Viscosity and dissipation - early stages

0.11 MB·English

by  P. Bozek

9 ∗
0 Viscosity and dissipation - early stages
0
2 Piotr Boz˙ek †

n
a
Institute of Physics, Rzesz´ow University, PL-35959Rzesz´ow, Poland
J

5 and
1

] Institute of Nuclear Physics PAN, PL-31342Krak´ow, Poland
h
t
-
l
c
u
A very early start up time of the hydrodynamic evolution is needed
n
in orderto reproduceobservationsfromrelativistic heavy-ioncollisionsex-
[
periments. At suchearly times the systems is still not locally equilibrated.

1 Another source of deviations from local equilibrium is the viscosity of the
v
fluid. We study these effects at very early times to obtain a dynamical
2
prescription for the transition from an early 2-dimensional expansion to a
7
nearly equilibrated 3-dimensional expansion at latter stages. The role of
2
viscosity at latter stages of the evolution is also illustrated.
2
.
1
PACS numbers: 25.75.-q, 25.75.Dw, 25.75.Ld
0
9
0
1. Introduction
:
v
i Recent hydrodynamic calculations modelling heavy-ion collisions can
X
reproduce experimentally measured soft observables : transverse momen-
r tum spectra, collective elliptic flow and Hanbury-Brown Twiss correlation
a
radii [1, 2] if the initial time of the collective expansion is pushed down to
τ = 0.25fm/c. This raises the question about the applicability of perfect
0
the fluid hydrodynamics at such small proper times. The mechanism of
the formation of the dense matter in the fireball is not understood up to
now. However, in all imaginable scenarios some time is required for the
formation of the matter constituents and for their subsequent equilibration.
In hydrodynamics, which is a coarse-grained description, the dynamics is
definedby thelocal thermodynamicalquantities, such as the energy density
∗ TalkpresentedattheWorkshoponParticleCorrelationsandFemtoscopy,September
2008,Cracow,Poland. SupportedbyPolishMinistryofScienceandHigherEducation
undergrant N202 034 32/0918.
† email: [email protected]
(1)
2 wpcf printed on January 15, 2009
and pressure. The details of the underlying microscopic degrees of freedom
are irrelevant. Although formally, perfect fluid thermodynamics requires
that local thermal equilibrium is maintained, phenomenological applicabil-
ity of the hydrodynamics in the description of heavy-ion collisions starts as
soon as the pressure becomes approximately isotropic. The dense matter
in the fireball can be described by the hydrodynamic model after the time
when the effective pressure in the system is similar in the longitudinal and
transverse directions. Complete kinetic equilibrium is not required, since
the model has other sources limiting the robustness of its predictions, such
as the uncertainties in the high temperature equation of state, in the initial
density, and in the freeze-out procedure.
When the deviations of the energy momentum tensor Tµν from its form
µν
in a perfect fluid T
0
Tµν = Tµν +πµν (1)
0
is small the evolution can be formulated as the hydrodynamics of a viscous
fluid [3, 4, 5, 6, 7, 8]. But, in the very early evolution the initial anisotropy
of the pressure is the main contribution that makes the matter to evolve
differently from the perfect fluid [9]. These early dissipative effects are
strong, since the initial pressure anisotropy is large.
2. Early dissipation
Theinitial anisotropy of thepressureand its relaxation towards theper-
fect fluid value cannot be reliably described with the second order Israel-
Stewart relativistic viscous fluidformalism [10]. Theapplicability of thethe
viscous fluid equations requires πµνπ ≪ p2, where πµν is the stress ten-
µν
sor. Instead, we propose an effective description of the transition from the
anisotropicsystemwithatwo-dimensionalpressuretothethree-dimensional
hydrodynamics [9]. The energy momentum tensor is the sum of the perfect
fluid energy momentum tensor and a stress correction
ǫ 0 0 0 0 0 0 0
0 p 0 0 0 π/2 0 0
Tµν =  +  . (2)
0 0 p 0 0 0 π/2 0
 0 0 0 p   0 0 0 −π 
   
   
The dissipative correction π quantifies the pressure anisotropy in the trans-
verseandlongitudinaldirections. Asimilarformofthestresstensorappears
in the hydrodynamics with shear viscosity for the case of the Bjorken flow
[4]. For large stress corrections the second order viscous hydrodynamics
equations for π cannot be reliably applied. Instead an effective equation
describing the relaxation of the pressureasymmetry is used. Neglecting the
wpcf printed on January 15, 2009 3
1.5
1.4
1.3
0
S 1.2
(cid:144)
S
1.1
1.0
0.9
0 2 4 6 8
Τ@fm(cid:144)cD
Fig.1. Relative increase of the entropy from dissipative processes in the early
stage of the collision for several initial times τ of the evolution. The dotted line
0
represents the entropy production from the Navier-Stokes shear viscosity tensor
with η = 0.1 s, the dashed line represents the increase of the entropy obtained
fromthe secondorderviscoushydrodynamicequationwith η =0.1 s,τ =6η/Ts,
π
and Π(τ ) = 4η , and the solid represents the relative entropy production due to
0 3τ0
the stress tensor term of the form Π(τ)=p(τ )exp(−(τ −τ )/τ ) [9].
0 0 0
shear viscosity we take
π(τ) = π(τ )e−(τ−τo)/τπ, (3)
0
where τ is a phenomenological parameter, in principle unrelated to the
π
relaxation time in the Isreal-Steward equation for the stress-tensor.
The dynamics is followed using a numerical solution of the relativistic
hydrodynamic equations
∂ Tµν = 0 (4)
µ
with some assumed symmetry of the fireball. Entropy production from the
dissipative relaxation of the pressure can be estimated in the Bjorken so-
lution. Depending on the ratio τ /τ , up to 30% increase of the entropy
π 0
is possible in the early phase. This additional entropy forces a retuning of
the initial conditions of the evolution to reproduce final particle multiplic-
ities. After this retuning is taken into account, most of the effect of the
early dissipation on final observables is canceled. However, we note that
the transverse momentum spectra of final particles are harder if the early
dissipative phase is present.
4 wpcf printed on January 15, 2009
D D
2 2
V V 10
e 100 e
G G
2(cid:144) 10 2(cid:144) 1
c c
@ 1 @
y y 0.1
d d
0.1
T T
p p
2 0.01 2 0.01
d d
(cid:144) (cid:144)
N N
d 0.0 0.5 1.0 1.5 2.0 2.5 3.0 d 0.0 0.5 1.0 1.5 2.0 2.5 3.0
p @GeV(cid:144)cD p @GeV(cid:144)cD
T T
Fig.2. π+ (left) and proton (right) spectra from hydrodynamic calculations (Solid
and dashed-dotted line are for the ideal hydrodynamics starting at τ = 1fm/c
0
and τ = 0.5fm/c respectively.The dotted and dashed lines are for the dissipative
0
evolution corresponding to τ = 1fm/c and τ = 0.5fm/c.). Data are from the
0 0
PHENIX Collaboration[11] for most central events (0-5%) [9].
3. Dissipation and viscosity
We use relativistic hydrodynamics with viscosity [10]. The stress tensor
πµν is the solution of a dynamical equation
ηT τ uβ
τ ∆µ∆νπαβ +πµν = η < ∇µuν > − πµν∂ π (5)
π α β 2τπ β ηT !
where
2
< ∇ u >= ∇ u +∇ u − ∆ ∇ uα , (6)
µ ν µ ν ν µ µν α
3
∇µ = ∆µν∂ (7)
ν
with uµ the fluid velocity, ∆ = g −u u , η the shear viscosity, τ the
µν µν µ ν π
relaxation time. We solve the equations numerically in a boost-invariant
geometry with an azimuthally asymmetric expansion in the transverse di-
rections. We use η/s = 1/4π, τ = 0.25fm/c and πzz(τ )/2 = πxx(τ ) =
0 0 0
πyy(τ ) = p/2. Compared to other calculations of the hydrodynamic model
0
with viscosity, we use a small initial time and a large value of the initial
stress correction π(τ ). The model encompasses both the relaxation of the
0
initial pressure anisotropy, and the latter interplay of the relaxation and
velocity gradients. To compare with perfect fluid results again a retuning
of the initial energy density is necessary to reproduce the final multiplic-
ity. The additional transverse push is strong, it has a contribution from
the initial stage of large pressure anisotropy and another one due to the
viscosity driven stress corrections. As a consequence of the prolongated
transverse push, the transverse momentum spectra get even harder for the
wpcf printed on January 15, 2009 5
0.6
0.5
0.4
p 0.3
(cid:144)
x
Px 0.2
0.1
0.0
2 4 6 8 10 12
t Hfm(cid:144)cL
Fig.3. Ratioofthe stresscorrectionto the pressureatthe center ofthe fireballfor
two initial conditions for πxx.
0.3
103 pions c=20-30% Tf=145MeV pions c=20-30% Tf=145MeV
T v2
p
2d102 visc. fluid η/s=1/4π
N/ 0.2 initial dissipation Π=Π0e-t/τ
d id. fluid
10
visc. fluid η/s=1/4π 0.1
initial dissipation Π=Π0e-t/τ
1 id. fluid
0
0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5
pT (GeV) pT (GeV)
Fig.4. Transverse momentum spectra (left) and elliptic flow coefficient (right)
for π+ for the perfect fluid (solid line), for the perfect fluid with initial pressure
anisotropy (dashed line) and for the viscous fluid (dashed-dotted line).
case when shear viscosity and initial anisotropy are combined than for the
case with only initial dissipation. A similar effect is observed for the elliptic
flow. The reduction of the azimuthal asymmetry is the strongest for the
viscosity+dissipation scenario of the fluid evolution.
4. Summary
We discuss dissipative effects in the very early phase of the collective
development of the fireball created in relativistic heavy-ion collisions. The
6 wpcf printed on January 15, 2009
initial anisotropy of the effective fluid pressure must dissipate. In the pro-
cess entropy is produced. After the retuning of the initial conditions to
accommodate for this additional entropy, the effect of the early dissipation
is most pronounced in the transverse momentum spectra of emitted parti-
cles. The initial dissipation of the pressure can be taken together with the
effect of the shear viscosity at latter stages. These corrections to the energy
momentum-tensor combine to increase the transverse push in the collective
flow and cause a significant reduction of the elliptic flow.
REFERENCES
[1] M.Chojnackietal., Phys.Rev.C78(2008)014905,arXiv:0712.0947[nucl-th].
[2] W. Broniowski et al., Phys. Rev. Lett. 101 (2008) 022301, arXiv:0801.4361
[nucl-th].
[3] H.SongandU.W.Heinz, Phys.Lett.B658(2008)279,arXiv:0709.0742[nucl-
th].
[4] D. Teaney, Phys. Rev. C68 (2003) 034913,nucl-th/0301099.
[5] R.Baier,P.RomatschkeandU.A.Wiedemann, Nucl.Phys.A782(2007)313,
nucl-th/0604006.
[6] R. Baier and P. Romatschke, Eur. Phys. J. C51 (2007)677,nucl-th/0610108.
[7] A.K. Chaudhuri, Phys. Rev. C74 (2006) 044904,nucl-th/0604014.
[8] A. Muronga and D.H. Rischke, (2004), nucl-th/0407114.
[9] P. Bozek, Acta Phys. Polon. B39 (2008) 1375, arXiv:0711.2889[nucl-th].
[10] W. Israel and J. Stewart, Annals Phys. 118 (1979) 341.
[11] PHENIX, S.S. Adler et al., Phys. Rev. C69 (2004) 034909,nucl-ex/0307022.