Relativistic Theory of Hydrodynamic Fluctuations

Joe KapustaUniversity of Minnesota

Nuclear Physics Seminar October 21, 2011Collaborators: Berndt Muller and Misha Stephanov

WMAP picture

WMAP 7 years

Fluctuations in temperature of cosmicmicrowave background radiation

lm

lmlmYaT ),(),( 2|| lml aC

WMAP7

Most cosmological information comes from fluctuations.Can something similar be done with heavy ion collisions?

Elliptic Flow• Nonzero impact parameter b > 0:

– Spatial anisotropy in the initial state– Momentum anisotropy in the final state

• coordinate space momentum space • Fourier analysis

=> v2 = elliptic flow

n

TnTTTT

nPvdPP

dNddPP

dN

cos212

xy z

Correlations as observed by ALICE

The Ridge! Jet fragmentation?

ALICE

Decomposition into 5 Fourier modes

Sources of Fluctuations in High Energy Nuclear Collisions

• Initial state fluctuations• Hydrodynamic fluctuations due

to finite particle number• Energy and momentum

deposition by jets traversing the medium

• Freeze-out fluctuations

Hydrodynamic FluctuationsHydrodynamic fluctuations (noise) have been applied to awide variety of physical, chemical, and biological systems.

There are fluctuations in high energy heavy ion collisionsdue to the finite size and finite particle content of the system.

Molecular Dynamics

Lubrication Equation

Stochastic Lubrication Equation

BBB JunJ

TuuwPgT

uHuuT 32

uTuTQuuguuH ,,

BBBB

B JT

susTw

TnJ

,

2

22

22

32

2 kkk

kk

kiji

jj

i uTTT

uT

uuuT

s

Relativistic Dissipative Fluid Dynamics

In the Landau-Lifshitz approach u is the velocity of energy transport.

Extend Landau’s theory of hydrodynamic fluctuations to the relativistic regime

STTT ideal

heatvis SSS

)(2)()( 432

visvis yxHHHHHHTySxS

)(2)()( 42heatheat yxuuHuuHuuHuuHTySxS

0)()( heatvis ySxS

Stochastic source

Similar expressions arise in the Eckart approach.

Procedure

• Solve equations of motion for arbitrary source function

• Perform averaging to obtain correlations/fluctuations

• Stochastic fluctuations need not be perturbative

Example: Boost Invariant Bjorken Model

00)( )( sinh,0,0,cosh ssTTu ss

)(),(

)(),( 2

00

TT

vsX s

ss

)',(~)',;(~'),(~

0

kfkGdkX

2|),;(~|e)()(34)(1),0(),(

0

f

ikffs kGdkTd

AXX s

f

Fluctuation

Solution

response function

noise

calculableand similarly for fluctuations in the local flow velocity…

'

21'

21e'21),';(~)( 22'

zzz

zzz

zzzzkGzT zz

223/2

000

0 3121 & / kk

Tsz

In the small viscosity limit

This leads to delta functions and their derivatives.

Cause: Space-time delta functions (white noise)in the original correlation functions.Cure: Use finite range correlations (colored noise).

)sinh(uz

)/(ln2 horizon sound at the off-cut are nsCorrelatio

0 fsv

horizon. sound theand origin theat ermssingular t are but there

)/(ln2 horizon sound at the off-cut are nsCorrelatio

0 fsv

Regular part of response function

)sinh(uz

)/(ln2 horizon sound at the off-cut are nsCorrelatio

0 fsv

Singular part of the responsefunction smeared withGaussians

),(sinh u z ss

Fluctuations in the local temperatureand flow velocity fields

give rise to a nontrivial 2-particlecorrelation function when the fluidelements freeze-out to free-streaminghadrons.

)sinh(uz

)/(ln2 horizon sound at the off-cut are nsCorrelatio

0 fsv

)(115

)()(

0

2

f

0

ffeff4

12

12

KsT

TTN

d

ddN

ddN

ddN

ddN

Magnitude and shape of correlationfunction depend on shear viscosity

3D viscous -smooth initialconditions

3D viscous -lumpy initialconditions

McGill group

Will hydrodynamic fluctuationshave an impact on our abilityto discern a critical point in thephase diagram (if one exists)?

quarks & gluons

baryons & mesons

What are the relevant degrees of freedom? What happens to confinement?

VffnfTT c

||)(),();,( 2210

Expansion away from equilibrium states using Landau theory

Vnf c0 0 along coexistence curve

The relative probability to be at a density other than the equilibrium one is

Vff

TPP

ll

l

||||

/exp)()(

222

cc nnn /)(

Volume = 400 fm3

Volume = 400 fm3

=(n-nc)/nc

mmmmmmm

imm OooooOO ',

*'

0

0 e)(

2

*n 2

)(cos )(

nn

ccpppp 2.18.0on with distributi top-flat a have to Take

Look in a window of longitudinal and transverserapidities to correlate position and momentum space.

(CMB: spherical symmetry / High Energy Nuclear Collisions: cylindrical symmetry)

Many Accelerators

• Relativistic Heavy Ion Collider (RHIC) low energy runs - BNL

• Facility for Antiproton and Ion Research (FAIR) - Germany

• SPS Heavy Ion and Neutrino Experiment (SHINE) - CERN

• Nuclotron-based Ion Collider Facility (NICA) - Dubna

Summary

• Fluctuations are interesting and provide essential information on transport coefficients.

• We are learning and are having fun.

• There is plenty of work for both theorists and experimentalists!

Supported by the Office Science, U.S. Department of Energy.