- Email: [email protected]
Convective stability of hot matter in ultrarelativistic heavy-ion collisions
Nuclear Physics A540 (1992) 659-674 North-Holland
NUCLE PHYSICS
Convective stability of hot matter in ultrarelativistic heavy-ion collisions* Wojciech Florkowski a,b, Bengt L. Friman", Gordon Baymd and P. Vesa Ruuskanend,e ° Gesellschaft für Schwerionenforschung GSI, Postfach 110552, D-6100 Darmstadt, Germany b H. Niewodniczanski Institute of Nuclear Physics, ul. Radzikowskiego 152, 31-342 Krak6w, Poland c Institut Ar Kernphysik, Technische Hochschule Darmstadt, D-6100 Darmstadt, Germany d Department of Physics, University ofIllinois at Urbana-Champaign, 1110 W. Green St., Urbana, Il. 61801, USA e Department of Physics, University or.lyvdskyld, P.O. Box 35, 40_=51 Jyvdsky1d, 1-inland Received 22 January 1992 Abstract: The convective stability of strongly interacting matter undergoing hydrodynamic flow in ultrarelativistic heavy-ion collisions is studied in both the quark-gluon plasma and hadron gas phases . We find that this stability depends on the form of the initial conditions assumed for the hydrodynamic flow. In the case of initial conditions corresponding to partial transparency the flow of the quark-gluon plasma is stable whereas the flow of the hadron gas is convectively unstable. The timescale for hydrodynamic oscillations around the (stable or unstable) equilibrium state is found to be larger than the expected lifetime of the system, suggesting that the flow in either case is close to neutral convective equilibrium. 1. Introductio&t A fully developed quark-gluon plasma formed in a central ultrarelativistic nucleusnucleus collision is expected, after initial equilibration, to undergo an essentially hydrodynamic expansion phase in which the mean free paths of the constituent particles are small compared to the length scales of the system, e.g., the nuclear radii, and the collision times are short compared to the expansion time 1-4 ) . An important question is whether the flow is hydrodynamically stable . In a hydrodynamic expansion, the particles in an accelerated fluid element experience an inertial force in the direction opposite to the acceleration . If the entropy per particle varies in the direction of the acceleration the situation is analogous to that of a gas in a gravitational field, e.g., an atmosphere, which is convectively unstable if the entropy per particle decreases with increasing altitude 5 ) . In an expanding quark-gluon plasma, with the entropy density or baryon density varying in the direction of expansion, a convective instability, analogous to that in an atmosphere, is possible . Such an instability, if it has sufficient * Research supported in part by NSF Grants PHY84-15064 and PHY89-21025, and by the Polish Mate Committee for Scientific Research under Grant No. 2.0204.91 .01 . 0375-9474/92/$05 .00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
. Florkowski et al. / Convective stability ti .e to develop, would lead to interesting experimental consequences, e.g., enhanced fluctuations in rapidity and other distributions . In this per we continue the investigation of the convective stability of hot strongly interacting matter initiated in ref. 5 ) . We derive the convective stability criterion for relativistic hydrodynamic flow and apply it to a one-dimensional hydrodynamic flow o the matter produced in ultrarelati:-Anti;, üeaw-ion collisions. We consider both the cases that a quark-gluon plasma is formed and that the produced matter is a hadron gas. The paper is organized as follows . In sect. 2 we formulate the convective stability condition. In sect . 3 we calculate the thermodynamic quantities that determine the stability properties for a given equation of state . The acceleration of the fluid, which enters the stability condition, is estimated by solving the hydrodynamic equations in sect. 4. Our main results are presented in sect. 5 and conclusions in sect. 6. 2. Condition for convective stability To derive the conditon for convective stability, let us first recall the argument for a fluid at rest in a uniform gravitational field. Later, we replace the gravitational force by the local inertial force experienced by an accelerated fluid. Imagine that a fluid element, originally at height z, undergoes an adiabatic displacement upwards (against the force) to height z + ® z. A necessary condition for the fluid to be stable against convection is that the net force on the fluid element tends to bring it back to its original position. In a relativistic fluid gravity couples to the enthalpy density w = E + p, where E is the energy density and p the pressure, and so the condition is that the gravitational force per unit volume, --gw (where g > 0), should be stronger than the buoyancy force, -dp/dz. Since in equilibrium the net force vanishes, stability requires that the fluid element be heavier than the fluid it displaces ; thus gw (p', a) > gw (p', a'), where a = s/nB is the entropy per net baryon at the original point, and a' = a(z + d z) and p' = p (z + dz). A small displacement d z produces a restoring force, -g (dwldz )Pd z; thus for small j z the stability condition is g (a wl0a )P (daldz) < 0. Using the standard thermodynamic relation (âw/aa) P = (T/cp )(8c/8T) P, where T is the temperature, and c. the specific heat per net baryon at constant pressure, and the fact that the specific heat is positive we can write the stability condition as ac l d <0 . (1) g C8T I dz P The derivative (&/8 T )P vanishes under several conditions. The first is if a gas has zero net baryon number (or more generally does not have a composition degree of freedom) ; then only one thermodynamic variable is independent. Convective instability is possible only if there exists an independent composition degree of freedom. It also vanishes in a gas where the energy density is proportional to the pressure. A gas of non-interacting massless particles is in neutral equilibrium with respect to convection . For a non-relativistic f id the energy density is replaced by the mass density p in the stability condition (1) . For "normal" matter that expands when heated at constant
W. Florkowski et al. / Convective stability
661
pressure, (ap/a T)P < 0. Thus one sees the familiar condition for an atmosphere stable with respect to convection, that dQ/dz > 0, i.e., that the entropy per particle increases with increasing altitude. In order to find the timescale characterizing the development of a convective instability we observe that the force acting on the adiabatically shifted fluid element is proportional to the distance from the original equilibrium point. The inertial mass of the element is equal to its enthalpy . If stable, the system has small oscillations with frequency D Q2 _ -
g
w
aw
du .
Càa )P dz
When the stability condition (1) is not fulfilled, 0 becomes purely imaginary, and the timescale for growth is determined by 191-1 . To generalize eq. (1) to the case of an expanding fluid (without gravity) we simply replace, according to the equivalence principle, the gravitational acceleration, -gZ, by minus the acceleration of the fluid element in its local rest frame (we denote this frame by primes) . Thus we find that stability requires
aE du dv~ < 0, Ca T ~ P dz dt
(3)
where v' is the fluid velocity in the z' direction . We can rewrite (3) in the explicitly Lorentz covariant form, ae agaua < 0 . (OT)p
(4)
Here a. = (at , V) and au is the four-acceleration, where V is the four velocity: V = y (1, v), v is the flow velocity and y = (1- v 2) -'~? In the local rest frame V = (1, 0, 0, 0), a,` = (0, dv'/dt'), and eq. (4), for flow in the z-direction, simplifies to the expression (3). Note that a"uu = 0. 3. Estimating (óe/OT), in strongly interacting matter In an atmosphere, the entropy generally resides in the baryons. In matter created in an ultrarelativistic heavy-ion collision, on the other hand, a substantial part of the entropy is associated with degrees of freedom that do not carry a net baryon number, and gradients in the entropy per baryon, Q, can arise effectively from gradients in the baryon number density, even if the entropy density is constant in space. To determine whether a given flow is convectively stable requires knowledge of the thermodynamic derivative (&/OT )P, as well as the entropy gradient along the acceleration . For a quark-gluon plasma at very high temperature or baryon density, the running coupling constant, ac = ac (T, #E,), a function of t-inperature, T, and baryon chemical
662
Florkonski et al. / ContAective stability
tentral, B, is small, so that the equation of state can be evaluated perturbatively . To first order in ac the pressure for two flavors 7 ) is
A) Pv MA)
= 377x2 90
I _ 110 ac ) 37 7r /
+
ac(T) =
(
I
_ 2ac
2
pB T
+
ué 2
.
(6) 162ít We omit the bag constant in eq. (6), as it does not affect the present results. When juB « T we need consider only the temperature dependence of the coupling constant ; the leading term of the renormalization group relation gives T4
7r
9
g2 (T)
_ 6a' 4r 291n (T/A) ' where A is il"e q.c.d. scale parameter. A straightforward calculation yields _) 2 L74 (8) s ` ''s 37 a T ) n - ~- T3 n 7r ' a positive result not dependant on the scale A ; here s is the entropy density of the non-interacting plasma at pB =_ 0. For general #B, the pressure is of the form P = 71f (#/ T, ac ), where the running coupling constant ac depends on both the temperature and baryon chemical potential . Then the energy density is given by (9) e = 3P -t- Ts af áa~ r~ac ( 0 T
) ju/ T "
where according to the beta-function equation for the scale dependence of the running coupling constant, âac _ _ 29 2 T
~ OT
(10)
67t a `
/ u/T
The convective stability of a perturbative quark-gluon plasma is independent of the dependence of the running coupling constant on #/T; with (10), (9), and (6), we again obtain
oe
(OR,
_
LI
(
7r2T3
2C
)
2 .
If the plasma contains strange in addition to up and down quarks it is no longer reasonable to neglect the quark masses. In this case (Oe/oT)p is non-zero due to the finite mass of a strange quark ms. We consider only the effect of ms > 0 for ps = 0, and neglect the corrections to the equation of state of order ac and treat the strange quarks by Boltzmann statistics. The pressure then has the form
pB
1 p22 T2 + ps (T" MB) = 90 7r2 T4 + 1627c2 p2
+ 4 [ d3p .t
(2a) 3
P2
+ ms
e
-
cp2 +`~ )IT
(12)
the latter integral is given by 23 (m,T/7r )2 K2 (ms / T), where K2 is the modified Bessel function . Using the equation of state (12) we find 6
( a£
_
ms T
7r 2
( ms)
K
2 \ T
(13)
W. Florkowski et aL / Convective stability
663
We note that (OE/OT ) P is positive in this case as well. In carrying out hydrodynamic calculations below, we consider both equations of state, (6) and (12 ), for the plasma phase. The derivative (ac/OT) P has not yet been calculated for general temperatures and baryon densities in Monte Carlo simulations of q.c.d. However, when the baryon chemical potential, #B, is small compared to the temperature, one can infer the stability from the properties of the zero baryon system. For hot matter with low baryon density we can expand the pressure around pB = 0: p(T,hB) =
po(T) + 2xB(T)pB,
where XB(T) =
(~nB ) ~B
T
=
T
«n 2 ) - (n)2)
(14) (15)
is the baryon number susceptibility at OB = 0; as we see from eq. (15), XB is essentially the equal-time long-wavelength density-density correlation function. The coefficient of the term linear in AB in eq. (14) vanishes since the baryon density nB = (ap/aps)T vanishes at pB = 0. Keeping only the lowest order terms in #B/ T we find
ae _- 3 - a In (XB/T2 ).) =s(' CaT) P cg lnT CS2
(16)
where s is the entropy density and 2
cs
=
(8 1nT a ln s
)MB=O
(17)
is the velocity of sound for JIB _-_ 0. The quantities appearing on the right-hand side of eq. (16) can be evaluated in lattice Monte Carlo simulations of q.c.d. at uB = 0. Gottlieb et al. 12 ) have calculated the quark-number susceptibility Xq = an q /aju q = 9X B in q.c.d. with two light flavours, as a function of ß = 6/g2. The calculations were done on an 83 x 4 lattice with a small quark mass, mqa = 0.05, where a is the lattice spacing (related to the temperature by TNT a = 1, where NT is the number of lattice sites in the time direction) . Gottlieb et al. give results for 9 2 (18) XL=a Xq= 7,2N2 XB T
thus
a In (XB/T2 ) = a In XL 8,0 alnT aß c)InT 12), Using relation (7) and estimating aX L /a ß ;zt~ 0.35 and X L 0.22 from fig. 1 of ref. we find a In (XB/ T 2)/8 In T ;:z~ 1 .2 for T = 1 .25 T,.* By comparison, a peaurbative * Because of the large uncertainties in the velocity of sound at T = Tc we choose a som"What higher temperature.
`V . Florkowski et al. / Convective stability
664 Iculation yields
B = §2 (1
- ta c /x) T2 and 0InQB/T2 ) _ 87 fl-2 (20) - 4 na 8 In T to second order in the coupling constant. The perturbative result (20), , 0.05, is much smaller than the Monte Carlo estimate, N 1.2. The latter result is, however, very sensitive to finite lattice-size effects. We estimate the sound velocity from the Monte Carlo results of Redlich and Satz 13). For T = 1 .25 Tc , their fig. 2 implies cs se, 0.22. From eq. (16) we find that (ec/e T)p is again positive, and over an order of magnitude larger than the perturbative result. One should not take this estimate too seriously; a more reliable estimate of (8E/8 T)p should extract both d In (X B/ T2) / d In T and cs from the same simulation, a calculation not, to our knowledge, available at present . We now consider the convective stability of matter in the hadronic phase. At temperatures small compared to the nucleon mass MN we treat the system as a gas of non-interacting pions and nucleons. For simplicity we neglect the rest mass of the pions and treat the nucleons by Boltzmann statistics. Then the pressure is given by Pn (T, luB) = 3 §ô n2 T4 and we find ( aE _ `~BT ) p -
2mN T + 12 T ( n )
~ 2 n2 T3 9°
3
2
-
N MN T
-
3/2
AB
[ex p (
)3/2
?
n
T-MN l
3 ( mN T
T
)5/2
+
exp
-JIB
T
MN
)] '
(21)
e-MNIT cosh(lcB/T) .
(22) Note that (8Elc)T)p is negative in the entire temperature range of interest (T < 600 Mev < 41N) ; a hadron gas described by the above equation of state has the same stability properties with respect to convection as a normal atmosphere. Unlike in the quark-gluon plasma phase, an estimate of (8E/8T)p cannot be reliably extracted from current lattice calculations at T < Tc. In models of ultrarelativistic nucleus-nucleus collisions based on an inside-outside cascade, the baryon number is concentrated in the fragmentation regions and is small in the central rapidity region. Unless the energy density is very much higher in the fragmentation regions, the entropy per baryon is lower in the fragmentation regions than in the central region, i .e ., dQ/dz < 0 for z > 0. For small fluid velocities, e.g., in the region close to z = 0, we have dQ/dz' -- dQ/dz and dv'/dt' ;z:', dv/dt. Consequently, in that case we would expect a convective instability in the plasma phase if the fluid is accelerated towards the central region, and an instability in the hadrollic phase if the fluid is accelerated towards the fragmentation regions . 4. Hydrodynamic equations We turn now to studying explicitly the development of instabilities in hydrodynamic flow in 1 + 1 dimensional dynamics . In this section we review the hydrodynamic
W. Florkowski et al. / Convective stability
66 5
equations and the method ofsolution, and in the following section we present numerical results. Hydrodynamic flow is governed by the conservation laws for energy, momentum and the baryon number, which read:
aOr
= 0,
(23)
Op n « = 0 .
(24)
The energy-momentum tensor Tu' is given by T~~
( + N) ur° u V - pgUV
(25)
nBup .
(26)
and the baryon current by nB =
Here guv is the metric tensor (with goo = 1) . We take into account here only the longitudinal expansion; then the transverse components of T"v vanish, and up can be regarded as a two-vector uu = (u`, u'). The acceleration is given by eq. (5). Eqs. (23)-(26) must be supplemented by an equation of state. We use eq. (6) or (12) for the quark-gluon plasma, and for the hadron gas eq. (21). From the pressure as a function of the temperature and the baryon chemical potential other thermodynamic quantities follow from the standard relations c + p = Ts + uBnB, and dp = AT + nBdpB . In the absence of dissipation, the conservation law for entropy is
aM (sue) = 0 .
(27)
The entropy per particle, Q, is conserved; thus the sound velocity Cs=( ap áC, ) Q ,
(2s)
directly connects gradients of pressure with gradients of the energy density. The equation of state of the quark-gluon plasma (6) gives cs = 110 to first order in ac. On the other hand, for the plasma containing strange quarks (12), and for a hadron gas (21 ), cs depends in general on both the temperature and the baryon chemical potential . However, this dependence is very weak, since the massless particles dominate in both . equations of state, i.e., cS z:~ 1/f In 1 + 1 dimensions it is convenient to use as variables the rapidity
-
y = !In
1 +v ,
(29)
the invariant time, z ---
t2 - z2,
(30)
and the space-time rapidity, +` il = !ln t1 -z ,
(31)
. Florkowski et al ® Convective stability
666
which have simple transformation properties under Lorentz boosts along the collision is. Initial the alization is expected to take place on a hyperbola in space-time, r = To [ref. 8)] . n terms of (29)-(31), the hydrodynamic equations are 9 ) C9 =0 ~)P w (óß +vaq)Y++
(32)
Y =0
(33)
C9
a +v a a aq
w ere
a +v
nB
a
+
a v a4 +
nB
WI-7-
s+s v' + ~
)y
~ = in(T/To),
v = tank(y - q) .
=0,
(34) (35) (36)
Eqs. (33) and (34) are the conservation laws of baryon number and entropy, while eq. (32) is Euler's equatioal . Eqs. (32)-(34) admit Bjorken's boost-invariant solution, in which the thermodynamic quantities are functions of T (or ~) only, and y = ri [ref. 8 ) ] . Generalized to finite baryon density, the Bjorken scaling solution is . To) nB ( (37) s(T) (LO ), n(T)
so T It is useful for numerical solution to cast eqs. (32)-(34) into characteristic form . e define the potentials 0t by the relations
dO:': =
d
f dy ; w using (28) we find the characteristic equations
a
CSW
v+cs
e
=0 +lfcsvari ) 40t (K +v~
Q =0,
(38) (39) (40)
which are easily integrated numerically . In the next section we solve these equations for various initial conditions in both a quark-gluon plasma and a hadron gas. In terms of ri and ~ , the convective stability condition takes the form n o ,6 ~ X12=- T 2 aY aY aQ )0 . (41) +v c w (9T, 4917 ) a n (a For the scaling solution, 0 vanishes. esults As illustration we present in this section numerical results for the convective stability of the longitudinal flow of matter created in ultrarelativistic heavy-ion collisions. At
W. Florkowski et al. / Convective stability
667
high collision energies the nuclei are expected to be partially transparent in the sense that the matter is created with strong longitudinal flow and the baryon number remains mainly in the fragmentation regions. Consequently, we assume as initial conditions for the hydrodynamic equations that the temperature and baryon density distributions at z = ro (~ = 0) are given by* = Toexp
[- (I!/IT)21 ~ no (q) = nBexp [-(r1 2 - nF) 2/ B1 . To(q)
(42) (43)
Here To is the initial temperature in the central region and nB is the initial baryon density in the fragmentation regions. The width of the temperature distribution is determined by ?IT. The baryon density distribution (43) is chosen to have peaks at r7 = fr1F to describe the baryon distribution in the fragmentation regions. The parameter -IB determines the depth of the hollow in between the peaks. We take the rapidity initially to be Yo(r1) = n .
(44)
Before presenting our numerical results, we discuss an approximate solution of the hydrodynamic equations (32)-(34) with the initial conditions (42)-(44) that is useful in selecting the parameters of the initial conditions in such a way that the final distributions resulting from the hydrodynamic flow are qualitatively similar to the experimental distributions. The hydrodynamicequations for one-dimensional flow simplify considerably in the limit when the entropy density is much larger than the baryon density. Depending on the equation of state, this may be the case even if the baryon chemical potential is comparable to the temperature. In such a situation we can use the following approximate relations w ;zz~ Ts and dp ;z:~ s d T in eqs. (39) and (40) to find the approximate characteristic equations a1nT 1 -cs aInT 2 1 -v2 _ay an +cs 1 _ sv2a rl _0 +v1_csv2 a c
ay + 1 -v2 aInT + 1 v - cs aY =0 . 2î 2 _ _ cs v2 a1l a 1 1 C v2 arl
(45)
(46)
As long as 6 - y - %l is small, 161 « 1, and the gradient of the temperature is not large, 1611 (a In T/ar7) j « c,21 (1 - cs) ~ 0.5, eqs. (45) and (46) reduce to the very simple forms ,0InT(4,g) + cs = 0, a~ alnT(~,g) + 08(~,rl) &(1 _ cs) = 0 . K + 017
(47) (48)
* At present CERN energies, the stopping of baryon number in Pb + Pb collisions could be so strong that the type of initial conditions we consider here would be inappropriate.
Florkouski et al. / Convective stability r approximately constant sound velocity and with initial conditions (42)-(44), these
cations have the following solution:
T (r,
= To(q) zo 'r (
csa
'
21 (1 cs ) J!T
T,
nB(r,q) = n O]B (q) o To
s
i _~
(49)
The conditions 131 < l and 16110 In T/8gl < 0.5 are satisfied in the space-time rapidity interval J l < q2 Note that 9, eq. (41), does not vanish for the above solution, as it does for longitudinal scaling solutions of Bjorken. Thus, depending on the equation of state, the flow can exhibit convective instability. The above approximate solution suggests that in the case of strong initial longitudinal flow the rapidity profiles ofthe temperature and the baryon number density change only slightly during the expansion. To the extent that the measured distributions resemble the initial ones, we can estimate the initial conditions from present experimental results at CEIZN. In collisions at 200 GeV per nucleon the measured rapidity distribution of charged particles '°) is approximately Gaussian, dN/dy - exp (-y2/2Q,2 ), with, e.g., 1.6 for O + W. Assuming that dN/dy - s - T3, and using the approximate solution (49) - (50) one obtains r1T r. V6-ay , 4, a value useful for present estimates. Similarly, to estimate the parameters in (43) we approximate the observed number of baryons per unit rapidity in terms of the initial baryon density by dNB (y) /dy nOB (y) rte, where S is the area of the transverse cross section of the slab containing 1/3 the matter. If we take To = 1 fm and S = 7rR2A , where RA 1 .2A fm is the nuclear radius, then the experimental data on the rapidity distribution of protons in S + S collisions at 200 GeV per nucleon ") suggest that nB - 0.2 fm-3, 1jF ti 2 and qB - 3 for central collisions. For central collisions of larger nuclei nB will presumably be bigger. To account for this we choose nB = 0.34 fm-3, twice the normal baryon density These values of the parameters r;T, n o , r7B and 17F will be used in all numerical examples below. We adjust To to keep the initial energy densities for various equations of s;a!-: similar. Let us first consider a quark-gluon plasma. We take the initial temperature To to be 250 MeV and the strongcoupling constant ac = 0.2. In Fig. 1 a we show the "laaiei cäi solution of the hydrodynamic equations for the space-time evolution of the thermodynamic variables and the rapidity. The full line shows the quantities t th` initial time T = To and the dashed line at time T = 2TO. We note that the dependence predicted by the approximate solution (49)-(50) agrees with the numerical results . In fig. lb we show the values of S22 at T = To and 2TO. In the interesting range of space-time rapidity the restoring frequency 9 is always real ; for these initial conditions the by d . od ynam~c flows of the quark-gluon plasma is convectively stable. The situation is similar when we use the equation of state (12) which includes strange quarks . Now ac = 0 and (&/a T)p is calculated from eq. (13). In this case we
W. Florkowski et al. / Convective stability P-9
w
4 t0
92
.5 0
i
0.0 QA
"-" 20
a4
n 1- m r N
0.3
5
op
00
%
0 .2
%,
0.1 0.5
0.05
N
669
r
11
a
-1
0
1
0.04
C 0.03 0.02 00
.
.
\ (a)
3
2
j
-1
0
1
2
3
-3
-2
11 = O.SW(t+Z)/(t-Z»
3
2
2
3
1q
2
= 0.5W(t+Z)/(t-Z»
3
8.0 ç 7.0 x 6.0
`r
5.0
`- 6 .0 ÓI 3.0 2.0 1.0 0.0
-1
0
1
i = 0.51n((t+z)/(t-Z') Fig. 1 . (a) Time evolution of thermodynamic quantities and rapidity (the lowest right picture) . The equation of state is assumed to be that of a quark-gluon plasma for two massless flavours. The solid line corresponds to the initial conditions at z = ro, whereas the dashed line represents the values at a later moment r = 2zo. The initial parameters are To = 250 MeV, nB = 0.34 fm-3 , qT = 4, qB = 3 and 1/F = 2. (b) Values of 0 corresponding to the expansion of a quark-gluon plasma (as in part (a) ).
670
Florkowski et al. ® Comertive stability c-2
0 x
1.0 .8 t \
-3
q
-2
-1
0
1
2
= 0.51n((t+z)/(t-z))
3
Fig. 2. Values of 9 characterizing the expansion of a quark-gluon plasma containing strange quarks. In this case To = 200 MeV and the .nass of a strange quark ms = 150 MeV.
choose an initial temperature To = 200 MeV. The values of SQ 2 are shown in fig. 2. Again j22 is positive. If the matter created in ultrarelativistic heavy-ion collisions forms a hadron gas, the general features of the space-time evolution of the thermodynamic quantities are similar to those in the quark-gluon plasma case . The results in fig. 3a are obtained with the same initial conditions as those in fig. 1 a except for the initial temperature To = 400 MeV. With the hadron equation of state, (21), Q2 becomes negative since in this case the thermodynamic derivative, (22), is negative and the fluid elements are accelerated outwards from the central region . We note however, that the timescale which characterizes the growth of the instability, 1 /I Q 1, extracted from fig. 3b, is on the order of 10-100 fm. This is of the same order as or longer than the expected expansion timescale, indicating that for the assumed initial conditions, convective instabilities would not develop strongly . As a third example, we discuss a situation closer to stopping, by letting the initial rapidity be less than ri. In this calculation we take yo = q, and use the perturbative 3 equation of state (6) for the plasma. The results are shown in figs . 4a and 4b. Even though both the velocity and the entropy per baryon number show stronger time dependence than for yo = rl, the general features of g22 (cf. figs. 1 b and 4b) are not changed . This change in the initial conditions affects mainly the magnitude of the acceleration ; the direction remains the same . Furthermore, the change in the velocity does not reverse the z-dependence of Q . As a result the sign of the factor al'a~Q in the stability condition (4) remains unchanged. If in heavy-ion collisions the stopping of the baryon number is stronger than that of the energy flow, the sign of dQ/dz could be reversed, giving rise to unstable flow in the plasma phase. Whether the timescale of the instability is longer than the plasma lifetime depends on the magnitude of dQ/dz.
W. Florkowski et al / Convective stability
fa
i
0
5
671
dpdi -00
0.40 ^ 0.95 a9o
a25 0.20 0_00
i
i .0
~~
~~%
a
i
.
0.05 0.04 0.02 3
2
-1
0
1
2
3
-3
-2
1 = 0.5W(t+z)!(t-z))
3
2 1 0 1 2 3 11 = 0.5tm(t+z)/(t-0
0.000
-0.005
-0.010
-0.015 -1
0
1
2
9 = 0.51n«t+z)/(t-z))
3
Fig. 3. (a) Similar situation to that in fig. 1a but here the matter is characterized by the equation of state of a hadron gas. The initial temperature is taken to be To = 400 MeV . The remaining initial parameters are as in fig. I a. (b) Values of Sl corresponding to the expansion of a hadron gas (as in part (a) ).
W Borkowski et al ® Convective stability
2
11
a -1 0 ' 2 3 = 0.5W(t+z)/(t-z))
-3
11
-2
-1
0
1
2
= 0.5ln«t+z)/(t-z»
3
2.0 0
0.5
0.0
~ (b) -s
n
-2
-1
0
1
2
= 0 .51n((t+z)/(t-z))
3
Fig . 4. (a) Evolution of a quark-gluon plasma under the same conditions as in fig. l a but with initial rapidity yo - 3 q. (b) Values of a corresponding to the expansion. of a quark-gluon plasma shown in part (a).
W. Florkowski et al. / Convective stability
673
6. Conclusions The crucial quantity in estimating the convective stability of matter is the derivative (8c/OT )p, which depends on the equation of state. In the case of. a quark-gluon plasma, the first order perturbative corrections to the ideal fluid equation of state lead to non-zero (ac/OT)p through the dependence of the running coupling constant on temperature. Similarly the finite mass of the strange quark results in a nonzero value of (OcIO T )p. In both cases the derivative is positive. Although the estimate of this derivative from lattice calculations differs in magnitude from the perturbative results, the sign is consistent. For a hadron gas of non-interacting nucleons and pions the derivative (OcIOT)p is negative. In this sense the plasma and the hadron gas have opposite stability properties, the latter behaving as a normal atmosphere. The stability of flows depends strongly on the initial conditions, which can change considerably with collision energy and the size of the nuclei . As we found in a onedimensional hydrodynamic expansion of the matter with initial conditions appropriate for partially transparent collisions with baryon number increasing towards fragmentation regions, the plasma flow is stable, while the hadron gas exhibits unstable behavior. In the examples studied, the typical timescale of the development of instability or of the oscillations around the equilibrium for the stable flo'%3d is 10-100 fm, larger than the expansion timescale, indicating that in these cases the flow is close to the neutral equilibrium with respect to convection . However, these conclusions on the timescale for growth of instabilities are very sensitive to the initial conditions, and magnitude of (OE1,1 T )r in the plasma phase. Experimentally, one may find larger accelerations and gradients giving rise to convective instabilities with shorter growth times. Author G.B. is grateful to J.D . Bjorken for early discussions that led to our considering this problem. W.F. would like to thank the Theory Group at GSI for very warm hospitality and B.L.F. and P.V.R. acknowledge the warm hospitality of the Physics Department of the University of Illinois, where this work was initiated . References 1) G. Baym, H. Monien, C.J. Pethick and D.G. Ravenhall, Phys. Rev. Lett . 64 (1990) 1867; Nucl. Phys. A525 (1991) 415c 2) A. Hosoya and K. Kajantie, Nucl. Phys. B250 (1985) 666 3) P. Danielewicz and M. Gyulassy, Phys. Rev . D31 (1985) 53 4) H. von Gersdorff, L. McLerran, M. Kataja and P.V. Ruuskanen, Phys. Rev. D34 (1986) 794 5) L.D. Landau and E.M. Lifshitz, Fluid mechanics (Pergamon, O-Jord, 1959) 6) B.L. Friman, G. Baym and P.V. Ruuskanen, Proc. XV Int. Worl.3hop on gross properties of nuclei and nuclear excitations, ed. H. Feldmeier, Hirschegg, Austria (1987) 7) S.A. Chin, Phys. Lett . B78 (1978) 552 8) J.D. Bjorken, Phys. Rev. D27 (1983) 140 9) K. Kajantie, R. Raitio and P.V. Ruuskanen, Nucl. Phys. B222 (1983) 152; M. Kataja, Z. Phys . C38 (1988) 419
74
i. Florkowski et al. / Convective stability
. St le et al (NÁ35 collaboration), Z. Phys. C38 (1988) 89; T. .Âk n et al. (HELIOS collaboration) Nucl. Phys. B333 (1990) 48; 1. Adamovich et al (EMU01 collaboration), Phys. Rev . Lett. 67 (1991) 1201; 1. Otterlund et al (EMU01 collaboration), Lund University Report LUIP 9105 (1991) 11) S. Wenig, GSI Report 90-23 (1990) 12) S. Gottlieb, VW. Liu, R.L. Renken, R.L. Sugar and D. Toussaint, Nucl. Phys. A498 (1989) 435c 13) It. Redlich and H. Satz, Phys. Rev . D33 (1986) 3747 10)
NUCLE PHYSICS
Convective stability of hot matter in ultrarelativistic heavy-ion collisions* Wojciech Florkowski a,b, Bengt L. Friman", Gordon Baymd and P. Vesa Ruuskanend,e ° Gesellschaft für Schwerionenforschung GSI, Postfach 110552, D-6100 Darmstadt, Germany b H. Niewodniczanski Institute of Nuclear Physics, ul. Radzikowskiego 152, 31-342 Krak6w, Poland c Institut Ar Kernphysik, Technische Hochschule Darmstadt, D-6100 Darmstadt, Germany d Department of Physics, University ofIllinois at Urbana-Champaign, 1110 W. Green St., Urbana, Il. 61801, USA e Department of Physics, University or.lyvdskyld, P.O. Box 35, 40_=51 Jyvdsky1d, 1-inland Received 22 January 1992 Abstract: The convective stability of strongly interacting matter undergoing hydrodynamic flow in ultrarelativistic heavy-ion collisions is studied in both the quark-gluon plasma and hadron gas phases . We find that this stability depends on the form of the initial conditions assumed for the hydrodynamic flow. In the case of initial conditions corresponding to partial transparency the flow of the quark-gluon plasma is stable whereas the flow of the hadron gas is convectively unstable. The timescale for hydrodynamic oscillations around the (stable or unstable) equilibrium state is found to be larger than the expected lifetime of the system, suggesting that the flow in either case is close to neutral convective equilibrium. 1. Introductio&t A fully developed quark-gluon plasma formed in a central ultrarelativistic nucleusnucleus collision is expected, after initial equilibration, to undergo an essentially hydrodynamic expansion phase in which the mean free paths of the constituent particles are small compared to the length scales of the system, e.g., the nuclear radii, and the collision times are short compared to the expansion time 1-4 ) . An important question is whether the flow is hydrodynamically stable . In a hydrodynamic expansion, the particles in an accelerated fluid element experience an inertial force in the direction opposite to the acceleration . If the entropy per particle varies in the direction of the acceleration the situation is analogous to that of a gas in a gravitational field, e.g., an atmosphere, which is convectively unstable if the entropy per particle decreases with increasing altitude 5 ) . In an expanding quark-gluon plasma, with the entropy density or baryon density varying in the direction of expansion, a convective instability, analogous to that in an atmosphere, is possible . Such an instability, if it has sufficient * Research supported in part by NSF Grants PHY84-15064 and PHY89-21025, and by the Polish Mate Committee for Scientific Research under Grant No. 2.0204.91 .01 . 0375-9474/92/$05 .00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
. Florkowski et al. / Convective stability ti .e to develop, would lead to interesting experimental consequences, e.g., enhanced fluctuations in rapidity and other distributions . In this per we continue the investigation of the convective stability of hot strongly interacting matter initiated in ref. 5 ) . We derive the convective stability criterion for relativistic hydrodynamic flow and apply it to a one-dimensional hydrodynamic flow o the matter produced in ultrarelati:-Anti;, üeaw-ion collisions. We consider both the cases that a quark-gluon plasma is formed and that the produced matter is a hadron gas. The paper is organized as follows . In sect. 2 we formulate the convective stability condition. In sect . 3 we calculate the thermodynamic quantities that determine the stability properties for a given equation of state . The acceleration of the fluid, which enters the stability condition, is estimated by solving the hydrodynamic equations in sect. 4. Our main results are presented in sect. 5 and conclusions in sect. 6. 2. Condition for convective stability To derive the conditon for convective stability, let us first recall the argument for a fluid at rest in a uniform gravitational field. Later, we replace the gravitational force by the local inertial force experienced by an accelerated fluid. Imagine that a fluid element, originally at height z, undergoes an adiabatic displacement upwards (against the force) to height z + ® z. A necessary condition for the fluid to be stable against convection is that the net force on the fluid element tends to bring it back to its original position. In a relativistic fluid gravity couples to the enthalpy density w = E + p, where E is the energy density and p the pressure, and so the condition is that the gravitational force per unit volume, --gw (where g > 0), should be stronger than the buoyancy force, -dp/dz. Since in equilibrium the net force vanishes, stability requires that the fluid element be heavier than the fluid it displaces ; thus gw (p', a) > gw (p', a'), where a = s/nB is the entropy per net baryon at the original point, and a' = a(z + d z) and p' = p (z + dz). A small displacement d z produces a restoring force, -g (dwldz )Pd z; thus for small j z the stability condition is g (a wl0a )P (daldz) < 0. Using the standard thermodynamic relation (âw/aa) P = (T/cp )(8c/8T) P, where T is the temperature, and c. the specific heat per net baryon at constant pressure, and the fact that the specific heat is positive we can write the stability condition as ac l d <0 . (1) g C8T I dz P The derivative (&/8 T )P vanishes under several conditions. The first is if a gas has zero net baryon number (or more generally does not have a composition degree of freedom) ; then only one thermodynamic variable is independent. Convective instability is possible only if there exists an independent composition degree of freedom. It also vanishes in a gas where the energy density is proportional to the pressure. A gas of non-interacting massless particles is in neutral equilibrium with respect to convection . For a non-relativistic f id the energy density is replaced by the mass density p in the stability condition (1) . For "normal" matter that expands when heated at constant
W. Florkowski et al. / Convective stability
661
pressure, (ap/a T)P < 0. Thus one sees the familiar condition for an atmosphere stable with respect to convection, that dQ/dz > 0, i.e., that the entropy per particle increases with increasing altitude. In order to find the timescale characterizing the development of a convective instability we observe that the force acting on the adiabatically shifted fluid element is proportional to the distance from the original equilibrium point. The inertial mass of the element is equal to its enthalpy . If stable, the system has small oscillations with frequency D Q2 _ -
g
w
aw
du .
Càa )P dz
When the stability condition (1) is not fulfilled, 0 becomes purely imaginary, and the timescale for growth is determined by 191-1 . To generalize eq. (1) to the case of an expanding fluid (without gravity) we simply replace, according to the equivalence principle, the gravitational acceleration, -gZ, by minus the acceleration of the fluid element in its local rest frame (we denote this frame by primes) . Thus we find that stability requires
aE du dv~ < 0, Ca T ~ P dz dt
(3)
where v' is the fluid velocity in the z' direction . We can rewrite (3) in the explicitly Lorentz covariant form, ae agaua < 0 . (OT)p
(4)
Here a. = (at , V) and au is the four-acceleration, where V is the four velocity: V = y (1, v), v is the flow velocity and y = (1- v 2) -'~? In the local rest frame V = (1, 0, 0, 0), a,` = (0, dv'/dt'), and eq. (4), for flow in the z-direction, simplifies to the expression (3). Note that a"uu = 0. 3. Estimating (óe/OT), in strongly interacting matter In an atmosphere, the entropy generally resides in the baryons. In matter created in an ultrarelativistic heavy-ion collision, on the other hand, a substantial part of the entropy is associated with degrees of freedom that do not carry a net baryon number, and gradients in the entropy per baryon, Q, can arise effectively from gradients in the baryon number density, even if the entropy density is constant in space. To determine whether a given flow is convectively stable requires knowledge of the thermodynamic derivative (&/OT )P, as well as the entropy gradient along the acceleration . For a quark-gluon plasma at very high temperature or baryon density, the running coupling constant, ac = ac (T, #E,), a function of t-inperature, T, and baryon chemical
662
Florkonski et al. / ContAective stability
tentral, B, is small, so that the equation of state can be evaluated perturbatively . To first order in ac the pressure for two flavors 7 ) is
A) Pv MA)
= 377x2 90
I _ 110 ac ) 37 7r /
+
ac(T) =
(
I
_ 2ac
2
pB T
+
ué 2
.
(6) 162ít We omit the bag constant in eq. (6), as it does not affect the present results. When juB « T we need consider only the temperature dependence of the coupling constant ; the leading term of the renormalization group relation gives T4
7r
9
g2 (T)
_ 6a' 4r 291n (T/A) ' where A is il"e q.c.d. scale parameter. A straightforward calculation yields _) 2 L74 (8) s ` ''s 37 a T ) n - ~- T3 n 7r ' a positive result not dependant on the scale A ; here s is the entropy density of the non-interacting plasma at pB =_ 0. For general #B, the pressure is of the form P = 71f (#/ T, ac ), where the running coupling constant ac depends on both the temperature and baryon chemical potential . Then the energy density is given by (9) e = 3P -t- Ts af áa~ r~ac ( 0 T
) ju/ T "
where according to the beta-function equation for the scale dependence of the running coupling constant, âac _ _ 29 2 T
~ OT
(10)
67t a `
/ u/T
The convective stability of a perturbative quark-gluon plasma is independent of the dependence of the running coupling constant on #/T; with (10), (9), and (6), we again obtain
oe
(OR,
_
LI
(
7r2T3
2C
)
2 .
If the plasma contains strange in addition to up and down quarks it is no longer reasonable to neglect the quark masses. In this case (Oe/oT)p is non-zero due to the finite mass of a strange quark ms. We consider only the effect of ms > 0 for ps = 0, and neglect the corrections to the equation of state of order ac and treat the strange quarks by Boltzmann statistics. The pressure then has the form
pB
1 p22 T2 + ps (T" MB) = 90 7r2 T4 + 1627c2 p2
+ 4 [ d3p .t
(2a) 3
P2
+ ms
e
-
cp2 +`~ )IT
(12)
the latter integral is given by 23 (m,T/7r )2 K2 (ms / T), where K2 is the modified Bessel function . Using the equation of state (12) we find 6
( a£
_
ms T
7r 2
( ms)
K
2 \ T
(13)
W. Florkowski et aL / Convective stability
663
We note that (OE/OT ) P is positive in this case as well. In carrying out hydrodynamic calculations below, we consider both equations of state, (6) and (12 ), for the plasma phase. The derivative (ac/OT) P has not yet been calculated for general temperatures and baryon densities in Monte Carlo simulations of q.c.d. However, when the baryon chemical potential, #B, is small compared to the temperature, one can infer the stability from the properties of the zero baryon system. For hot matter with low baryon density we can expand the pressure around pB = 0: p(T,hB) =
po(T) + 2xB(T)pB,
where XB(T) =
(~nB ) ~B
T
=
T
«n 2 ) - (n)2)
(14) (15)
is the baryon number susceptibility at OB = 0; as we see from eq. (15), XB is essentially the equal-time long-wavelength density-density correlation function. The coefficient of the term linear in AB in eq. (14) vanishes since the baryon density nB = (ap/aps)T vanishes at pB = 0. Keeping only the lowest order terms in #B/ T we find
ae _- 3 - a In (XB/T2 ).) =s(' CaT) P cg lnT CS2
(16)
where s is the entropy density and 2
cs
=
(8 1nT a ln s
)MB=O
(17)
is the velocity of sound for JIB _-_ 0. The quantities appearing on the right-hand side of eq. (16) can be evaluated in lattice Monte Carlo simulations of q.c.d. at uB = 0. Gottlieb et al. 12 ) have calculated the quark-number susceptibility Xq = an q /aju q = 9X B in q.c.d. with two light flavours, as a function of ß = 6/g2. The calculations were done on an 83 x 4 lattice with a small quark mass, mqa = 0.05, where a is the lattice spacing (related to the temperature by TNT a = 1, where NT is the number of lattice sites in the time direction) . Gottlieb et al. give results for 9 2 (18) XL=a Xq= 7,2N2 XB T
thus
a In (XB/T2 ) = a In XL 8,0 alnT aß c)InT 12), Using relation (7) and estimating aX L /a ß ;zt~ 0.35 and X L 0.22 from fig. 1 of ref. we find a In (XB/ T 2)/8 In T ;:z~ 1 .2 for T = 1 .25 T,.* By comparison, a peaurbative * Because of the large uncertainties in the velocity of sound at T = Tc we choose a som"What higher temperature.
`V . Florkowski et al. / Convective stability
664 Iculation yields
B = §2 (1
- ta c /x) T2 and 0InQB/T2 ) _ 87 fl-2 (20) - 4 na 8 In T to second order in the coupling constant. The perturbative result (20), , 0.05, is much smaller than the Monte Carlo estimate, N 1.2. The latter result is, however, very sensitive to finite lattice-size effects. We estimate the sound velocity from the Monte Carlo results of Redlich and Satz 13). For T = 1 .25 Tc , their fig. 2 implies cs se, 0.22. From eq. (16) we find that (ec/e T)p is again positive, and over an order of magnitude larger than the perturbative result. One should not take this estimate too seriously; a more reliable estimate of (8E/8 T)p should extract both d In (X B/ T2) / d In T and cs from the same simulation, a calculation not, to our knowledge, available at present . We now consider the convective stability of matter in the hadronic phase. At temperatures small compared to the nucleon mass MN we treat the system as a gas of non-interacting pions and nucleons. For simplicity we neglect the rest mass of the pions and treat the nucleons by Boltzmann statistics. Then the pressure is given by Pn (T, luB) = 3 §ô n2 T4 and we find ( aE _ `~BT ) p -
2mN T + 12 T ( n )
~ 2 n2 T3 9°
3
2
-
N MN T
-
3/2
AB
[ex p (
)3/2
?
n
T-MN l
3 ( mN T
T
)5/2
+
exp
-JIB
T
MN
)] '
(21)
e-MNIT cosh(lcB/T) .
(22) Note that (8Elc)T)p is negative in the entire temperature range of interest (T < 600 Mev < 41N) ; a hadron gas described by the above equation of state has the same stability properties with respect to convection as a normal atmosphere. Unlike in the quark-gluon plasma phase, an estimate of (8E/8T)p cannot be reliably extracted from current lattice calculations at T < Tc. In models of ultrarelativistic nucleus-nucleus collisions based on an inside-outside cascade, the baryon number is concentrated in the fragmentation regions and is small in the central rapidity region. Unless the energy density is very much higher in the fragmentation regions, the entropy per baryon is lower in the fragmentation regions than in the central region, i .e ., dQ/dz < 0 for z > 0. For small fluid velocities, e.g., in the region close to z = 0, we have dQ/dz' -- dQ/dz and dv'/dt' ;z:', dv/dt. Consequently, in that case we would expect a convective instability in the plasma phase if the fluid is accelerated towards the central region, and an instability in the hadrollic phase if the fluid is accelerated towards the fragmentation regions . 4. Hydrodynamic equations We turn now to studying explicitly the development of instabilities in hydrodynamic flow in 1 + 1 dimensional dynamics . In this section we review the hydrodynamic
W. Florkowski et al. / Convective stability
66 5
equations and the method ofsolution, and in the following section we present numerical results. Hydrodynamic flow is governed by the conservation laws for energy, momentum and the baryon number, which read:
aOr
= 0,
(23)
Op n « = 0 .
(24)
The energy-momentum tensor Tu' is given by T~~
( + N) ur° u V - pgUV
(25)
nBup .
(26)
and the baryon current by nB =
Here guv is the metric tensor (with goo = 1) . We take into account here only the longitudinal expansion; then the transverse components of T"v vanish, and up can be regarded as a two-vector uu = (u`, u'). The acceleration is given by eq. (5). Eqs. (23)-(26) must be supplemented by an equation of state. We use eq. (6) or (12) for the quark-gluon plasma, and for the hadron gas eq. (21). From the pressure as a function of the temperature and the baryon chemical potential other thermodynamic quantities follow from the standard relations c + p = Ts + uBnB, and dp = AT + nBdpB . In the absence of dissipation, the conservation law for entropy is
aM (sue) = 0 .
(27)
The entropy per particle, Q, is conserved; thus the sound velocity Cs=( ap áC, ) Q ,
(2s)
directly connects gradients of pressure with gradients of the energy density. The equation of state of the quark-gluon plasma (6) gives cs = 110 to first order in ac. On the other hand, for the plasma containing strange quarks (12), and for a hadron gas (21 ), cs depends in general on both the temperature and the baryon chemical potential . However, this dependence is very weak, since the massless particles dominate in both . equations of state, i.e., cS z:~ 1/f In 1 + 1 dimensions it is convenient to use as variables the rapidity
-
y = !In
1 +v ,
(29)
the invariant time, z ---
t2 - z2,
(30)
and the space-time rapidity, +` il = !ln t1 -z ,
(31)
. Florkowski et al ® Convective stability
666
which have simple transformation properties under Lorentz boosts along the collision is. Initial the alization is expected to take place on a hyperbola in space-time, r = To [ref. 8)] . n terms of (29)-(31), the hydrodynamic equations are 9 ) C9 =0 ~)P w (óß +vaq)Y++
(32)
Y =0
(33)
C9
a +v a a aq
w ere
a +v
nB
a
+
a v a4 +
nB
WI-7-
s+s v' + ~
)y
~ = in(T/To),
v = tank(y - q) .
=0,
(34) (35) (36)
Eqs. (33) and (34) are the conservation laws of baryon number and entropy, while eq. (32) is Euler's equatioal . Eqs. (32)-(34) admit Bjorken's boost-invariant solution, in which the thermodynamic quantities are functions of T (or ~) only, and y = ri [ref. 8 ) ] . Generalized to finite baryon density, the Bjorken scaling solution is . To) nB ( (37) s(T) (LO ), n(T)
so T It is useful for numerical solution to cast eqs. (32)-(34) into characteristic form . e define the potentials 0t by the relations
dO:': =
d
f dy ; w using (28) we find the characteristic equations
a
CSW
v+cs
e
=0 +lfcsvari ) 40t (K +v~
Q =0,
(38) (39) (40)
which are easily integrated numerically . In the next section we solve these equations for various initial conditions in both a quark-gluon plasma and a hadron gas. In terms of ri and ~ , the convective stability condition takes the form n o ,6 ~ X12=- T 2 aY aY aQ )0 . (41) +v c w (9T, 4917 ) a n (a For the scaling solution, 0 vanishes. esults As illustration we present in this section numerical results for the convective stability of the longitudinal flow of matter created in ultrarelativistic heavy-ion collisions. At
W. Florkowski et al. / Convective stability
667
high collision energies the nuclei are expected to be partially transparent in the sense that the matter is created with strong longitudinal flow and the baryon number remains mainly in the fragmentation regions. Consequently, we assume as initial conditions for the hydrodynamic equations that the temperature and baryon density distributions at z = ro (~ = 0) are given by* = Toexp
[- (I!/IT)21 ~ no (q) = nBexp [-(r1 2 - nF) 2/ B1 . To(q)
(42) (43)
Here To is the initial temperature in the central region and nB is the initial baryon density in the fragmentation regions. The width of the temperature distribution is determined by ?IT. The baryon density distribution (43) is chosen to have peaks at r7 = fr1F to describe the baryon distribution in the fragmentation regions. The parameter -IB determines the depth of the hollow in between the peaks. We take the rapidity initially to be Yo(r1) = n .
(44)
Before presenting our numerical results, we discuss an approximate solution of the hydrodynamic equations (32)-(34) with the initial conditions (42)-(44) that is useful in selecting the parameters of the initial conditions in such a way that the final distributions resulting from the hydrodynamic flow are qualitatively similar to the experimental distributions. The hydrodynamicequations for one-dimensional flow simplify considerably in the limit when the entropy density is much larger than the baryon density. Depending on the equation of state, this may be the case even if the baryon chemical potential is comparable to the temperature. In such a situation we can use the following approximate relations w ;zz~ Ts and dp ;z:~ s d T in eqs. (39) and (40) to find the approximate characteristic equations a1nT 1 -cs aInT 2 1 -v2 _ay an +cs 1 _ sv2a rl _0 +v1_csv2 a c
ay + 1 -v2 aInT + 1 v - cs aY =0 . 2î 2 _ _ cs v2 a1l a 1 1 C v2 arl
(45)
(46)
As long as 6 - y - %l is small, 161 « 1, and the gradient of the temperature is not large, 1611 (a In T/ar7) j « c,21 (1 - cs) ~ 0.5, eqs. (45) and (46) reduce to the very simple forms ,0InT(4,g) + cs = 0, a~ alnT(~,g) + 08(~,rl) &(1 _ cs) = 0 . K + 017
(47) (48)
* At present CERN energies, the stopping of baryon number in Pb + Pb collisions could be so strong that the type of initial conditions we consider here would be inappropriate.
Florkouski et al. / Convective stability r approximately constant sound velocity and with initial conditions (42)-(44), these
cations have the following solution:
T (r,
= To(q) zo 'r (
csa
'
21 (1 cs ) J!T
T,
nB(r,q) = n O]B (q) o To
s
i _~
(49)
The conditions 131 < l and 16110 In T/8gl < 0.5 are satisfied in the space-time rapidity interval J l < q2 Note that 9, eq. (41), does not vanish for the above solution, as it does for longitudinal scaling solutions of Bjorken. Thus, depending on the equation of state, the flow can exhibit convective instability. The above approximate solution suggests that in the case of strong initial longitudinal flow the rapidity profiles ofthe temperature and the baryon number density change only slightly during the expansion. To the extent that the measured distributions resemble the initial ones, we can estimate the initial conditions from present experimental results at CEIZN. In collisions at 200 GeV per nucleon the measured rapidity distribution of charged particles '°) is approximately Gaussian, dN/dy - exp (-y2/2Q,2 ), with, e.g., 1.6 for O + W. Assuming that dN/dy - s - T3, and using the approximate solution (49) - (50) one obtains r1T r. V6-ay , 4, a value useful for present estimates. Similarly, to estimate the parameters in (43) we approximate the observed number of baryons per unit rapidity in terms of the initial baryon density by dNB (y) /dy nOB (y) rte, where S is the area of the transverse cross section of the slab containing 1/3 the matter. If we take To = 1 fm and S = 7rR2A , where RA 1 .2A fm is the nuclear radius, then the experimental data on the rapidity distribution of protons in S + S collisions at 200 GeV per nucleon ") suggest that nB - 0.2 fm-3, 1jF ti 2 and qB - 3 for central collisions. For central collisions of larger nuclei nB will presumably be bigger. To account for this we choose nB = 0.34 fm-3, twice the normal baryon density These values of the parameters r;T, n o , r7B and 17F will be used in all numerical examples below. We adjust To to keep the initial energy densities for various equations of s;a!-: similar. Let us first consider a quark-gluon plasma. We take the initial temperature To to be 250 MeV and the strongcoupling constant ac = 0.2. In Fig. 1 a we show the "laaiei cäi solution of the hydrodynamic equations for the space-time evolution of the thermodynamic variables and the rapidity. The full line shows the quantities t th` initial time T = To and the dashed line at time T = 2TO. We note that the dependence predicted by the approximate solution (49)-(50) agrees with the numerical results . In fig. lb we show the values of S22 at T = To and 2TO. In the interesting range of space-time rapidity the restoring frequency 9 is always real ; for these initial conditions the by d . od ynam~c flows of the quark-gluon plasma is convectively stable. The situation is similar when we use the equation of state (12) which includes strange quarks . Now ac = 0 and (&/a T)p is calculated from eq. (13). In this case we
W. Florkowski et al. / Convective stability P-9
w
4 t0
92
.5 0
i
0.0 QA
"-" 20
a4
n 1- m r N
0.3
5
op
00
%
0 .2
%,
0.1 0.5
0.05
N
669
r
11
a
-1
0
1
0.04
C 0.03 0.02 00
.
.
\ (a)
3
2
j
-1
0
1
2
3
-3
-2
11 = O.SW(t+Z)/(t-Z»
3
2
2
3
1q
2
= 0.5W(t+Z)/(t-Z»
3
8.0 ç 7.0 x 6.0
`r
5.0
`- 6 .0 ÓI 3.0 2.0 1.0 0.0
-1
0
1
i = 0.51n((t+z)/(t-Z') Fig. 1 . (a) Time evolution of thermodynamic quantities and rapidity (the lowest right picture) . The equation of state is assumed to be that of a quark-gluon plasma for two massless flavours. The solid line corresponds to the initial conditions at z = ro, whereas the dashed line represents the values at a later moment r = 2zo. The initial parameters are To = 250 MeV, nB = 0.34 fm-3 , qT = 4, qB = 3 and 1/F = 2. (b) Values of 0 corresponding to the expansion of a quark-gluon plasma (as in part (a) ).
670
Florkowski et al. ® Comertive stability c-2
0 x
1.0 .8 t \
-3
q
-2
-1
0
1
2
= 0.51n((t+z)/(t-z))
3
Fig. 2. Values of 9 characterizing the expansion of a quark-gluon plasma containing strange quarks. In this case To = 200 MeV and the .nass of a strange quark ms = 150 MeV.
choose an initial temperature To = 200 MeV. The values of SQ 2 are shown in fig. 2. Again j22 is positive. If the matter created in ultrarelativistic heavy-ion collisions forms a hadron gas, the general features of the space-time evolution of the thermodynamic quantities are similar to those in the quark-gluon plasma case . The results in fig. 3a are obtained with the same initial conditions as those in fig. 1 a except for the initial temperature To = 400 MeV. With the hadron equation of state, (21), Q2 becomes negative since in this case the thermodynamic derivative, (22), is negative and the fluid elements are accelerated outwards from the central region . We note however, that the timescale which characterizes the growth of the instability, 1 /I Q 1, extracted from fig. 3b, is on the order of 10-100 fm. This is of the same order as or longer than the expected expansion timescale, indicating that for the assumed initial conditions, convective instabilities would not develop strongly . As a third example, we discuss a situation closer to stopping, by letting the initial rapidity be less than ri. In this calculation we take yo = q, and use the perturbative 3 equation of state (6) for the plasma. The results are shown in figs . 4a and 4b. Even though both the velocity and the entropy per baryon number show stronger time dependence than for yo = rl, the general features of g22 (cf. figs. 1 b and 4b) are not changed . This change in the initial conditions affects mainly the magnitude of the acceleration ; the direction remains the same . Furthermore, the change in the velocity does not reverse the z-dependence of Q . As a result the sign of the factor al'a~Q in the stability condition (4) remains unchanged. If in heavy-ion collisions the stopping of the baryon number is stronger than that of the energy flow, the sign of dQ/dz could be reversed, giving rise to unstable flow in the plasma phase. Whether the timescale of the instability is longer than the plasma lifetime depends on the magnitude of dQ/dz.
W. Florkowski et al / Convective stability
fa
i
0
5
671
dpdi -00
0.40 ^ 0.95 a9o
a25 0.20 0_00
i
i .0
~~
~~%
a
i
.
0.05 0.04 0.02 3
2
-1
0
1
2
3
-3
-2
1 = 0.5W(t+z)!(t-z))
3
2 1 0 1 2 3 11 = 0.5tm(t+z)/(t-0
0.000
-0.005
-0.010
-0.015 -1
0
1
2
9 = 0.51n«t+z)/(t-z))
3
Fig. 3. (a) Similar situation to that in fig. 1a but here the matter is characterized by the equation of state of a hadron gas. The initial temperature is taken to be To = 400 MeV . The remaining initial parameters are as in fig. I a. (b) Values of Sl corresponding to the expansion of a hadron gas (as in part (a) ).
W Borkowski et al ® Convective stability
2
11
a -1 0 ' 2 3 = 0.5W(t+z)/(t-z))
-3
11
-2
-1
0
1
2
= 0.5ln«t+z)/(t-z»
3
2.0 0
0.5
0.0
~ (b) -s
n
-2
-1
0
1
2
= 0 .51n((t+z)/(t-z))
3
Fig . 4. (a) Evolution of a quark-gluon plasma under the same conditions as in fig. l a but with initial rapidity yo - 3 q. (b) Values of a corresponding to the expansion. of a quark-gluon plasma shown in part (a).
W. Florkowski et al. / Convective stability
673
6. Conclusions The crucial quantity in estimating the convective stability of matter is the derivative (8c/OT )p, which depends on the equation of state. In the case of. a quark-gluon plasma, the first order perturbative corrections to the ideal fluid equation of state lead to non-zero (ac/OT)p through the dependence of the running coupling constant on temperature. Similarly the finite mass of the strange quark results in a nonzero value of (OcIO T )p. In both cases the derivative is positive. Although the estimate of this derivative from lattice calculations differs in magnitude from the perturbative results, the sign is consistent. For a hadron gas of non-interacting nucleons and pions the derivative (OcIOT)p is negative. In this sense the plasma and the hadron gas have opposite stability properties, the latter behaving as a normal atmosphere. The stability of flows depends strongly on the initial conditions, which can change considerably with collision energy and the size of the nuclei . As we found in a onedimensional hydrodynamic expansion of the matter with initial conditions appropriate for partially transparent collisions with baryon number increasing towards fragmentation regions, the plasma flow is stable, while the hadron gas exhibits unstable behavior. In the examples studied, the typical timescale of the development of instability or of the oscillations around the equilibrium for the stable flo'%3d is 10-100 fm, larger than the expansion timescale, indicating that in these cases the flow is close to the neutral equilibrium with respect to convection . However, these conclusions on the timescale for growth of instabilities are very sensitive to the initial conditions, and magnitude of (OE1,1 T )r in the plasma phase. Experimentally, one may find larger accelerations and gradients giving rise to convective instabilities with shorter growth times. Author G.B. is grateful to J.D . Bjorken for early discussions that led to our considering this problem. W.F. would like to thank the Theory Group at GSI for very warm hospitality and B.L.F. and P.V.R. acknowledge the warm hospitality of the Physics Department of the University of Illinois, where this work was initiated . References 1) G. Baym, H. Monien, C.J. Pethick and D.G. Ravenhall, Phys. Rev. Lett . 64 (1990) 1867; Nucl. Phys. A525 (1991) 415c 2) A. Hosoya and K. Kajantie, Nucl. Phys. B250 (1985) 666 3) P. Danielewicz and M. Gyulassy, Phys. Rev . D31 (1985) 53 4) H. von Gersdorff, L. McLerran, M. Kataja and P.V. Ruuskanen, Phys. Rev. D34 (1986) 794 5) L.D. Landau and E.M. Lifshitz, Fluid mechanics (Pergamon, O-Jord, 1959) 6) B.L. Friman, G. Baym and P.V. Ruuskanen, Proc. XV Int. Worl.3hop on gross properties of nuclei and nuclear excitations, ed. H. Feldmeier, Hirschegg, Austria (1987) 7) S.A. Chin, Phys. Lett . B78 (1978) 552 8) J.D. Bjorken, Phys. Rev. D27 (1983) 140 9) K. Kajantie, R. Raitio and P.V. Ruuskanen, Nucl. Phys. B222 (1983) 152; M. Kataja, Z. Phys . C38 (1988) 419
74
i. Florkowski et al. / Convective stability
. St le et al (NÁ35 collaboration), Z. Phys. C38 (1988) 89; T. .Âk n et al. (HELIOS collaboration) Nucl. Phys. B333 (1990) 48; 1. Adamovich et al (EMU01 collaboration), Phys. Rev . Lett. 67 (1991) 1201; 1. Otterlund et al (EMU01 collaboration), Lund University Report LUIP 9105 (1991) 11) S. Wenig, GSI Report 90-23 (1990) 12) S. Gottlieb, VW. Liu, R.L. Renken, R.L. Sugar and D. Toussaint, Nucl. Phys. A498 (1989) 435c 13) It. Redlich and H. Satz, Phys. Rev . D33 (1986) 3747 10)