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Relativistic Theory of Hydrodynamic Fluctuations
Joe KapustaUniversity of Minnesota
Nuclear Physics Seminar October 21, 2011Collaborators: Berndt Muller and Misha Stephanov
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WMAP picture
WMAP 7 years
Fluctuations in temperature of cosmicmicrowave background radiation
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lm
lmlmYaT ),(),( 2|| lml aC
WMAP7
Most cosmological information comes from fluctuations.Can something similar be done with heavy ion collisions?
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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
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Correlations as observed by ALICE
The Ridge! Jet fragmentation?
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ALICE
Decomposition into 5 Fourier modes
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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
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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.
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Molecular Dynamics
Lubrication Equation
Stochastic Lubrication Equation
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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.
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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.
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Procedure
• Solve equations of motion for arbitrary source function
• Perform averaging to obtain correlations/fluctuations
• Stochastic fluctuations need not be perturbative
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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…
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'
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).
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)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
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)sinh(uz
)/(ln2 horizon sound at the off-cut are nsCorrelatio
0 fsv
Singular part of the responsefunction smeared withGaussians
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),(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.
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)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
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3D viscous -smooth initialconditions
3D viscous -lumpy initialconditions
McGill group
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Will hydrodynamic fluctuationshave an impact on our abilityto discern a critical point in thephase diagram (if one exists)?
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quarks & gluons
baryons & mesons
What are the relevant degrees of freedom? What happens to confinement?
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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 /)(
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Volume = 400 fm3
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Volume = 400 fm3
=(n-nc)/nc
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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)
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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
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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.