Lawrence Berkeley National Laboratory Recent Work
Title Fluidity and supercriticality of the QCD matter created in relativistic heavy ion collisions
Permalink https://escholarship.org/uc/item/8x54z018
Journal Physical Review C - Nuclear Physics, 81(1)
ISSN 0556-2813
Authors Liao, J Koch, V
Publication Date 2010-01-07
DOI 10.1103/PhysRevC.81.014902
Peer reviewed
eScholarship.org Powered by the California Digital Library University of California arXiv:0909.3105v3 [hep-ph] 8 Jan 2010 aet nitrrtto of interpretation on caveats uain:tecretsau ae nti prahis approach this on based an status that current the cal- culations: hydrodynamic viscous on constraints stringent rather eetdb ahrsalrtoo ha-icst over shear-viscosity the of Firstly, ratio small rather entropy-density, a by resented of energies mass con- QCD transition. the enforcing finement plasma ultimately one the magnetic of co- the components with the magnetic features and which electric existing scenario” for “magnetic [4][5][6], the suggested expla- been example microscopic have interacting, Several behavior this strongly [3]. for is nations fluidity ideal collisions nearly mat- these the with in that conjecture, produced the ter to led has observation This ueeto nuepcel ag litcanisotropy mea- elliptic the large example unexpectedly For an matter. of this surement unex- of and intriguing properties many pected (RHIC) revealed Collider years, Ion the Heavy over have, Relativistic the Synchrotron at Proton and Super (SPS) CERN collisions, be the ion at heavy can experiments of matter and means by QCD laboratory dense the in and created strong Hot in one questions physics. and is interaction challenges interesting matter most QCD the of of characterization quantitative the † ∗ yais[] tlatfrlwtases oet ( momenta hydro- transverse of low framework for the least at within [2], reproduced dynamics be can [1] lcrncades [email protected] address: Electronic lcrncades [email protected] address: Electronic ruet o h eryielfliiya etrof center at fluidity ideal nearly the for Arguments h xlrto fteQDpaedarma elas well as diagram phase QCD the of exploration The pe limit upper v fgo udt tcranlwrba nry ial ae o based Finally ion energy. heavy beam in lower created certain be at to fluidity matter good the of of diagram. fluidity phase better QCD even the an on Critical-End-Point the to respect ASnmes 23.h 57.q 47.75.+f 25.75.-q, 12.38.Mh, numbers: PACS evolution fireball the for description hydrodynamic a of ity ev o olsosi oprsnwt aiu te fluids other various with comparison in collisions ion heavy olsosa etro aseeg of energy consi mass fluidity of o its center fluidity has at good regime seemingly collisions the supercritical between its relationship diversit in possible we remarkable fluid a a perspectives, for that universality hydrodynamic certain and compar shows which to thermodynamic way both proper the from examining After supercriticality. 2 nti ae edsustefliiyo h o n es C ma QCD dense and hot the of fluidity the discuss we paper this In aamaue tRI aeprovided have RHIC at measured data .INTRODUCTION I. η/s √ aecm rmdffrn directions. different from come have , s η/s 0 GV uniaieyrep- quantitatively AGeV, 200 = ≤ liiyadsprrtclt fteQDmatter QCD the of supercriticality and Fluidity 6 ula cec iiin arneBree ainlLabo National Berkeley Lawrence Division, Science Nuclear / (4 v 2 rae nrltvsi ev o collisions ion heavy relativistic in created π aahwvrsol be should however data a est[] (Some [2]. set be may ) S001,1CcornRa,Bree,C 94720. 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And given a sufficiently large external scale, This paper is organized as follows: In Section II we hydrodynamics is always applicable. For example we ob- will discuss the difference and relation between a non- serve sound propagation in both water and air, and yet relativistic(NR) fluid and a relativistic(R) fluid. In one would be inclined to assign a better fluidity to water Section III we will then propose a new fluidity mea- than to air. Therefore a meaningful definition of flu- sure, which is applicable for both relativistic and non- idity needs to measure the effective mean free path in relativistic systems. Based on this new measure we will terms of a length scale inherent to the substance itself. compare various fluid systems and demonstrate the im- One obvious choice is the interparticle distance, or more provement of fluidity in a fluid’s supercritical regime. In generally, the density-density correlation length. This al- Section IV we will use Lattice QCD results to construct lows the comparison of substances with vastly different equal-pressure lines on the QCD phase diagram. We then inherent length scales, such as water vs. a Quark-Gluon discuss the relationship between fluidity and supercriti- Plasma. Obviously, the minimum wavelength (in me- cality for heavy ion collisions, the evolution of fluidity ters) for sound propagation even in a weakly interacting with beam energies, and its implications for the search of QGP is still considerably shorter than that of a strongly the QCD Critical-End-Point(CEP). Finally in Section V interacting atomic or molecular substance. However, if the applicability of hydrodynamics in heavy ion collisions we ask the question “what is the minimal wavelength for at various energies will be discussed. sound propagation measured in units of the interparticle distance?” then a comparison between these substances becomes meaningful, and the strongly interacting sub- II. RELATIVISTIC AND NON-RELATIVISTIC stance will likely exhibit a better fluidity. FLUIDS Following these general considerations, in this paper we discuss the fluidity of the hot and dense QCD mat- When comparing a relativistic (R) with a non- ter created at RHIC by comparing it with normal non- relativistic (NR) fluid, one needs to carefully keep track of relativistic fluids and by applying valuable insights from the mass terms, which customarily are neglected in non- those fluids. While such a comparison is not new (see relativistic thermodynamics. In non-relativistic thermo- e.g. [11][12]), we differ from all previous approaches in a dynamics, the basic thermodynamic relation few distinct points. First of all, we carefully examine the E = TS pV + µ N (1) difference between the relativistic and non-relativistic flu- NR − NR ids. This includes their different inertia: the relativistic does not take into account the mass of the particles. Here inertia, i.e. the enthalpy, reduces to the mass density in ENR refers to the kinetic and interaction energy of the the non-relativistic regime. This also includes the choice particles. Similarly, the chemical potential, which repre- of reference scale, as discussed above: while the temper- sents the increase of energy by the addition of one extra ature may serve as a reasonable estimator for the inter- particle, does not account for the particle’s mass. The particle distance for relativistic fluids, it certainly does relativistic version of the basic thermodynamic relation, not in the non-relativistic limit. Based on these consid- erations, we propose a new measure of fluidity for com- ER = TS pV + µRN, (2) paring various fluids. Furthermore we emphasize, for the − first time, the possible relevance of the so-called super- on the other hand, takes into account the particle masses. critical fluid for the matter produced at RHIC and discuss Its non-relativistic limit can be obtained by simply in- important implications for the expected fluidity for the cluding the mass terms in both the energy and the chem- matter produced in heavy ion collisions at different center ical potential of mass energies. We will also elaborate on potential con- sequences for the search of the QCD Critical-End-Point ER = ENR + m (3) via a beam energy scan. µR = µNR + m. (4) As just discussed, a closely related, but different, ques- As we will discuss below (see also [17, 18]), the thermody- tion is the applicability of hydrodynamics in heavy ion namic quantity entering hydrodynamics is the enthalpy collisions at various beam energy √s. For the descrip- density, w defined as tion of the dynamical evolution of system within hydro- dynamics to be valid the length scale characterizing the wR = ǫ + p = Ts + µR n (5) variations of the system needs to be large compared to the effective mean free path of the (quasi) particles within the where ǫ is the energy density, p the pressure, s the fluid [2][15]. Thus, good fluidity of the underlying matter entropy-density, and n is the particle-density. As we may not guarantee applicability of hydrodynamics, if the will show, the non-relativistic limit of hydrodynamics in- typical gradients of the flow field are large, e.g. when the volves the non-relativistic limit of the enthalpy, wR, in- system size is very small. Based on an analysis of sound cluding the mass term wave attenuation, we will provide quantitative criteria ≪ for the applicability of hydrodynamics, and evaluate the w = Ts + (µ + m)n T m mn ρ (6) NR → ≡ 3 and it is dominated by the mass density. In the ultra- Next we examine the non-relativistic limit by taking 2 relativistic limit T µR, on the other hand, the enthalpy βmc in function I[y], Eq.10, (and its derivative density is given by≫ I′ dI→[y] ∞/dy): ≡ T ≫µR 2 3 w T s. (7) f NR : nmc + n(kBT ) ln(nλ ) 1 → → − 2 ln 3 The kinematic viscosity, which is defined as the ratio of µ NR : mc + (kB T ) (nλ ) → the shear viscosity over the enthalpy density 3 ǫ NR : nmc2 + n(k T ) (13) η → 2 B ν = (8) w 2 1/2 with λ (2π~ /mkBT ) . Obviously, in the non- usually serves as a measure for dissipation [17]. While relativistic≡ regime the energy associated with the rest the widely used ratio of shear-viscosity over entropy den- mass dominates the chemical potential, the free energy sity, η/s is indeed related with the kinematic viscosity for density, the energy density, and the enthalpy density. ultra-relativistic systems, it misses the dominant mass Contrary to that, neither entropy density s = (ǫ f)/T term for a non-relativistic fluid. nor the pressure p = µn f, have an explicit dependence− In the following, we further demonstrate this point of the mass term, as intuitively− expected. both in thermodynamics and in hydrodynamics2.
B. Hydrodynamics A. Thermodynamics We now turn to the hydrodynamics in the relativis- We start with the example of a classical (Boltzmann), tic and non-relativistic regime. Since the non-relativistic relativistic free gas at temperature T and fixed (net- limit of relativistic hydrodynamics is discussed in text- )particle density n = n n ¯ . Following standard sta- P − P books [17, 18], we will be brief here and just remind tistical mechanics in e.g. [16], the Helmholtz free energy ourselves of the essential points, in particular how the density is given by relativistic inertia i.e. the enthalpy density w is replaced in the non-relativistic limit by the mass density ρc2. We 2 2 2 I [βmc ] start with the non-relativistic Navier-Stokes (N-S) equa- f(T,n)= n (kB T ) 1+ − ˜3 3 s nλ tion : 3 3 nλ˜ nλ˜ 2 2 + ( 2 ) +4 ~ p η I[βmc ] I[βmc ] ~ ~ ji +ln (9) [∂t + ~v ]~v = ▽ + j Σ (14) q2 · ▽ − ρ ρ▽ ji 2 1/3 with the non-relativistic shear tensor Σ = ∂j vi + ∂ivj with β 1/(kBT ) and λ˜ = (2π ) (β~c). Here we have − ≡ 2 ~ introduced the re-scaled momentum integral,p ˜ = βpc , 3 δji ~v. The corresponding relativistic N-S equation is ▽ · 4 ∞ given by : 2 2 −√p˜2+(βmc2)2 I[βmc ] dp˜p˜ e . (10) 1 ~v ≡ 0 2 ~ ~ Z γ [∂t + ~v ]~v = 2 [ p + ∂0p] Using the usual thermodynamic relations we obtain ex- · ▽ −w/c ▽ c η pressions for other quantities, such as the chemical po- ~ νi + 2 ∂ν Σ (15) tential and the energy density: w/c
3 3 nλ˜ nλ˜ 2 with the relativistic shear tensor Σµν = c [∂µuν + ∂ν uµ 2 + ( 2 ) +4 ∂f I[βmc ] I[βmc ] 2 2 −2 µ = = (k T ) ln (11) (u ∂)uµuν + (uµuν gµν )(∂ u)], γ = 1/ 1 v /c , B q · 3 − · − ∂n 2 u = γ(1, ~v/c), and ∂ = 1 ∂ . In the non-relativistic µ 0 c t p limit one has γ 1, and w/c2 ρ and thus in leading ∂f order of v/c recovers→ the non-relativistic→ Navier-Stokes ǫ = f T − ∂T equation [17]. 2 ′ To further elaborate the point, let us consider the prop- 2 I 2 I agation and attenuation of a sound wave of frequency ω = n (kBT ) 1+ 3 (βmc ) (12) ˜3 − · I s nλ
3 In this discussion of N-S equation we neglect bulk viscosity and 2 To better keep track of the mass terms, in the following two assume constant shear viscosity across the fluid. subsections we make explicit the dependence on the speed of 4 It is well-known that for relativistic hydrodynamics the derivative light c, the Boltzmann constant kB and the Planck constant ~, expansion to only first order, i.e. the N-S form, has causality while in the rest of the paper we use natural units with these problem and a consistent treatment requires higher orders in constants taken to be unity. derivative expansion, see e.g. [2]. 4 and wave vector k = 2π/λs (λs is the wavelength ) in not the dynamical viscosity η itself but rather the so- the presence of dissipation, characterized by the shear called kinematic viscosity η/ρ (see e.g. [17]). Indeed by viscosity η. The dispersion relation for the wave is given looking at actual data for water in the liquid and vapor by phase at the same pressure one finds that liquid water has about one order of magnitude bigger shear viscosity, 4 3 η η, but nonetheless “wins the fluidity contest” since its i 2 w/c2 , R fluid ω = cs k k 4 (16) kinematic viscosity, η/ρ, is about two orders of magni- − 2 × 3 η ( ρ , NR fluid ) tude smaller than that of the vapor phase. We note that, the expression η/w for a relativistic fluid is a relativistic To take into account possible bulk viscosity ζ, one sim- version of the kinematic viscosity (see also related dis- ply makes the replacement 4 η 4 η + ζ. Furthermore 3 → 3 cussions in [3]). To conclude, we emphasize that for a the speed of sound cs is given by non-relativistic fluid η/ρ is different from η/s (as is evi- dent from the thermodynamics discussion) and only η/ρ ∂P ∂(ǫ/c2) , R fluid serves as a good measure of fluidity (as is evident from cs = (17) the hydrodynamics discussion). This also implies that q ∂P , NR fluid ∂ρ using η/s to compare the fluidity between a relativistic q fluid (like the QGP, the AdS/CFT plasma) and a non- As one can see, the same correspondence (w,ǫ)R relativistic fluid (like Helium, water or cold Fermion gas 2 → (ρc )NR appears again as in the thermodynamics [20] ) may not be as informative as one would expect. Eq.(13).
C. Discussion on η/s III. A NEW FLUIDITY MEASURE
We end this section by a discussion of the ratio of shear- In this section we propose a new fluidity measure that viscosity over entropy-density-ratio, η/s, which has at- is suitable for comparison between relativistic and non- tracted a considerable interest in various fields of physics. relativistic fluids. In the following we will analyze the We first recall, from the perspective motivated by propagation of sound modes in the presence of dissipa- studying the QGP via heavy ion collisions, why η/s tion (viscosity) and ask ourselves under what condition is a useful measure for the relativistic fluid, as was the dissipative (viscous) terms in the equations prevent first pointed out in the seminal paper [19]. Consider a sound from propagating. The advantage of this strategy Bjorken-type longitudinal expansion of the quark-gluon is that we stay entirely within the framework of viscous plasma (with low baryonic density) formed in heavy ion hydrodynamics and do not need to make additional as- collisions: its system size is limited by cτ at early time. sumptions, such as the applicability of kinetic theory. As With the presence of shear viscosity η, there will be dis- a consequence our fluidity measure will be expressed in sipative effect (e.g. entropy generation) characterized by terms of well defined quantities such as the shear viscos- η/w the ratio τ (roughly the Knudsen number) which, by ity, the speed of sound, etc. thermodynamic relation Eq.(7) w Ts, leads to η 1 . We start with the sound dispersion relation in Eq.(16). ≃ s Tτ To ensure a controllable dissipative correction to the ideal By requiring the imaginary part of the frequency, mω, hydrodynamics, one requires firstly η/s is small enough to be small in magnitude as compared with its realI part and, secondly, T τ is of the order 1 or bigger. The former eω, we obtain: shall be a property of the underlying QGP while the lat- R ter means the viscous effect is most severe at early time mω I << 1 and the hydrodynamic evolution may not be a good ap- | eω | → proximation at too early time. (Note in this discussion R 2π 4π we adopted natural relativistic units with c = 1). One λs = >> Lη (18) k 3 can draw two conclusions from this discussion: (a) η/s η 2 (w/c ) cs , R fluid can serve as a good measure of fluidity for a relativistic L η (19) η ≡ , NR fluid fluid and the smaller it is the better the fluidity; (b) the ρcs ability of η/s to serve such a role is actually inherited from η/w. The above equation implies that if a sound wave has its The discussion above, however, leads to the observa- wavelength λs comparable (or even smaller) than Lη, it tion that η/s for a non-relativistic fluid does NOT nec- will be quickly damped on (or shorter than) a time scale essarily provide a good measure for its fluidity, because of its period and spatially on a length scale about (or the role of the relativistic η/w corresponds to the non- shorter than) its wavelength, which essentially means it relativistic η/ρ. This is certainly not a new lesson: it has can not propagate away in the medium. Therefore, the been known from Navier and Stokes’s time that what physical meaning of the length Lη introduced above is really matters for usual non-relativistic fluids’ fluidity is to provide a measure for the minimal wavelength of a 5 sound wave to propagate in such a viscous fluid5. A diameter of a pipe or the size of a moving object inside more quantitative criteria will be discussed in Section V the fluid, etc. In the context of relativistic fluids created and given in Eq.(27). in heavy ion collisions, there have been scaling studies Furthermore the length Lη has the meaning of an effec- of collective flow for lower energy collisions based on the tive mean-free-path (MFP) in terms of microscopic fluid Reynolds number in [13] and more recently for higher en- particle motion. This becomes transparent in a weakly ergy collisions based on the Knudsen number in [14] (see coupled gas: taking the non-relativistic gas as an exam- also related discussions in [15]). The external length scale ple, the shear viscosity according to kinetic transport is in those numbers, however, is not an intrinsic property of the fluid that we would like to invoke for comparing flu- η ρvT lMF P (20) ids across vastly different scales. Instead, a de-correlation ∼ length of certain density-density spatial correlator gives while the speed of sound is a natural scale of short-range order in the system. In most cases this de-correlation length is simply set by the ∂P k T inter-(quasi-)particle distance. For a non-relativistic fluid c = B v (21) s T “particles” and their number density n are well-defined, s ∂ρ ∼ r M ∼ and thus the inter-particle distance is also a well-defined with vT the thermal velocity. These lead to the combi- length scale: nation 1 η Ln 1 (23) Lη = lMF P (22) ≡ n 3 ρcs ∼ For relativistic fluid it is less straightforward. For ex- Unlike the mean-free-path which is conceptually intuitive ample a QGP with µB = 0 and thus nB = 0 can still but practically not easily computable or measurable (e.g. have substantial numbers of quarks and gluons. For for fluids), the length Lη is well-defined by macroscopic such a relativistic fluid a simple estimate can be made properties of the fluid and thus of practical use. s via the entropy density, i.e. n 4kB . Another way We emphasize that involving the speed of sound in the is to calculate the de-correlation∼ length of e.g. the cor- definition of length scale Lη is essential. This can be seen relator < T (~x, 0)T (~0, 0) > as has been done in re- 2 00 00 already from a dimensional argument: η/ρ or η/(w/c ) cent lattice work [22] which finds a short range order 2 −1 has the dimension of [Length] [Time] . To turn this about 0.6/T for QGP in 1 2Tc, in reasonable agree- into a length scale one needs to divide it by a quantity ment with estimate from n− s . There could still −1 4kB of the dimension [Length] [Time] i.e. a velocity. The be academic examples where∼ the entropy density can natural choice here is the speed of sound as a charac- be infinite while the short-range order does not van- teristic of the macroscopic matter. The speed of light, ish: AdS/CFT offers such an example in which the en- c, on the other hand is not suitable here since it is nei- 2 3 tropy density goes as s Nc T (in the large ther a property of any specific substance nor should it be 1/3∝ → ∞ Nc limit) implying 1/n 0 while spatial correlators of any relevance to non-relativistic fluids, such as water. → like
1 2.5 H ê ê h
L 2 s 0.6 c
r 1.5 ê h H 1 m u
= 0.4 m n ini
L 0.5 ê M foL h
L 0 0.2 0.5 1 1.5 2 2.5 3 0 50 100 150 200 ê T Tc Molar Mass Hg êmol L
FIG. 1: (Color online) (left panel) Fluidity measure = Lη /Ln versus T/Tc for fifteen different substances at fixed critical F pressure P = Pc, see text for more details. The sharp peaks centered at Tc are due to the actual divergence of the shear viscosity at the critical point. (right panel) The minimum of the fluidity measure for various substances (as obtained from F 4 respective curves in the left panel) versus their molar masses, with the stars (from left to right) for H2, He, H2O, D2O, Ne, N2, O2, Ar, CO2, Kr, Xe, and the diamonds(from left to right) for C4H10, C8H18, C12H26, C4F8.
their respective critical pressure Pc, including: Hydrogen To conclude the study of critical fluids, there appears 4 (H2), Helium-4 ( He), Water (H2O), Deuterium oxide to be certain universality of the newly proposed fluidity (D O), Neon (Ne), Nitrogen (N ), Oxygen (O ), Ar- measure , indicating that “a good fluid is a good fluid” 2 2 2 F gon (Ar), Carbon Dioxide (CO2), Krypton (Kr), Xenon despite many other details regarding the microscopic de- (Xe), Isobutane (C4H10), Octane (C8H18), Dodecane grees of freedom. (C12H26), Octafluorocyclobutane (C4F8). These sub- stances cover a wide range of molar mass, chemical struc- ture and complexity, with their respective critical tem- B. Supercritical Fluids perature Tc and pressure Pc differing by orders of magni- tude. Despite such huge differences, their fluidity curves While much attention has been paid to critical fluids, resemble each other not only in shape but even quan- the fluidity of a fluid with significantly larger pressure titatively. In particular in their good liquid regime — P >>Pc has been little discussed in the context of heavy roughly the “valley” region at 0.7 1 Tc — they ion collisions and QCD matter: so let us next explore ∼ − all show amazingly similar fluidity. To further expose this region. In the literature, this region is often referred the similarity, in the right panel of Fig.1 we show the to as the “supercritical fluid” region [21], defined on a value of the fluidity measure at its minimum versus typical substance’s T P phase diagram (with critical F − the substances’ molecular molar masses. From this plot, point Tc, Pc) as: roughly two bands can be identified: the green stars spread in a narrow band of (0.3, 0.45) while span- supercritical : T >Tc & P > Pc (25) ning two-orders-of magnitudeF in ∈ molar mass which in- clude all 11 non-organic substances; the red diamonds In particular we want to explore the deeply supercritical with roughly twice bigger , include 4 organic substances region with P >>Pc. Taking water as an example, in F Fig.2(left) we plot the fluidity measure versus T/Tc for with much more complicated molecular structures (e.g. F chains) which, not surprisingly, lead to more dissipation. fifteen different fixed pressure values ranging from 5 MPa Even so, the splitting in fluidity between the two bands is all the way to 1000 MPa (note for water Pc = 22 MPa). merely a Oˆ(1) factor rather than any order-of-magnitude The dashed (green) curve is for fixed critical pressure difference. Nevertheless as one can imagine, with increas- P = Pc = 22 MPa, while the four solid (red) curves ing chemical complexity and molecular mass, the fluidity above it are for P < Pc, and the ten solid (blue) curves of more complex systems like e.g. engine oil may deviate below it are for P > Pc. As can be seen from the plot, significantly from what are shown in the figure. the curves change shape gradually and the fluidity be- comes better and better with increasing P . The “valley” 7 L
Å 3 Å
Å 20 1 3 Å L n o ê A
2.5 H 1 H 15 ê L L
2 s s c c r r 1.5 10 ê ê h h H 1 H =
= 5 n L
0.5 h ê L h 0 L 0 0.5 1 1.5 2 0.5 1 1.5 2 T ê Tc T ê Tc
FIG. 2: (Color online) The dimensionless fluidity measure (left panel) and the length scale, Lη (in units of A)˚ (right F panel) versus T/Tc for water at fifteen different fixed pressure values. In both panels, the dashed (green) curve is for fixed critical pressure P = Pc = 22 MPa, while the four solid (red) curves above it are for P < Pc, and the ten solid (blue) curves below it are for P > Pc, with the respective pressure values for each curve (from top down) being P = 5, 10, 15, 20, 22, 25, 30, 40, 50, 100, 150, 200, 250, 500, 1000 MPa.
300 where remains small and relatively flat becomes much F wider and eventually flattens out substantially above Tc 250 for P >> Pc. To quantify the remarkable fluidity of L the supercritical fluid, we note that the minimum on the B P = Pc curve has a value for the fluidity min(Pc) 0.33
k F ≈ 200 while the minimum for the P = 1000MPa 45Pc curve p ≈ is at min(45Pc) 0.11, getting smaller by a factor of 4 F ≈ ê 3 and remaining close to the minimum within a rather Ñ 150 broad temperature region! H
s 100 It is also interesting to examine whether the length
ê scale Lη itself shows similar trends, as Lη is the essential
h scale for discussing the applicability of hydrodynamics 50 where it is to be compared with the external scale char- acterizing the variation of flow field. In the right panel of 0 Fig.2 we plot Lη itself for the same conditions and find 0.5 0.75 1 1.25 1.5 1.75 2 that, similar to the fluidity measure , the scale Lη also becomes considerably smaller in supercriticalF water. T ê Tc It should be mentioned that the same observation is also true for other fluids that we examined, like Helium-4, FIG. 3: (Color online) The shear-viscosity-entropy-density- Nitrogen, etc. Furthermore such behavior is not specific ratio η/s (in unit ~/(4πkB )) versus T/T c for water at fifteen to our fluidity measure. In Fig.3 we plot the widely used different fixed pressure values. The dashed (green) curve ratio of shear viscosity over entropy-density, η/s, which is for fixed critical pressure P = Pc = 22 MPa, while the show the same qualitative behavior. four solid (red) curves above it are for P < Pc, and the ten solid (blue) curves below it are for P > Pc, with the respec- The main lessons, as we emphasize again, are (a) for tive pressure values for each curve (from top down) being P = a given substance, the best fluidity is not necessarily 5, 10, 15, 20, 22, 25, 30, 40, 50, 100, 150, 200, 250, 500, 1000 MPa. achieved close to the critical point and (b) when going deeper into the supercritical regime its fluidity becomes much better. 8
2 accurately determined. We simply use Ln = 1/T as an estimate. This gives the fluidity = √3/(4π) 0.138. Hadronic Gas? For the Cold Fermi Atom gas,F its shear viscosity≈ has Water üPc been measured near its Feshbach resonance by Thomas 1.5 et al in [26] for a certain range of system energy by an- Helium üPc alyzing the damping of collective modes in the atomic
n cloud (see also related work in [27]). As a benchmark,
L 1 we take the lowest viscosity found from the measurement ê (see Fig.4 in [26]) which is η 0.214 ~ n. The speed of ≈ h sound has also been measured by the same group in [28], L from which we take the value cs 0.3632 vF near the Fes- sQGP? ≈ ~ 0.5 hbach resonance. Since the Fermi velocity vF = kF /m, the mass density ρ = mn, and the Fermi momentum Water ü11 *P c k /n1/3 = (3π2)1/3, we obtain for the fluidity measure Cold Fermi Atom? F 0.214/(0.3632 (3π2)1/3) 0.191. AdS êCFT? F ≈ · ≈ 0 Water ü45 *Pc The question marks in Fig.4 indicate that the current estimates presented above may carry sizable uncertain- 0.5 1 1.5 2 2.5 ties, and we expect the knowledge on these systems will T ê Tc become more accurate with time. We finally come to the comparison in Fig.4. As one can see, the critical fluids (water and Helium-4 at Pc) FIG. 4: (Color online) Comparison of fluidity measure for have a somewhat worse fluidity than the QCD, AdS/CFT various fluids (see text for more details). The curves with and Cold Fermi Atom systems. Supercritical water at question marks indicate current estimates of the respective P = 11Pc, on the other hand, already has a fluidity fluidity with possible uncertainty, while the curves for Helium comparable to the QGP while at P = 45Pc the fluidity at Pc and for water at Pc, 11Pc, 45Pc are from actual data. of supercritical water appears to be better than that of the recently discussed “nearly perfect fluids”.
C. Comparison of Various Fluids IV. A SUPERCRITICAL QCD FLUID AT RHIC? Finally we attempt to compare various fluids of great current interests in terms of the new fluidity measure The fluidity study in the previous section, in particular F we have proposed and studied above. The results are on the supercritical fluid, naturally leads to the follow- shown in Fig.4. Below we give the details of how the ing interesting question: Are we observing a supercritical curves for QCD, Cold Fermi Atom, and AdS/CFT are QCD fluid at RHIC? obtained. As is well known, what is usual referred to as Tc For the QCD system, we have used a parametrization 170 190MeV in QCD is not a true second-order phase≈ of the viscosity η by Hirano and Gyulassy in [24]: one transition∼ temperature but rather the temperature for a 3 uses η/Tc T/Tc for hadronic gas (H.G.) below Tc, and rapid crossover at µ = 0 [25]. There is, however, a (hypo- η/T 3 (T/T≈ )3[1+ (T )ln(T/T )]2 for the quark-gluon thetical) Critical-End-Point(CEP) at (T ,µ ) on c ≈ c W c CEP CEP plasma(sQGP) in the temperature interval T/Tc [1 3] the QCD phase diagram marking the end of a plausible with ∈ − first-order phase transition at low T but high µ (see e.g. [ (T )/(4π)] = (9β2) [80π2K ln(4π/g2(T ))]−1 [29] and references therein). Accurate determination of W 0 · SB interpolating to pQCD results for T >>Tc, where the the CEP from lattice QCD [30–34] and unambiguous ob- parameters are given as β0 = 10, KSB = 12 and the servation of the CEP from heavy ion collisions [12, 35, 36] running coupling are among the most interesting and exciting goals of QCD [g2(T )]−1 = (9/8π2)ln(2πT/Λ) + research. While currently the position of the CEP is not (4/9π2)ln(2ln(2πT/Λ)) (with Λ 190MeV ) ( see well constrained, it is expected to have lower temperature [24][11] for more details). The≈ enthalpy density than that of the QCD crossover transition at vanishing w = ǫ + p and speed of sound cs are taken from recent baryon density, TCEP < Tc. Being very aware on the lattice results by Karsch et al [25] for 2+1 flavor QCD present uncertainty of the actual location of the QCD with mπ 220MeV . As we mentioned before, Ln is critical end point, let us, solely for demonstration pur- ≈ 1/3 estimated by 1/(s/4kB) with the entropy density also poses and for the sake of the argument, assume that its taken from [25]. location is close to the estimate of ref. [32], which sug- For the strongly coupled AdS/CFT system, the shear gests that TCEP 0.94Tc,µCEP 1.8TCEP . This loca- viscosity is well known to be η/s = 1/(4π) [9]. As we tion is indicated≈ in Fig.5 as the filled≈ black circle along also pointed out before, there is a short-range order at with a question mark. the length scale L 1/T however the pre-factor is not Next if we adopt the definition of supercriticality as in n ∼ 9
critical pressure P (CEP ) (value on the blue curve) are ó ó ó ó ó ó ó óóóóó óóóóóóóóóóóóóó óóóóóóóóóóóóóó óóóóóóóóóóóóóó (from bottom to top) 0.05, 0.08, 0.13, 0.23, 0.27, 0.35, 400 LHC óóóóóóóóóóóóóó ç ç ç ç ç ç çççç 0.63, 1, 2.7, 10, 18, respectively. Of course with improved ççççççççççççç ççççççççççççç ççççççççççççç ççççççççççççç lattice results for pressure, CEP and (higher order) sus- ççççççççççççç RHIC çç ceptibilities a more accurate equal-pressure map for QCD
L 300 ô ô ô ô ô ô ôô matter will become available. ôôôôôôôôôôôôô ôôôôôôôôôôôôô ôôôôôôôôôôôô ôôôôôôôôôôôôô ôôôôôôôôôôôôô í í í ôôôôôôôôôôôô í í íííííííííí ôôôôôôôôôôôôô ò ííííííííííííí ôôôôôôôôôô ò ò ò ò òòòòòòòòò ííííííííííííí á òòòòòòòòòòòòò ííííííííííííSPS á á áááááááááá òòòòòòòòòòòòò íííííí We now discuss a few implications for heavy ion colli- ìí ì íììíì ìí ìíìíìáìáìááìáááááááá òòòòòòòòò ííííííííííí MeV í ííííììììì ááááá ò íííí à à à à ààààààà íííííííìíììíìíìíìììììììáááááááááá òòòòòòòòò ííííííííííCEP? ààààààààààà íííííííììíìíìíìííììììáìáìááááááááá òòòòòòòòòòòò ííííííííííí H 200 àààààààà íííííííìíìììììììáááááááá òòòòò ííííí sions experiments. As a precaution, the following discus- æ àà íí ì áá ò í æ æ æ ææææææææ ààààààààààà íííííííííìíìììììáááááá òòòòòò íííííí sions are by no means intended to provide quantitative T statements but rather qualitative yet interesting ideas. (I) At current RHIC energy (√s = 200 AGeV), Fig.5 in- 100 dicates that most of the QGP phase of the created mat- ter is likely in the supercritical region. This may very well be the reason why such good fluidity has been ob- 0 served. If a relationship between good fluidity and su- 0 100 200 300 400 percriticality is indeed true for the QCD matter just as it is for water, then there will be a sensitive dependence mq HMeV L of the fluidity on the actual position of the CEP. For example, the matter created at RHIC might have most of its evolution being in the supercritical region in case FIG. 5: (Color online) Schematic isobaric contours (i.e. with TCEP and PCEP are considerably lower than indicated constant pressure along each line) on the QCD T µ phase in the Fig.5. On the other hand, the matter created at diagram with the filled black circle indicating a possible− posi- SPS (√s 20 GeV) seems to have its initial pressure tion of the hypothetical Critical-End-Point (CEP); the dashed and temperature≃ already rather close to the critical one blue horizontal short line stretching from the CEP to the right which would result in poor fluidity. In principle these re- is included to indicate the T = TCEP boundary (see text for gions (supercritical, near-CEP, below-CEP) of very dif- more details). The filled red, blue and green circles indicate ferent fluidity can be accessed experimentally by tuning estimates of the initial (T, µ) reachable at SPS, RHIC and the center of mass energy √s which changes the initial LHC, respectively. densities (see e.g. [41]) as well as the freeze-out points (see e.g. [42][43]) in such collisions. To which extend Eq.(25) for QCD, we need to know the constant-pressure this difference in fluidity transforms into measurable ef- lines on the QCD phase diagram typically plotted in fects, such as non-hydrodynamic behavior and thus less terms of T µ. To schematically construct these lines, we elliptic flow, is a non-trivial question. As we will discuss make use of− the Taylor expansion of the pressure P (T,µ) in the next section, besides the fluidity the actual size with respect to µ: of the system is essential for the applicability of hydro- dynamics. Our estimates (see Fig.6) show that even for 1 µ 2 the very good fluidity assumed for a strongly interacting P (T,µ)= T 4 P (T )+ χB(T ) × 0 2 2 T QGP, denoted as “sQGP” in Fig.4, a hydrodynamic de- 4 scription for the fireball expansion is a best marginal at 1 B µ + χ4 (T ) + ... (26) SPS energies. The reason is that, according to our esti- 24 T mate, the system starts close to the phase transition and, 2n 4 thus, reaches the hadronic phase while still comparatively B ∂ (P/T ) with χ 2n the baryonic susceptibilities 2n ≡ ∂(µ/T ) µ=0 small in size. Consequently, it may be difficult to tell if [32, 33, 37, 38, 40]. In order to construct the equal- the system enters a regime of reduced fluidity from flow pressure lines shown in Fig.5 we have used the lattice studies alone. Of course if our estimates are wrong and B B QCD results from Cheng et al for P0(T ) [25] and χ2 ,χ4 one finds evidence that the initial entropy density reached [37]. is considerably larger, a change in the fluidity may show The blue curve in Fig.5 going right through the indi- up in the systematics of elliptic flow measurements, such cated CEP point indicates the constant-pressure line with as its dependence on beam energy and system size. fixed critical pressure P = P (CEP ) which, together with (II) Based on the possibility of a mostly supercritical the T = TCEP line (dashed blue horizontal one stretch- QCD matter to be created in experiments at the Large ing from the CEP to the right) form the boundary above Hadron Collider (LHC) (see Fig.5), one is led to pre- which there is the QCD supercritical region. We empha- dict an even better fluidity to be observed at LHC than size that above the boundary the pressure P T 4 in- at RHIC. And since for the same fluidity the conditions creases very rapidly. As a result the QGP quickly∼ enters for hydrodynamics are more favorable for LHC energies deeply into the supercritical regime where P P (CEP ). than for RHIC, we would expect nearly ideal hydrody- To give an idea: the pressures of the lines in≫ units of the namic evolution at LHC. 10
V. APPLICABILITY OF HYDRODYNAMICS IN the system size is limited mainly by its longitudinal ex- HEAVY ION COLLISIONS tent and the scale governing the variation in the fluid can be estimated to be about 2τ. At late time when τ be- comes larger than the nuclear radius R the limiting In this section we turn to a discussion of applicability ∼ A of hydrodynamics in heavy ion collisions, particularly for scale is set by the transverse size of the fireball which is SPS, RHIC and LHC. As already discussed in the intro- slightly larger than RA and slowly grows. For the purpose duction, the fluidity of the underlying matter is not the of our rough estimate, we adopt a simple prescription: L 2τ for τ < R and L 2R for τ R with R = 8 fm only determining factor: the “best” (in terms of fluidity) s ≈ s ≈ ≥ fluid may cease to flow if put into a sufficiently narrow for AuAu and PbPb collisions. For the collision dynam- pipe. In other words, the applicability of hydrodynamics ics which determines the temperature evolution s(τ) (or depends equally on both the internal properties of the inversely τ(s)), we simply use the Bjorken flow relation substance, i.e. its fluidity or L , and the external set- s(τ) = s0(τ0) τ0/τ, which up to τ R is a reasonable F η · ∼ tings, i.e the typical length scale Ls of the variation of the approximation. For the initial condition we follow esti- flow field in the system under consideration 6 : one useful mates used in hydrodynamic modelling (see e.g. [41]) and criteria, equivalent to the famous Knudsen number, is a assume equilibration at τ0 = 1 fm, with fireball center en- −3 ratio of the two Lη/Ls. tropy densities (in central collisions) to be s0 = 24 fm −3 −3 In order to derive a quantitative criterion, we again at SPS, s0 = 70 fm at RHIC, and s0 = 154 fm at use the example of sound dispersion discussed before. By LHC. requiring that a propagating sound wave has to complete at least one full period before its amplitude damps by Finally we turn to the discussion of the applicability a factor e−1 due to the imaginary part, we arrive at a of hydrodynamics given the criteria of Eq.27. The ratio 2 L /L in units of 3/(8π2) is plotted in Fig.6(right) for minimal wave length λ = 8π L . For a sound wave η s min 3 η SPS(red diamonds), RHIC(blue stars), and LHC(green the length scale characterizing the flow field variations boxes). The thick dashed horizontal line indicates the simply the wavelength, L λ. Thus we arrive at the s equality of the condition expressed by Eq.27: If the sys- following criterion for the applicability≈ of hydrodynamics: tem is considerably below this line a hydrodynamic de- scription should be a reasonable approximation. If it L 3 η finds itself considerably above corrections to hydrody- < 2 0.038 (27) Ls 8π ≈ namics from higher orders in derivatives or an even non- hydrodynamic description are called for. To quantify the The length Lη(T ) for QCD (at µ = 0), shown in previous statement, if the system finds itself at a value Fig.6(left), is determined as described section III C. It 2 of (Lη/Ls)/(3/8π ) = 2, such as the points for SPS relies on a parametrization for η(T ) from the Hirano- energies on Fig.6(left), then the amplitude of a sound Gyulassy [24], and we note that the actual shear viscosity mode after propagating the distance of one wavelength of QCD matter, once it is known, may be quite different. will be reduced by a factor of e2 7, and for a value of The temperature dependence of Lη shown in the plot 2 ≃ 5 (Lη/Ls)/(3/8π ) = 5 the reduction would be e 150! features a steep drop from the hadronic phase to about Given our estimate and the assumptions which it is≃ based T 1.1T , followed by a rather constant value for tem- ≃ c on, a hydrodynamic description is not a good approxi- peratures above, T & 1.1Tc. One may expect Lη becomes mation for SPS, whereas for RHIC and even more so for large again at very high temperatures T >>Tc where a LHC energies, it seems to be more or less justified8. Such weakly coupled QGP describable by pQCD takes over. In differences from SPS to RHIC and LHC lie in the differ- principle Lη also depends on the baryon-number chemi- ent time evolution due to the different initial densities: cal potential, µB, and this dependence shall be taken into when cooling down into the region where Lη starts ris- account when discussing collisions at low energy (such as ing abruptly near and below Tc, the SPS fireball still has SPS) where µB could be sizable. However currently very a rather small size (longitudinally) while the RHIC and little is known about such dependence, and we here use LHC fireballs already becomes large with a size about 2R the zero µ result also for SPS. and hence have their hydrodynamic evolution extended We next give a rough estimate for the typical length much longer. While our estimates contain uncertainties scale Ls of flow field variance for heavy ion collisions at and should by no means considered to be precise, we have different √s 7. At early times, right after the collision,
the estimation easier. Generally speaking when going from cen- 6 We re-emphasize that the length scale Lη is a well defined quan- tral to peripheral collisions one expects hydrodynamics to be less tity depending only on macroscopic variables that are measurable and less applicable. and/or calculable. Furthermore unlike the mean-free-path which 8 We note that even for RHIC and LHC the first fm or so of the depends on quasi-particle picture (see e.g. discussions in [15]), evolution is above the “criteria line” due to the rather small sys- the length scale Lη remains a useful and meaningful scale even tem size. Whether this has a profound effect (such as in entropy for strongly coupled systems. generation) or not is unclear at this time (see a very interesting 7 In this discussion we focus only on central collisions which make discussion in [44]). 11 L
5 2 8 p 4 8 4 ê
L SPS 3 CuCu
3 H m 2 RHIC LHC f ê H L h 2 1 L s L
1 ê 0.5 0.5 h L 0.25 0.1 H 0.75 1 1.25 1.5 1.75 2 2.25 2.5 0.75 1 1.25 1.5 1.75 2 T ê Tc T ê Tc
η 2 FIG. 6: (Color online) (left panel) The length scale Lη = (w/c )cs (in unit fm) versus T/Tc for QCD at µ = 0, with η from parametrization in [24] and w, cs from lattice data [25]. The dashed horizontal lines at 0.1 fm and 0.5 fm are included to guide the eyes. (right panel) The criteria (27) for applicability of hydrodynamics versus T/Tc at SPS(red diamonds), RHIC(blue stars), and LHC(green boxes), with the thick dashed horizontal line indicating the borderline below which hydrodynamics may be a good description according the criteria (see text for more details). A line for CuCu(orange triangles) collisions at RHIC energy (200 AGeV) is also presented. provided a semi-quantitative picture for evaluating the thermodynamics and hydrodynamics perspectives. A applicability of hydrodynamics with varying √s, which new fluidity measure = Lη/Ln is then proposed, which implies at LHC the fireball expansion shall be even better shows certain universalityF in the good fluid regime for a described by hydrodynamics as compared to RHIC. remarkable diverse set of critical fluids. We have further Besides the beam energy √s, a change of the system demonstrated that the fluidity is enhanced in the super- size should also affect the conditions for the applicability critical fluid regime on a fluid’s phase diagram. These of hydrodynamics. For example, RHIC has done exper- studies inspired us to conjecture that the seemingly good iments with both AuAu collisions and CuCu collisions, fluidity of the QCD matter at RHIC may actually be re- both at full energy, and it would be interesting to examine lated to its supercriticality with respect to the Critical- and compare the applicability of hydrodynamics for these End-Point on the QCD phase diagram. This observation, two different systems. In Fig.6 (right) we have included if true, has far-reaching consequences for heavy ion col- a calculation of the same applicability measure for CuCu lisions experiments: (a) the loss of such good fluidity at (orange triangle) collisions at 200 AGeV, with a initial certain lower beam energy which is sensitively related to −3 center entropy density s0 = 45 fm and a transverse the position of the long sought CEP; (b) an even better size parameter R = 5 fm. The curve indicates that hy- fluidity may be expected at higher beam energy, which drodynamics is marginally applicable for such collisions, will soon be tested by the LHC heavy ion program. Fi- as both the initial density and the transverse size of the nally we have analyzed the applicability of hydrodynam- matter formed in CuCu collisions are smaller that those ics for the fireball evolution in heavy ion collisions at in AuAu collisions at the same energy. various energies. Our analysis was based on a model parametrization for the shear-viscosity and a rough esti- mate of the relevant length scales governing the fireball VI. SUMMARY expansion. Given these assumptions we find that hydro- dynamics should be applicable for collisions at RHIC and LHC energies but not for SPS energies. In summary, we have discussed the fluidity of the hot and dense QCD matter as produced in ultrarelativistic heavy ion collisions in comparison with various other, well known fluids. In particular we have suggested its possible supercriticality based on insights gained from studying more conventional fluids like water. We have discussed several aspects relevant for a proper compari- son of non-relativistic and relativistic fluids, both from 12
Acknowledgments ported by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Divisions of Nu- The authors are grateful to Ulrich Heinz, Roy Lacey, clear Physics, of the U.S. Department of Energy under Dirk Rischke, Thomas Schaefer, Edward Shuryak, and Contract No. DE-AC02-05CH11231. Nu Xu for valuable communications. The work is sup-
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