How much entropy is produced in strongly coupled Quark-Gluon Plasma
  (sQGP) by dissipative effects?
How much entropy is produced in strongly coupled
Quark-Gluon Plasma (sQGP) by dissipative effects?
M.Lublinsky and E.Shuryak
Department of Physics and Astronomy, State University of New York, Stony Brook NY 11794-3800, USA
(Dated: November 4, 2018)
We argue that estimates of dissipative effects based on the first-order hydrodynamics with shear
viscosity are potentially misleading because higher order terms in the gradient expansion of the
dissipative part of the stress tensor tend to reduce them. Using recently obtained sound dispersion
relation in thermal N=4 supersymmetric plasma, we calculate the resummed effect of these high
order terms for Bjorken expansion appropriate to RHIC/LHC collisions. A reduction of entropy
production is found to be substantial, up to an order of magnitude.
PACS numbers:
Hydrodynamical description of matter created in high
energy collisions have been proposed by Landau [1] more
than 50 years ago, motivated by large coupling at small
distance, as followed from the beta functions of QED and
scalar theories known at the time. Hadronic matter is of
course described by QCD, in which the coupling runs in
the opposite way. And yet, recent RHIC experiments
have shown spectacular collective flows, well described
by relativistic hydrodynamics. More specifically, one ob-
served three types of flow: (i) outward expansion in trans-
verse plane, or radial flow, (ii) azimuthal asymmetry or
“elliptic flow” [2, 3], as well as recently proposed (iii)
“conical flow” from quenched jets [4]. These observation
lead to conclusion that QGP at RHIC is a near-perfect
liquid, in a strongly coupled regime [5]. The issue we
discuss below is at what “initial time” τ0 one is able to
start hydrodynamical description of heavy ion collisions,
without phenomenological/theoretical contradictions.
Phenomenologically, it was argued in [2, 3] that elliptic
flow is especially sensitive to τ0. Indeed, ballistic motion
of partons may quickly erase the initial spatial anisotropy
on which this effect is based. In practice, hydrodynamics
at RHIC is usually used starting from time τ0 ∼ 1/2fm,
otherwise the observed ellipticity is not reproduced.
Can one actually use hydrodynamics reliably at such
short time? How large is τ0 compared to a relevant “mi-
croscopic scales” of sQGP? How much dissipation occurs
in the system at this time? As a measure of that, we will
calculate below the ratio of the amount of entropy pro-
duced at τ > τ0 to its “primordial” value at τ0, ∆S/S0.
To set up the problem, let us start with a very crude
dimensional estimate. If we think that the QCD effective
coupling is large αs ∼ 1 and the only reasonable micro-
scopic length is given by temperature [14], then the rele-
vant micro-to-macro ratio of scales is simply T0τ0. With
T0 ∼ 400MeV at RHIC, one finds this ratio to be close
to one. We are then lead to a pessimistic conclusion: at
such time application of any macroscopic theory, thermo-
or hydro-dynamics, seems to be impossible, since order
one corrections are expected.
Let us then do the first approximation, including the
explicit viscosity term to the first order. Zeroth order (in
mean free path) stress tensor used in the ideal hydrody-
namics has the form
T (0)µν = (ǫ+ p)uµuν + p gµν (1)
while dissipative corrections are induced by gradients of
the velocity field. The well known first order corrections
are due to shear (η) and bulk (ξ) viscosities
δT (1)µν = η(∇µuν +∇νuµ −
∆µν∇ρuρ) + ξ(∆µν∇ρuρ)(2)
In this equation the following projection operator onto
the matter rest frame was used:
∇µ ≡ ∆µν∂ν , ∆µν ≡ gµν − uµuν (3)
The energy-momentum conservation ∂µ Tµν at this order
corresponds to Navier-Stokes equation.
Because colliding nuclei are Lorentz-compressed, the
largest gradients at early time are longitudinal, along the
beam direction. The expansion at this time can be ap-
proximated by well known Bjorken rapidity-independent
setup [6], in which hydrodynamical equations depend on
only one coordinate – proper time τ =
t2 − x2.
ǫ + p
= − 1
1− (4/3)η + ξ
(ǫ+ p)τ
where we have introduced the entropy density s = (ǫ +
p)/T . Note that for traceless Tµν (conformally invariant
plasma), the bulk viscosity ξ = 0.
For reasons which will become clear soon, let us com-
pare this eqn to another problem, in which large longi-
tudinal gradients appear as well, namely sound wave in
the medium. The dispersion relation (the pole position)
for a sound wave with frequency ω and wave vector q is,
at small q
ω = csq −
q2Γs, Γs ≡
Notice that the right hand side of (4) contains precisely
the same combination of viscosity and thermodynamical
http://arxiv.org/abs/0704.1647v1
parameters as appears in the sound attenuation problem:
the length Γs, which measures directly the magnitude
of the dissipative corrections. At proper times τ ∼ Γs
one has to abandon the hydrodynamics altogether, as
the dissipative corrections cannot be ignored.
For the entropy production (4) the first correction to
the ideal case is (1−Γs/τ). Since the correction to one is
negative, it reduces the rate of the entropy decrease with
time. Equivalently statement is that the total positive
sign shows that some amount of entropy is generated
by the dissipative term. Danielewicz and Gyulassy [7]
have analyzed eq. (4) in great details considering vari-
ous values of η. Their results indicate that the entropy
production can be substantial.
Our present study is motivated by the following ar-
gument. If the hydrodynamical description is forced to
begin at early time τ0 which is not large compared to
the intrinsic micro scale 1/T , then limiting dissipative
effects to the first gradient only (δT
µν ) is parametrically
not justified and higher order terms have to be accounted
for. Ideally those effects need to be resummed. As a first
step, however, we may attempt to guess their sign and
estimate the magnitude.
Formally one can think of the dissipative part of the
stress tensor δTµν as expended in a series containing all
derivatives of the velocity field u, δT 1µν being the first
term in the expansion. In general 3+1 dimensional case
there are many structures, each entering with a new and
independent viscosity coefficient. We call them “higher
order viscosities” and the expansion is somewhat similar
to twist expansion. For 1+1 Bjorken problem, the ap-
pearance of the extra terms modifies eq. (4), which can
be written as a series in inverse proper time
∂τ (sτ)
s (τ T )
(τ T )2
(Tτ)2n
We have put T here simply for dimensional reasons:
clearly Tτ is a micro-to-macro scale ratio which deter-
mines convergence of these series and the total amount of
produced entropy. Similarly, the sound wave dispersion
relation becomes nonlinear as we go beyond the lowest
order:
ω = ℜ[ω(q)] + iℑ[ω(q)] ; (7)
2 π T
2 π T
2 π T
)2n+1
= − 4πη
Based on T-parity arguments we keep only odd (even)
powers of q for the real (imaginary) parts of ω. The co-
efficients cn, rn and ηn are related since they originate
from the very same gradient expansion of Tµν . Although
both the entropy production series above and sound ab-
sorption should converge to sign-definite answer, the co-
efficients of the series may well be of alternating sign (as
we will see shortly).
Clearly, keeping these next order terms can be use-
ful only provided there is some microscopic theory which
would make it possible to determine the values of the
high order viscosities. For strongly coupled QCD plasma
this information is at the moment beyond current theo-
retical reach, and we have to rely on models. A particu-
larly useful and widely studied model of QCD plasma is
N = 4 supersymmetric plasma, which is also conformal
(CFT). The AdS/CFT correspondence [8] (see [9] for re-
view) relates the strongly coupled gauge theory descrip-
tion to weakly coupled gravity problem in the background
of AdS5 black hole metric. Remarkably, certain informa-
tion on higher order viscosities in the CFT plasma can be
read of from the literature and we exploit this possibility
below.
The viscosity-to-entropy ratio (η/s = 1/4π) deduced
from AdS [10] turns out to be quite a reasonable ap-
proximation to the values appropriate for the RHIC data
description. Thus one may hope that the information on
the higher viscosities gained from the very same model
can be well trusted as a model for QCD. Admittedly hav-
ing no convincing argument in favor, we simply assume
that the viscosity expansion of the QCD plasma displays
very similar behavior, both qualitative and quantitative,
as its CFT sister.
Our estimates are based on the analysis of the quasi-
normal modes in the AdS black hole background due to
Kovtun and Starinets [11]. The dispersion relation for
the sound mode, calculated in ref. [11], is shown in Fig.1.
The real and imaginary parts of ω correspond to the ex-
pressions given in (7). At q → 0 they agree with the
leading order hydrodynamical dispersion relation (5).
The first important observation is that the next order
coefficient η2 is negative, reducing the effect of the first
one when gradients are large. The second is that |ℑ[ω]|
has maximum at q/2πT ∼ 1, and at large q the imaginary
part starts to decrease. This means that the expansion
(7) has a radius of convergence q/2πT ∼ 1.
In order to estimate the effect of higher viscosities on
the entropy production in the Bjorken setup we first iden-
tify τ in (6) with 2π/q in (7). Second we identify the coef-
ficients cn with ηn. Both sound attenuation and entropy
production in question are one dimensional problems as-
sociated with the same longitudinal gradients and pre-
sumably the same physics. In practice we use the curve
for the imaginary part of ω (Fig. 1) as an input for the
right hand side of (6).
The numerical results are shown in Figs. 2 and 3 in
which we compare our estimates with the “conventional”
shear viscosity results from (4). To be fully consistent
with the model we set η/s = 1/4π. We also set the initial
temperature T0 = 300MeV while the standard equation
of state s = 4 kSB T
3. For the coefficient kSB we use the
“QCD” value
kSB =
2(Nc − 1)2 +
; nf = 3; Nc = 3
Fig. 2 presents the results for entropy production as a
1 2 3 4 5
w q qq
1 2 3 4 5
Imw q
FIG. 1: Sound dispersion (real and imaginary parts) obtained
from the analysis of quasinormal modes in the AdS black hole
background. The result and figure are taken from Ref.[11].
function of proper time for two initial times τ0 = 0.2 fm
and τ0 = 0.5 fm. The dashed lines correspond to the first
order result (4) while the solid curves include the higher
order viscosity corrections. Noticeably there is a dra-
matic effect toward reduction of the entropy production
as we start the hydro evolution at earlier times (the ef-
fect is almost invisible on the temperature profile). This
is the central message of the present paper.
Fig. 3 illustrates the relative amount of entropy pro-
duced during the hydro phase as a function of initial time.
If the fist order hydrodynamics is launched at very early
times, the hydro phase produces too large amount of en-
tropy, up to 250%. (Such a large discrepancy is not seen
in the RHIC data.) In sharp contrast, the results from
the resummed viscous hydrodynamics is very stable, and
does not produce more than some 25% of initial entropy,
even if pushed to start from extremely early times. The
right figure displays the absence of any pathological ex-
plosion at small τ0.
It is worth commenting that we carried the analysis
using the minimal value for the ratio η/s = 1/4π. We
expect that if this ratio is taken larger, the discrepancy
between the first order dissipative hydro and all orders
will be even stronger.
Before concluding this paper we note that a practi-
cal implementation of relativistic viscous hydrodynamics
had followed Israel-Stewart second order formalism (for
recent publications see [12]) in which one introduces ad-
0 2 4 6 8 10
0 2 4 6 8 10
(fm)τ
τ (fm)
= 0.2 (fm)0τ = 0.5 (fm)0τ
FIG. 2: Entropy production as a function of proper time for
initial time τ0 = 0.2 fm (left) and τ0 = 0.5 fm (right). The
initial temperature T0 = 300MeV. The dashed (blue) curves
correspond to the first order (shear) viscosity approximation
Eq.(4). The solid curve (red) is the all order dissipative re-
summation Eq.(6).
0 0.2 0.4 0.6 0.8 1 1.2
0 0.2 0.4 0.6 0.8 1 1.2
1.25T   = 300 (MeV)0 T   = 300 (MeV)0
τ0 (fm) τ0 (fm)
τs 0 0
∆ (     )
FIG. 3: Fraction of entropy produced during the hydro phase
as a function of initial proper time. The initial temperature
T0 = 300MeV. The left (blue) points correspond to the first
order (shear) viscosity approximation. The right (red) points
are for the all order resummation.
ditional parameter , the relaxation time for the system.
Then the dissipative part of the stress tensor is found as
a solution of an evolution equation, with the relaxation
time being its parameter. For the Bjorken setup, the dis-
sipative tensor thus obtained has all powers in 1/τ and
might resemble the expansion in (6) and (7). The use of
AdS/CFT may shed light on the interrelation between
the two approaches: the first step in this direction has
been made recently [13], resulting in numerically very
small relaxation time.
Finally, why can it be that macroscopic approaches like
hydrodynamics can be rather accurate at such a short
time scale? Trying to answer this central question one
should keep in mind that 1/T is not the shortest micro-
scopic scale. The inter-parton distance is much smaller,
∼ 1/(T ∗N1/3dof ) where the number of effective degrees of
freedom Ndof ∼ 40 in QCD while Ndof ∼ N2c → ∞ in
the AdS/CFT approach.
In summary, we have argued that the higher order dis-
sipative terms strongly reduce the effect of the usual vis-
cosity. Therefore an “effective” viscosity-to-entropy ratio
found from comparison Navier-Stokes results to experi-
ment, can even be below the (proposed) lower bound of
1/4π. We conclude that it is not impossible to use a hy-
drodynamic description of RHIC collision starting from
very early times. In particular, our study suggests that
the final entropy observed and its “primordial” value ob-
tained right after collision should indeed match, with an
accuracy of 10-20 percent.
Acknowledgment
We are thankful to Adrian Dumitru whose results (pre-
sented in his talk at Stony Brook) inspired us to think
about the issue of entropy production during the hydro
phase. He emphasized to us the important problem of
matching the final entropy measured after late hydro
stage with the early-time partonic predictions, based on
approaches such as color glass condensate. This work is
supported by the US-DOE grants DE-FG02-88ER40388
and DE-FG03-97ER4014.
[1] L. D. Landau, Izv. Akad Nauk SSSR, ser. fiz. 17 (1953)
51. Reprinted in Collected works by L.D.Landau.
[2] D. Teaney, J. Lauret and E. V. Shuryak, Phys. Rev. Lett.
86, 4783 (2001) [arXiv:nucl-th/0011058]. “A hydrody-
namic description of heavy ion collisions at the SPS and
RHIC,” arXiv:nucl-th/0110037.
[3] P. F. Kolb and U. W. Heinz, “Hydrodynamic
description of ultrarelativistic heavy-ion collisions,”
arXiv:nucl-th/0305084.
[4] J. Casalderrey-Solana, E. V. Shuryak and D. Teaney, J.
Phys. Conf. Ser. 27, 22 (2005) [Nucl. Phys. A 774, 577
(2006)] [arXiv:hep-ph/0411315].
[5] E.V.Shuryak, Prog. Part. Nucl. Phys. 53, 273 (2004) [
hep-ph/0312227].
[6] J. Bjorken, Phys. Rev. D27(1983)140
[7] P. Danielewicz and M. Gyulassy, Phys. Rev. D 31 (1985)
[8] J. M. Maldacena, Adv. Theor. Math. Phys. 2,
231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)]
[arXiv:hep-th/9711200].
[9] O. Aharony, S. S. Gubser, J. M. Maldacena,
H. Ooguri and Y. Oz, Phys. Rept. 323, 183 (2000)
[arXiv:hep-th/9905111].
[10] P. Kovtun, D. T. Son and A. O. Starinets, Phys. Rev.
Lett. 94, 111601 (2005) [arXiv:hep-th/0405231].
[11] P. K. Kovtun and A. O. Starinets, Phys. Rev. D 72,
086009 (2005) [arXiv:hep-th/0506184].
[12] U. W. Heinz, arXiv:nucl-th/0512049. R. Baier and
P. Romatschke, arXiv:nucl-th/0610108. R. Baier, P. Ro-
matschke and U. A. Wiedemann, Nucl. Phys. A 782, 313
(2007) [arXiv:nucl-th/0604006].
[13] M. P. Heller and R. A. Janik, arXiv:hep-th/0703243.
[14] Note we have ignored e.g. ΛQCD .
http://arxiv.org/abs/nucl-th/0011058
http://arxiv.org/abs/nucl-th/0110037
http://arxiv.org/abs/nucl-th/0305084
http://arxiv.org/abs/hep-ph/0411315
http://arxiv.org/abs/hep-ph/0312227
http://arxiv.org/abs/hep-th/9711200
http://arxiv.org/abs/hep-th/9905111
http://arxiv.org/abs/hep-th/0405231
http://arxiv.org/abs/hep-th/0506184
http://arxiv.org/abs/nucl-th/0512049
http://arxiv.org/abs/nucl-th/0610108
http://arxiv.org/abs/nucl-th/0604006
http://arxiv.org/abs/hep-th/0703243
