Perfect Fluid: Flow measurements are described by ideal hydro
Problem: all fluids have viscosity
Ask: is viscosity small or flow strong?
I. Viscosity and its consequences I. Viscosity and its consequences
II. Radial flow fluctuations: Dissipated II. Radial flow fluctuations: Dissipated by shear viscosityby shear viscosity
III. Contribution to transverse momentum III. Contribution to transverse momentum fluctuationsfluctuations with Abdel-Azizwith Abdel-Aziz
in progressin progress
How Can We Measure Viscosity?How Can We Measure Viscosity?
Sean Gavin Sean Gavin Wayne State University Wayne State University
How Can We Measure Viscosity?How Can We Measure Viscosity?
Sean Gavin Sean Gavin Wayne State University Wayne State University
ViscosityViscosity
flow vx(z)
vvvvT ijijijjiij ⋅∇−⎟⎠
⎞⎜⎝
⎛ ⋅∇−∇+∇−= ζδδη32
viscous contribution to stress tensor, flow velocity v << 1
shear viscosity η resists shear
z
vT x
zx ∂∂
−= η
bulk viscosity bulk viscosity ζζ resists expansion resists expansion“hubble” flow
€
rv = h
r r
ijij hT δζ3−=
important:important: typically ζ << η
How Ideal is the Perfect Fluid?How Ideal is the Perfect Fluid?
How can we measure viscosity of parton and hadron fluids?
• measure flow through a fixed geometrymeasure flow through a fixed geometry
elliptic and radial flowelliptic and radial flowTeaney et al.; Muronga et al.; Heinz et al.; Baier et al.
• fluid response to external probefluid response to external probe
jet phenomena, Mach conejet phenomena, Mach cone Stocker; Casalderry-Solana, Shuryak & Teaney
• attenuation of sound wavesattenuation of sound waves
suggest:suggest: dissipation of fluctuations dissipation of fluctuations
Radial Flow FluctuationsRadial Flow Fluctuations
€
vr
shear viscosity shear viscosity drives velocity drives velocity toward the averagetoward the average
zvT rzr ∂∂−= η
damping of radial flow fluctuations viscosity
viscous friction arises as neighboring fluid elements flow past each another
small local variations in radial flow in each event
r
z
Evolution of Evolution of FluctuationsFluctuations
€
∂∂t
− Γs∇2 ⎛
⎝ ⎜
⎞
⎠ ⎟gt = 0
diffusion equationdiffusion equation for momentum current
momentum diffusion length Pes +
=Γη shear viscosity η
energy density e, pressure P
r
z
€
gt ≡ T0r − T0r ≈ e + P u
€
Tzr ≈ −η∂u
∂z≈ −
η
e + P
∂gt
∂z
€
∂∂t
T0r +∂
∂zTzr = 0
momentum current momentum current for small fluctuations
momentum conservation
u
u(z,t) ≈ vr vr
shear stress
viscous diffusion + Bjorken flowviscous diffusion + Bjorken flow
V = 2Γs/0
0 2 4 6
/0
2
2
2 y
gg tst
∂∂Γ
=∂∂
€
V = 2Γs
1
τ 0
−1
τ
⎛
⎝ ⎜
⎞
⎠ ⎟ V
random walk in rapidityrandom walk in rapidity y vs. proper time
€
V = y 2∫ gt dy
€
d
dτΔV =
2Γs
τ 2,
momentum diffusion length Γs η(e+P)
formation at 0
Viscosity Broadens Rapidity Viscosity Broadens Rapidity DistributionDistribution
analogous effect in charge diffusion
Stephanov & Shuryak; Abdel-Aziz & Gavin; Koide; Sasaki et al.; Wolchin; Teaney & Moore; Bass, Pratt & Danielowicz
Hydrodynamic Momentum CorrelationsHydrodynamic Momentum Correlations
€
rg = gt (x1)gt (x2) − gt (x1) gt (x2)
momentum density correlation functionmomentum density correlation function
equilibriumequilibrium )( 212
, xxpnr teqg −= δ
difference difference rg rg r g,eq satisfies diffusion satisfies diffusion equationequation
Van Kampen, Stochastic Processes in Physics and Chemistry, (Elsevier, 1997); Gardiner, Handbook of Stochastic Methods, (Springer, 2002)
width in relative rapidity grows from initial value :
• fluctuations diffuse through volume• global equilibrium – diffusion drives r rg
€
2 = σ 02 + 2ΔV (τ f )
variance minus thermal contributionvariance minus thermal contribution
multiplicity N
mean pt€
R =N 2 − N
2− N
N2
correlation function:correlation function:
€
R∝ r p1, p2( )∫∫ dp1dp2
Pruneau, Voloshin & S.G.
ttt ppp −≡δ
€
δpt1δpt 2 ∝ δpt1δpt 2 r p1, p2( )∫∫ dp1dp2
Dynamic FluctuationsDynamic Fluctuations
( ) 221 )singles(pairs, −=ppr
∑≠
≡ji
tjtitt ppN
pppairspairs
21
1δδδδ
pptt Fluctuations Energy IndependentFluctuations Energy Independent
Au+Au, 5% most central collisions
sources of pt fluctuations: thermalization, flow, jets?
• central collisions thermalized
• energy independent bulk quantity jet contribution small
Hydrodynamic Density CorrelationsHydrodynamic Density Correlations
€
R = dp1dp2
r p1, p2( )
N2∫
€
r p1, p2( ) = dx1dx2∫ f1 f2 − f1 f2 −δ12 f1( )
€
R = dx1dx2
Δrn (x1, x2)
N2∫
€
r = r − req
hydro:hydro: stress-energy tensor T and current j number density n = j0 and momentum density gi = T0i
multiplicity fluctuations probe densitydensity
density correlation density correlation functionfunction
phase space density f(x,p) fluctuates:
€
n(x) = dp f x, p( )∫
€
rn = n(x1)n(x2) − n(x1) n(x2)
Measured: NA49; PHENIX; PHOBOS
Hydrodynamic Momentum CorrelationsHydrodynamic Momentum Correlations
€
N pair δpt1δpt 2 = dp1dp2 pt1 − pt( ) pt 2 − pt( ) r p1, p2( )∫
€
rg = gt (x1)gt (x2) − gt (x1) gt (x2)
€
gt (x) = dp pt f x, p( )∫
momentum density correlation functionmomentum density correlation function
pt fluctuations probe transverse momentum transverse momentum densitydensity
€
S = dx1dx2
Δrg x1, x2( )
N pair
∫ = δpt1δpt 2 + pt
2 R
1+ R
measured δpt1δpt2 plus HIJING R rg and rn are comparablecomparable
observableobservable::
€
rg = rg − rg,eq
€
N pair δpt1δpt 2 = dx1dx2 Δrg (x1,x2) − pt
2Δrn (x1,x2)[ ]∫
How Much Viscosity?How Much Viscosity?• flow data doesn’t require small viscosity
• Reynolds number must be large enough for ideal flow
0
0.5
1
1.5
2
0 10 20
(fm)
V
sticky liquid ΓwQGP ~ ΓHRG ~ 2 fm
perfect liquid ΓsQGP ~ (4 Tc)-1, ΓHRG ~ 2 fm
sticky
perfect
Abdel-Aziz, S.G - in progress
Hirano & Gyulassy
€
Γs =η
e + P
momentum diffusion
diffusion for extreme scenarios:
€
Re =flow
dissipation~
e + P
ηvrR
radial flow speed and length scales vr, R
Rapidity Dependence of Momentum Rapidity Dependence of Momentum Fluctuations Fluctuations
momentum correlation function near midrapiditymomentum correlation function near midrapidity
€
rg η r,η a( )∝ e−η r2 / 2σ 2
e−η a2 / 2Σ2
)(220
2fV +=
• relative rapidity ηr η1η2
€
N S ∝ rg dy1dy2Δ
∫∫
00.10.20.30.40.50.60.70.80.9
1
0 1 2 3 4
€
N S
N Smax
initial
sticky
perfect
Abdel-Aziz, S.G - in progressfluctuationsfluctuations in rapidity window
initial ~ 0.25 balance function
~ 1 fm q ~ 3 fm h ~ 9 fm f ~ 20 fm
• weak dependence on ηa η1η22
Summary: Summary: small viscositysmall viscosity or strong or strong flow?flow?
Summary: small viscosity or Summary: small viscosity or strong strong flowflow??
viscosity broadens momentum correlation function in viscosity broadens momentum correlation function in rapidity rapidity
pptt fluctuations measure these correlations fluctuations measure these correlations
testing the perfect liquid testing the perfect liquid viscosity info viscosity info• diffusion coefficient shear viscosity• compare rapidity width of momentum fluctuations for different projectile sizes and energies
• cross-check: combine with other indirect viscosity measures
schematic calculation -- lots to do:schematic calculation -- lots to do:• Maxwell/Muronga type corrections• O(R-1) corrections• angular correlations
Summary: small viscosity or strong Summary: small viscosity or strong flow?flow?
130 GeV
s1/2=200 GeV
blue-shift: • average increases
• enhances equilibrium contribution
€
∝Teff2 ∝
1+ vr
1− vr
thermalization
flow added
M. Abdel-Aziz & S.G.
participants
Thermalization Thermalization + Flow + Flow
20 GeV€
δpt1δpt 2 ∝ δT1δT2 r12∫
• blue-shift cancels in ratio
€
δpt1δpt 2
pt
2 ≈ constant
€
pt ∝Teff ∝1+ vr
1− vr
⎛
⎝ ⎜
⎞
⎠ ⎟
1/ 2
Hydrodynamic Momentum CorrelationsHydrodynamic Momentum Correlations
€
N pair δpt1δpt 2 = dp1dp2 pt1 − pt( ) pt 2 − pt( ) r p1, p2( )∫€
gt (x) = dp pt f x, p( )∫
momentum density momentum density correlationscorrelations
pt fluctuations probe transverse momentum densitytransverse momentum density
€
S = dx1dx2
Δrg x1, x2( )
N pair
∫ = δpt1δpt 2 + pt
2 R
1+ R
measured δpt1δpt2 plus HIJING R rg and rn are comparablecomparable
observableobservable::
€
= dx1dx2 Δrg (x1,x2) − pt
2Δrn (x1, x2)[ ]∫
density density correlationscorrelations
Hydrodynamic Density CorrelationsHydrodynamic Density Correlations
€
R = dp1dp2
r p1, p2( )
N2∫ = dx1dx2
Δrn (x1, x2)
N2∫
€
r p1, p2( ) = dx1dx2∫ f1 f2 − f1 f2 −δ12 f1( )
€
r = r − req
hydro:hydro: stress-energy tensor T and current j number density n = j0 and momentum density gi = T0i
multiplicity fluctuations probe densitydensity
density correlation density correlation functionfunction
phase space density f(x,p) fluctuates:
€
n(x) = dp f x, p( )∫
€
rn = n(x1)n(x2) − n(x1) n(x2)
Measured: NA49; PHENIX; PHOBOS
Hydrodynamic Momentum CorrelationsHydrodynamic Momentum Correlations
€
N pair δpt1δpt 2 = dp1dp2 pt1 − pt( ) pt 2 − pt( ) r p1, p2( )∫
€
rg = gt (x1)gt (x2) − gt (x1) gt (x2)
€
gt (x) = dp pt f x, p( )∫
momentum density correlation functionmomentum density correlation function
pt fluctuations probe transverse momentum transverse momentum densitydensity
€
S = dx1dx2
Δrg x1, x2( )
N pair
∫ = δpt1δpt 2 + pt
2 R
1+ R
measured δpt1δpt2 plus HIJING R rg and rn are comparablecomparable
observableobservable::
€
rg = rg − rg,eq€
= dx1dx2 Δrg (x1, x2) − pt
2Δrn (x1,x2)[ ]∫