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)[ ]∫