, formula (44), for different values of the parameter h , which,10 Sdylog
Figure 2: The curve dS/dy with t = const. (cf. formula (43)) comparedwith the Landau approximation (cf. (44), dashed lines). The kinematicalend-points at y max correspond to z − = 0 ( y ≥ t = const. implies adifferent relation between l ± and y than the condition τ ∼ const. , which isthe freeze-out condition considered by Landau [4]. As discussed in the nextsubsection, this leads to a rather different shape of the distribution dS/dy . To investigate the relation to the Landau solution and its comparison withthe Bjorken one we consider the freeze-out at a fixed proper time. More precisely, Landau discusses the limitation of the 1+1 dimensional motion by itstransition to the 3+1 dimensional hydrodynamical motion and shows that it appears at τ ∼ const. playing the role of a freeze-out surface. z + , z − ) = z + z − = τ = const. (45) z + dz − + z − dz + = 0 ; Φ ± = z ∓ (46)and thus dS = he − ( g − l + − l − ) / e y z − + e − y z + l − e y z − + l + e − y z + dy . (47)This is a general formula. When supplemented by (4) and (20), it expresses dS in terms y and τ .When h → z ± = h F ± l ± (48)to obtain dS = h l + + l − l + l − e − g − ( l + − l − ) dy . (49)For l ± → ∞ and fixed y one recovers the Landau result (44).The result given by (47) is plotted in Fig.3 where dS/dy , is displayedfor different values of the parameter h and compared with the approximateformula dSdy = S e √ L − y (50)obtained by Landau [4]. The parameter L was adjusted to obtain the correctslope at y = 0.One observes some deviations from the perfect Gaussian which was con-sidered in a simplified version [12] of the Landau model (and which agrees-if the multiplicity distribution is assumed proportional to the entropy- withthe data [7]). Note that at fixed τ and h → , the distribution becomessignificantly flatter, going smoothly to the Bjorken limit at h = 0 . Instead of considering the freeze-out surface at the limit where the transversemotion becomes relevant (cf. [4]), a natural conjecture is to consider freeze-out at a fixed temperature, i.e. when the temperature reaches the valuewhere pions are expected to become liberated, e.g. [1, 11].12
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Figure 3: The curve dS/dy with τ = const. for various values of τ /h. Fulllines: Eq.(49); Dashed lines: Eq.(50). Note that, for clarity, the curves wereshifted by log τ /h .Denoting the initial temperature by T and the freeze-out temperatureby T F we haveΦ( z + , z − ) ≡ g (cid:2) l + l − (cid:3) − g − l + l − = g log( T /T F ) ≡ gθ = const. (51)Hence Φ + = 12 [( g + 1) l + − ( g − l − ] l ′ + = ( g + 1) l + − ( g − l − h exp ( l )Φ − = 12 [( g + 1) l − − ( g − l + ] l ′− = ( g + 1) l − − ( g − l + h exp ( l − ) (52)where we have used the relation following from (20): l ′ = d ( √ log F ) dz = F ′ F √ log F = 12 hF = e − l / h . (53)13herefore using (29) and (41) we have dS ∼ e − ( g − l + − l − ) / l + + l − ( g + 1) l + l − − g − ( l + l − ) . (54)Now, the relations1 + g (cid:2) l + l − (cid:3) − g − l + l − = gθ ; ; l − l − = 2 y (55)imply l − = g + 12 θ − y + g − p θ − y /g ; l = 2 y + l − (56)giving ( g + 1) l + l − − g −
12 ( l + l − ) = 2 g p θ − y /g ; g −
14 ( l + − l − ) = g − h θ − p θ − y /g i l + + l − = √ y h θ − p θ − y /g i − / . (57)Thus we finally obtain dS ∼ e g − h √ θ − y /g − θ i (cid:16) θ − p θ − y /g (cid:17) − / ydy p θ − y /g . (58)One sees that this formula exibits a singularity by its transition to the3+1 dimensional hydrodynamical motion at y = gθ which is of courseunphysical and reflects the singular initial conditions of our approach and theexpected limitation of hydrodynamic models to the more central scatteringregion.It is, however, interesting to observe that the hypersurface T =const isonly partly space-like. It becomes time-like at the rapidity determined fromthe condition Φ − = 0, i.e., ( g + 1) l − = ( g − l + , giving (c.f. (55)) y m = 2 gg + 1 θ ; l = ( g + 1) θ. (59)This is illustrated in Fig.4, where two profiles θ = const. are displayed. The singularity in dS/dy does not come from a true singularity in the kinematicaldomain but is due to a maximum value reached by the rapidity y as a function of z − . θ = const. for two values of θ = log T /T (continuouslines). The comparison is made with fixed τ (dashed lines). Note that adifferent value of the parameter h has been chosen to obtain a comparablerange in space-time.The numerical estimates show that -for large enough θ ( θ ≥ y is not signifi-cant in the region y ≤ y m . This is shown in Fig.5 where dS/dy is plotted forseveral values of θ ≥
2. One sees that all distributions are close to Gaussians.The slope, however, is rather small, certainly smaller than required by data,unless one considers a larger value of the parameter g, i.e. a smaller value ofthe speed of sound. We have investigated longitudinal hydrodynamic expansion of a perfect fluidforming an infinitely thin layer at the initial time and satisfying the equationof state with an arbitrary sound velocity. We proposed a generalized in-15
Sdylog Figure 5: The curve dS/dy with θ = const. for various values of θ , y < θ/ h defining a dynamical scale in configuration space.(iii) The resulting entropy distribution in rapidity, dS/dy , was evaluatedand shown to depend significantly on the assumed condition for the freeze-out.(a) For freeze-out at a fixed proper-time the density is close to (but withsome deviation, particularly at small τ, from) a Gaussian which is tradi-tionally attributed to Landau solution. It tends smoothly to the Bjorken16oost-invariant solution for h → T /T F is large enough.(iv) It is worthwhile to note that the freedom in the choice of the value ofthe sound velicity may be helping in phenomenological applications of theseresults to data.Compared to other recent (1 + 1) hydrodynamical models [9, 10], oursolution is mainly characterized by the smooth, one-parameter dependenttransition between the Bjorken and Landau hydrodynamical models and byits analytic simplicity. It would be useful to study further the classificationof all the solutions in a unified framework.More generally, there is a clear need for an extension of our investigationto include more flexible initial conditions, relaxing the point-like character ofthe fluid at the beginning of the evolution. This, however, demands a moresophisticated analysis (e.g. an application of the general recipe of [13]) andgoes beyond the scope of this paper.On a theoretical ground, it would be interesting to have a physical in-terpretation of the generalized Ansatz (4), which appears as a mathematicalharmonic property ∂ + ∂ − y = 0 of the hydrodynamical flow. In particular,an extension to this case of the Gauge/Gravity correspondence applied inRef.[14] to the Bjorken Boost-invariant flow, would be insightful.17 ppendix. Solution of the equation (20) We rewrite (20) as ( z − ζ ) = h Z FF dv √ log v (60)where F = f /H and h = H/A .Changing the variable of integration:log v = u ; 2 udu = dv/v ; dv/u = 2 e u du (61)we arrive at z − ζ = 2 hF Z √ log F √ log F e u − log F du = 2 HFA h D (cid:16)p log F (cid:17) − D (cid:16)p log F (cid:17)i (62)This integral is the so called Dawson’s integral: D ( x ) = e − x R x e u du . Forlarge √ log F it approaches (2 √ log F ) − and thus we obtain z − ζ ≈ HFA √ log F (63)The asymptotic expansion of D ( x ) is D ( x ) = 12 x ∞ X n =0 Γ( n + 1 / /
2) 1 x n (64)For small x one can evaluate this integral effectively by the series expan-sion: D ( x ) = e − x Z x ∞ X n =0 x nn ! = xe − x ∞ X n =0 x n (2 n + 1) n ! (65)Thus we have in this case z − z = 2 HA "p log( F ) ∞ X n =0 (log F ) n (2 n + 1) n ! − p log( F ) ∞ X n =0 (log F ) n (2 n + 1) n ! (66)18 cknowledgements We thank Wojtek Florkowski, Jean-Yves Ollitrault and Kacper Zalewski foruseful comments. This investigation was partly supported by the MEiN re-search grant 1 P03B 045 29 (2005-2008) and by 6 Program of EuropeanUnion “Marie Curie Transfer of Knowledge” Project: Correlations in Com-plex Systems “COCOS” MTKD-CT-2004-517186.
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