Viscous fluid dynamics in Au Au collisions

1
Viscous fluid dynamics in AuAu collisions A. K.
Chaudhuri Variable Energy Cyclotron Centre,
Kolkata
2
Introduction why do we need
dissipative hydrodynamics? Relativistic fluid
dynamics very briefly
Israel-Stewarts, Chapman-Enskogs and Grads
14-moment method for dissipative
dynamics Hydrodynamics with only
shear viscosity in 21 dimensions
Comparison with experiments
AuAu differential v2 as a function of
centrality, pT-spectra, centrality
dependence of dNch/dy, ltpTgt, pT integrated v2.
CuCu differential
v2 Summary conclusions
3
There are enough evidences that AuAu collisions
has produced some form of sQGP
high pT suppresion
disappearance of away side jet
elliptic flow in non-central coll. explained in
hydro models.
elliptic flow is not explained in transport
models unless partonic cross-section is large
4

freeze-out at TF
hadronic phase TltTc
mixed phase at Tc
QGP phase
and fairly unknown EOS pp(e)
ti,e(x,y,h), v(x,y,h)
inputs required ideal hydro
TF
ideal hydrodynamics explains a large volume of
RHIC data, e.g. pT-spectra, elliptic flow upto
pT1.5 GeV. Not HBT.
and shear viscosity h and relaxation time tp
5 components of pmn
dissipative hydro additionally (shear viscosity
only)
Increased parameter space reduces the efficacy of
viscous dynamics.
5
ADS/CFT conjecture all substances have h/s1/4p
as lower limit. The estimate is not related to
QCD! Question Can we say h/s1/4p small?
sound attenuation length (equivalent to mean
free path)
Validity of hydrodynamics require
In RHIC collisions, Tt not a small quantity.
e.g. Ti0.35 GeV, ti 1 fm. For hydrodynamics
to be valid at RHIC, h/s ltlt 0.26,
only 3 times bigger than ADS/CFT
lower bound! ADS/CFT lower bound h/s0.08 not
a very small quantity in the context of strong
interaction scale.
Viscosity is warranted in AuAu collisions at
RHIC. Incentive probe earlier time than in
ideal dynamics strongly non-equilibrium-----gt
slightly non-equilibrium----gtequilibrium
Further
incentive can provide for a better prescription
of freeze-out. Can give phenomenological limit
of transport coefficients.

6
Israel-Stewarts theory based on gradient
expansion of entropy density. Assumption system
is slightly off-equilibrium and
gradients of equilibrium variables (n,e,u) are
small.
2nd order theory expansion contain terms
quadratic in deviation from equilibrium
variables. Causality maintained.
1st order theory expansion contain terms linear
in deviation from equilibrium variables. Violate
causality.
(ignoring heat conduction and bulk viscosity)
7
In kinetic theory one solve the Boltzmann
transport equation Non-linear collision term
preclude any analytical solution.
Chapman-Enskog method lltlt R hydrodynamic
regime. f(x,p) is expressed as a function of
hydrodynamics variables (particle density (n),
energy density (e) and velocity (u) ), and their
gradients. Expand f(x,p) in terms of el/R to set
up a hierarchy equation. Find f(x,p) by
successive approximation. One obtain 1st order
results. Grads 14-Moment method l R
transient regime. f(x,p) is expanded in terms of
all its moments. Such an expansion lead to
infinite set of coupled equations. In Grads
14-moment approximation, the expansion is
truncated after 14 moments. In addition to
(equilibrium) 5 moments, n,e and u, 9 additional
moments (heat flow and viscous pressure tensor)
are included. One obtain relaxation equations for
the dissipative flow. Note Grads 14-moment
method is not an extension of Chapman-Enskog
method.
8
Israel-Stewart
kinetic theoryRomatschke et al
Kinetic theory gives two extra terms. Israel and
Stewart developed the theory on gradient
expansion (gradients of n, e and u are small).
pmn is assumed to be small and terms involving
pmn and gradients are neglected. It will be
wrong to say that Israel-Stewartss theory assume
acceleration free fluid. Spirit of the theory
must be understood. Romatschke and Baier (and
also Heinz) argue that the term is needed to
maintain the transversality condition. However,
with proper algorithm, one should (and can)
maintain that condition, without the extra
term. The other term is related to curl or
vorticity. In Israel-Stewarts fluid is
vorticity free.
9
Complete 2nd order theory is a numerically
challenging problem. It require simultaneous
solution of 14 (41)(531) partial
differential equations. Simplify (i) consider
baryon free fluid (ii) include only shear
viscosity (iii) assume longitudinal
boost-invariance. The energy-momentum
conservation equations and relaxation equations
are solved in (t,x,y,h) co-ordinate system
assuming boost-invariance. With boost-invariance
one energy-momentum conservation equation become
redundant. Shear stress tensor has 5 independent
components, which reduces to 3 in boost-invariant
system. Any 3 component of pmn should suffice.
After several trials we find that pxx, pyy and
pxy are the best choice. The dependent components
are obtained by using the condition, (i)traceless
ness and (ii)transversality of pmn. The
evolution, by design, then satisfy those
conditions!
10
3 energy-momentum conservation equations
3 relaxation equations
are solved by AZHYDRO-KOLKATA, developed at the
Cyclotron Centre, Kolkata.
11
An important issue in 2nd order theory time
derivatives. Given a EOS, energy density and
velocity at any time ti, the energy-momentum
conservation equations can be integrated to
obtain its value at the next time step, ti1. The
procedure works fine for ideal fluid, but poses a
problem for viscous fluid. Shear stress tensor
components contain time derivative of
velocities. To propagate the solutions from ti to
ti1 one needs to know velocities at time step,
ti and ti1 . Strictly speaking, one require
iterative solution! Possible way out (i) use the
previous step derivation. We do it. Implicitly
assume velocity at ti is average of ti-1 and ti1
(ii) use the ideal fluid equation of motion
to convert the time derivative to spatial
derivative. Correct in 1st order.
12
Initial conditions initial time ti and
transverse profile of energy density
ei(x,y)and vx(x,y),vy(x,y)
transverse profile for 3 independent shear
stress tensors(pxx,pyy,pxy). They must be
obtained by confronting the experimental
data. ti 0.6 fm, ei 30GeV/fm3 , vxvy0
Glauber model parameterisation with

25 hard collisions. independent shear
stress tensor components. Two choices can be
made, (i)pmn0 (effect of viscosity is
minimised)
pmn attained boost- invariant values
Viscosity ADS/CFT lower bound, h/s0.08.
Relaxation time Boltzmann gas value,
Equation of state EOS-Q(Heinz-Kolb). 1st order
phase transition at Tc164 MeV. (even though
lattice simulations indicate otherwise).
13
AuAu _at_ b0 fm


Evolution of energy density Both ideal and
viscous fluid are initialised similarly.
Initially both viscous and ideal fluid have same
energy density contours. As the fluid evolve, the
contours starts to differ. Energy density evolve
more slowly viscous fluid. It is expected.
Viscosity oppose expansion and cooling.
14
AuAu _at_ b0 fm
Evolution of shear stress tensor Initially
pxx,pyy are identical and symmetric. Immediately
after the evolution starts, spatial distribution
of pxx and pyy starts to differ and asymmetry is
developed. Late in evolution viscous pressure
is more on the periphery than on the centre.
Velocity gradients are more at the
periphery. pxx and pyy are related by x?y and
y?x transformation. AZHYRO-KOLKATA maintain the
symmetry.
15
Temperature evolution in AuAu _at_ b0 fm
In viscous fluid temperature evolve slowly, life
time of the fluid increases. Note It is
important that significant time elapsed before
freeze-out. In 2nd -order/cross-over, without the
mixed phase, fluid freezes-out fast and expt.
data are not reproduced. EOS for Hadronic
resonance gas possibly will not work in 2nd
order/cross-over.
16
Transverse momentum and elliptic flow
Hydrodynamics give e( or T), and vx, vy. The
information need to be converted into particle
spectra to connect with experiment. Cooper-Frye
prescription
In viscous dynamics, system is not in equilibrium
and f(x,p) can not be approximated by the
equilibrium distribution function
In a slightly off-equilibrium system
With only shear viscosity
non-equilibrium correction increases
quadratically with momentum. Large pT
particles are more affected by viscosity than low
pT particles.
Invariant distribution
Elliptic flow

17
Non-equilibrium vs. equilibrium contribution
Viscous hydrodynamics
applicable till
non-eq. correction
valid over larger pT for lower TF .
two opposing effects (i)F(x,p)?1/T6 ,decreases
as TF is lowered. (ii)Lower TF, fluid evolve
longer pmn decreases, F(x,p) increases.

Solution to freeze-out A reasonable choice for
freeze-out could be F(x,p)0.5. Choose a pT range
up to which hydrodynamics is applicable. At each
space-time, if F(x,p) gt 0.5, the fluid
freeze-out. High pT particles freeze-out at
higher T?
pmn decreases faster than T6
18
Teaney/PRC681st-order, blast wave model
chaudhuri/0801.3180nucl-th 2nd order
SongHeinz/0712.3715 2nd order
h/s0.84
h/s0.08
h/s0.08
though direct comparison is not possible, our
result is similar to Teaneys. 1st order
h/s0.84 non-eq. contr. eq. contr. at pT1.8
GeV. 2nd order h/s0.08 non-eq. contr. eq.
contr. at pT2 GeV. (viscous effects are enhanced
in 2nd order). However, the different trend
obtained by Song Heinz is very disturbing.
19
Fitting data increased parameter space makes
data fitting complex in viscous dynamics. We
assume viscous fluid also have same initial
condition as in ideal dynamics,
and
vary TF from 160-130 MeV and compare with the
PHENIX data on pT dependence of v2 in 16-23
AuAu collisions.
(i)eq. contribution ve, marginally changed from
TF130-160 MeV. (ii)non-eq. contribution ve.
Contribute less with lowering TF.
(iii)TF160-140 MeV, viscous dynamics produces
less v2 than in expt. (iv)TF130 MeV, data are
explained upto pT3.6 GeV. Small non-eq.
correction. (v)v2 do not show saturation
(though rate of increase slows down)
20
chaudhuri/0801.3180nucl-th

pT integrated v2 10-20 less in viscous
dynamics.
Centrality dependence mid-central collisions
reasonably explained. central collisions under
predicted. peripheral collisions over predicted.
Min.bias v2 well reproduced (even saturation).
Moderate description of v2 with ADS/CFT lower
bound on viscosity. Note description of
differential v2 is much worse in ideal dynamics.
21
Saturation of elliptic flow
Chaudhuri/PLB659
PHENIX p0, g
In p0 or photon, v2 is not saturated! Even min.
bias data do not show saturation. v2 veer about
pT3 GeV.
At large pT jets can be important. A quenching
jet can contribute negative v2. Possibly
jethydro can account for saturation.
22
pT spectra of p-,K and proton
chaudhuri/0801.3180nucl-th
(normalised by N1.4)
ADS/CFT lower bound is consistent with pT
spectra. Comparable description could not be
obtained in ideal dynamics.
23

• pT-spectra is explained well up to 40-50
  centrality collision.
• Differential v2 is explained only in mid-central
  collisions. Why?
• Elliptic flow being a ratio is a more sensitive
  probe than the pT-spectra.
• Incorrect initial condition is thermalisation
  time same in in central or peripheral collisions?
  In central collisions, density
• being more, one expect rather rapid
  thermalisation than in peripheral collisions.
  Equilibration time should not be same in central
  and peripheral
• collisions.

24
centrality dependence of dN/dch and mean ltpTgt

Normalised yield reproduces multiplicity data
for Npart 100.
ltpTgt is reproduced for Npart100.

ADS/CFT lower bound on viscosity is consistent
with centrality dependence of multiplicity and
mean pT in collisions with Npart 100. Possible
message For Npart lt 100-120 initial medium is
not a QGP.
25
J/Y suppression in AuAu
Chaudhuri/0711.2133nucl-th



J/Y suppression pattern changes for
Npartgt120-150. Can be explained by melting of
J/Y beyond a threshold density.
26
initial pmn and fluid evolution
viscous effect is enhanced if initially pmn is
non-zero.
shear stress tensor is less in evolution with
zero pmn.
pT-spectra and v2 indicate the same.
Initial shear stress tensor can be tuned to
match experimental data.
27
Relaxation time and fluid evolution
lesser the relaxation time, more is the viscous
effect on the evolution. pT-spectra and v2 also
indicate the same. It is expected. Relax. Eq. in
1D
larger relx. time, more is the instantaneous value
of shear stress tensor.
28
Particle yield and v2 in CuCu and AuAu
collisions at b5.2 fm (20-30).
ti0.6 fm,
Sini110fm-3.
particle yield in AuAu is 2-3 times more than
in CuCu.
nearly same elliptic flow.
29
Elliptic flow in CuCu collisions
AuAu v2 in mid-central collisions are
reasonably explained. CuCu even v2 in
mid-central collisions are not explained.
Centrality dependence of differential v2 may
be crucial In understanding initial and final
condition of the Matter produced in AuAu
collisions.
30
Summary If the deconfinement-confinement phase
transition is 1st order, with Tc164 MeV, then
minimally viscous (h/s1/4p) QGP fluid,
initialised as,
if freeze-out out at TF130 MeV, a host RHIC data
from AuAu collisions for blt8fm are
explained. ADS/CFT lower bound of viscosity is
consistent with RHIC data.
31
Need to be done (i)viscous hydrodynamics with
2nd order phase transition/cross-over. Viscous
dynamics extend the evolution and can possibly
explain the RHIC data even with 2nd order phase
transition. (Mikolaj Chojnacki did this type of
calculation for ideal hydro successfully. EOS is
quite different than rasonance gas in the
hadronic sector.) (ii) freeze-out prescription
Viscous dynamics can be effectively utilised to
built a consistent freeze-out prescription. Until
it is done, one will have ambiguity in initial
condition. One can very well produce a different
set of initial conditions, with a different
freeze-out temperature. Only when consensus is
reached on the final state, one can try to
compute the initial state. (iii) hydrodynamic
after burner switch over to transport
prescription after a decoupling temperature.
(Energy conservation problem.) (iv) Systematic
analysis of pT-spectra, v2, HBT radii etc. to
extract QGP viscosity.
32
BACKUPS
33
Present calculation differ from the calculations
done by SongHeinz, presumably due to the extra
term in their relaxation equation.
34
SongHeinz0712.3715nucl-th
They solved relax. eq. for ptt, ptx,pty and phh
. Relx. eq. contain the extra term from Kinetic
theory. They choose to study CuCu collisions and
did not compare with experiment.
v2 fall sharply for pTgt2.2GeV. In consistent
with RHIC data?
pTgt3GeV, vis. dynamics produces less pT than in
ideal hydro.
evidently viscous effects are very strong at the
freeze-out surface.
35
Saturation of elliptic flow
Chaudhuri/nucl-th/0705.1059
PHENIX p0, g
In p0 or photon, v2 is not saturated! Even min.
bias data do not show saturation. v2 veer about
pT3 GeV.
At large pT jets can be important. A quenching
jet can contribute negative v2. Possibly
jethydro can account for saturation, if it is
there.
36
Romatschke and Romatschke/0706.1522 extra term
in relx. eq. Glauber model parameterisation pmn0
initially. initial time ti1 fm, Freeze-out
TF150 MeV.
min.bias v2 require h/s0.03. Trend indicate that
with TFlt150, h/s0.08 may explain the data.
mean pT is overestimated for low mass particles
e.g. pion or kaon. Marginal viscous effect on
ltpTgt.
Multiplicity is reproduced for kaon.
Underestimate pion or proton multiplicity.
Marginal viscous effect. Viscosity reduces dN/dy ?
37
Testing AZHYDRO-KOLKATA 2nd order theory do
not possess analytical solution,even in a
restricted condition. General procedure for
checking the code (when analytical solutions are
not found) (i)results must be stable against
step length change, (ii)any symmetry present is
not destroyed, (iii)no unphysical maxima or
minima, (iv) the numbers should look
reasonable. We do one more check. For
boost-invariant longitudinal motion, centre of
the fluid will be least affected, by the
transverse motion, even in 21 dimensions. 2nd
order theory for 1D scaling expansion is well
studied. 1D theory require solution of 2
ordinary differential equation. AZHYDRO-KOLKATA
results are tested against it. Further checked
that as viscosity gradually decreases, ideal
dynamics results are reproduced.
38
chaudhuri/0801.3180nucl-th
TESTING AZHYDRO-KOLKATA
fluid at the centre follow 1D scaling expansion
recover ideal hydro results as viscosity
gradually reduces.
maintain symmetry, stable against change in
integration step length
AZHYDRO-KOLKATA pass all the tests.
39
Effect of viscosity on pT spectra and elliptic
flow

TF150 MeV
TF150 MeV
v2 decreases in viscous fluid. more viscous the
fluid, less is v2.
Yield at large pT increases. More viscous the
fluid, more is the increase in large pT yield.
Viscous dynamics appear to remedy the drawbacks
of ideal hydrodynamics. At large pT, ideal
hydrodynamics under-predict the spectra and
over-predict elliptic flow.
40
Chapman-Enskog method Boltzmann eq.
length scale l
length scaleR
in hydrodynamic region lltltR. Expand f in terms of
el/R

hierarchy eq.
Equilibrium distribution function
the 5 parameters of the equilibrium distribution
are identified with Chemical potential,
hydrodynamic velocity and temperature. Resulting
condition of fits can be satisfied if f(x,p) is
function of equilibrium variables and their
gradients. It can be obtained successive
approximation. One generally obtain the 1st term.
41
14-moment method
identify 5 parameters of f(0)(x,p) with chemical
potential, velocity and temperature. The
condition of fit impose certain condition on A, B
and C which lead to relaxation equations.
42

Id. Hydro and RHIC data(Heinz, Kolb)
integrated v2 deviates from expt. in peripheral
collisions.
spectra well explained up to pT1.5 GeV.
v2 explain the species dependence. Do not show
the saturation at high pT.
Suggest viscous effects at large pT and in
peripheral collisions! .
43
HYDROCascade Hirano,Bass,Dumitru,Nonaka,Heinz...
switch over from hydrodynamics to transport
model after a decoupling temperature (accounts
for dissipation in the hadronic stage). CGC
initial condition impressive reproduction of
dN/dy pT spectra. Over-predict elliptic flow.
30 reduction in v2 due to hadronic dissipation.
Suggest viscous effects in the QGP phase.
44
In Israel-Stewart theory, entropy 4-current in a
non-equilibrium state is assumed to be,
the parameters a and bl are not connected with
the actual state. The pressure p is also not the
actual pressure. a and bl has meaning only on the
equilibrium surface. There meaning need not be
extended to non-equilibrium states also. However,
in 1st order, it is possible to fit a fictitious
local equilibrium state, point by point such that
the pressure can be identified with the actual
thermodynamic pressure. The condition of fit
determine to underlying distribution phase space
function.
45
Viscous hydrodynamics in 21 dimensions with
boost-invariance







46
Energy-momentum conservation equations in 21
dimensions
where
of the required viscous pressures only 3 are
independent,
47
Getting e, and vx and vy



direction of flow (vx,vy) is not the direction
(Ttx,Tty) as in ideal case. One need a two
dimensional search to find e, vx and vy.
However, we did find that approximating ptt,ptx
and pty with previous step velocities changes
results marginally.
48
Choice of independent shear stress
components Shear stress tensor has 5 independent
components, which reduces to 3 for
boost-invariance. Any 3 component of pmn should
suffice. After several trials we find that pxx,
pyy and pxy are best choice. The 4 dependent
components can be obtained simply by multiplying
the independent components by vx and vy. The
procedure ensure that constraints on pmn is
satisfied at every space-time point.
any other combination of dependent pressure
tensors require division by fluid velocity.
Initially, fluid velocity is (assumed) zero.
Velocity grow slowly with time. Division by
small number can led to unrealistically large
value for shear stress tensors and ruin the
computation. This is purely a computational
problem and is avoided with the present choice of
independent shear pressure tensors.
49
Relaxation equations for

they are solved simultaneously with the 3
energy-momentum conservation equations.
50
AuAu _at_ b0 fm
AuAu _at_ b0 fm
velocity evolve faster. also attain higher value
in viscous dynamics.
temperature evolve slowly, life time of the fluid
increases. For minimally viscous fluid,
difference is not large.
51

Brief theory of dissipative hydrodynamics Israel
and Stewart Ann. Physics,118(1979) Simple
fluid is fully specified by primary variables Nm,
Tmn and Sm, and an unspecified number of
additional variables. Primary variables satisfy
conservation laws and the entropy law.

define a time like hydrodynamic 4 velocity u
(u21) and projector
In equilibrium, an unique hydrodynamic
4-velocity u exists such that,
An equilibrium state is fully specified by
5-parameters, (n, e,u) or (am/T,bmum/T). They
span a 5-dim. hyperspace S0 (a,bm).
52
in a state close to an equilibrium state,

Qm undetermined quantity of 2nd order in the
deviations
entropy production can be expressed as

dissipative flow X thermodynamic forces

for shear viscosity only
2nd order theory
1st order theory
Causality is violated. If in a fluid cell,
thermodynamic force happen to vanish,
dissipative flows stops immediately.
relaxation equation for the shear stress tensor
makes the theory causal.
53
Kndusen number a ratio of mean free path to the
size of the system. small ltlt 1, continuum
mechanics e.g fluid dynamic applicable large 1,
gt 1, mean free path comparable to system size and
continuum mechanics or fluid dynamical approach
is invalidated. Renoylds number ratio of
inertial over viscous force. flow is laminar when
Renolds number is small, i.e. system is
viscous. Flow is turbulent when Reynolds number
is high, inertial force dominate the flow
leading turbulence, eddies, vorticity