Shear viscosity and thermal fluctuations in heavy-ioncollisions
Clint Young with Joseph Kapusta
University of Minnesota
June 18, 2014
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Fluctuation-dissipation and the importance of noiseThermal noise in heavy-ion collisionsWhere does thermal noise come from?
Fluctuation-dissipation and the importance of noiseThe relations between Green functions
Fluctuating hydrodynamical variables in BES-IIThe shape of second-order noiseThe variance of momentum eccentricity of centralAu+Au
√sNN=19.6 GeV
Conclusions
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Thermal noise in heavy-ion collisions
Kapusta, Muller, Stephanov: Thermal noise exists in viscous relativisticfluids, represented by Sµνheat and Sµνvisc. (in the Eckart frame) and Sµν andIµ in the (Landau-Lifshitz frame).
K (∆η) ∝⟨dNdη (η + ∆η)dNdη (η)
⟩−⟨dNdη
⟩2, the two-particle correlation as a
function of rapidity gap, has a contribution from thermal noise.
Analytical calculations possible for the Bjorken expansion of anultrarelativistic gas.
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Thermal noise in ultrarelativistic gases
0 1 2 3 4 5 6-0.5
0.0
0.5
1.0
1.5
2.0
2.5
DΗ
KHD
ΗL
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The relations between Green functions
The effect of perturbations on every physical quantity is determined by aspecific Green function:
I Given δH(t) =∫
d3xj(x, t)φ(x, t),⟨δφ⟩
=∫
d4x ′GR(x − x ′)j(x ′),
where GR(x) ≡ −iθ(t)⟨
[φ(x), φ(0)]⟩
.
I For the same δH(t), transition rates determined with
G>(t) ≡ −i∫
dt⟨φ(t) ˆφ(0)
⟩≡ G<(−t).
I Variances determined with GS(t) ≡ 12
⟨{φ(t), φ(0)}
⟩.
In x and t, several Green functions exist, each with their own domain ofapplicability.
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The relations between Green functions
Relationships become clearer in Fourier space:
ImGR(ω) = − i
2
[−i
∫dt θ(t)e iωt
⟨[φ(t), φ(0)]
⟩−i
∫dt θ(t)e−iωt
⟨[φ(t), φ(0)]
⟩∗ ]= −1
2
[ ∫dt θ(t)e iωt
⟨[φ(t), φ(0)]
⟩+
∫dt θ(t)e−iωt
⟨[φ(0), φ(t)]
⟩]= −1
2
∫dt e iωt
⟨[φ(t), φ(0)]
⟩= −1
2(1− e−ω/T )
∫dt e iωt
⟨φ(t)φ(0)
⟩= − i
2(1− e−ω/T )G>(ω).
Hermiticity and the KMS relation lead to algebraic relations betweenGreen functions in Fourier space.
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The relations between Green functions
Going between coordinate and momentum space relates physics to factsfrom complex analysis:
Coordinate space Momentum spaceCausality Analyticity of GR(ω) in the upper-half plane
KMS relation Detailed balance
Fluctuation-dissipation GS(ω) = −(1 + 2nB(ω))ImGR(ω)
In particular the correlation function GS ≡ 12
⟨{φ(t), φ(0)}
⟩(fluctuations)
have an easy relationship to ImGR(ω) (the dissipation) in momentumspace.
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Viscosity and thermal noise
“Why didn’t I notice thermal noise in liquids before?”
12
⟨{δui
T (x), δujT (0)}
⟩≈ T
e+p
(π(e+p)η|t|
)3/2exp
(− (e+p)|x|2
4η|t|
)23δ
ij :
For non-relativistic gases, e + p ≈ ρ, making T/ρ tiny.
For ultrarelativistic gases, 4η|t|e+p often small compared to length scales of
interest.
In heavy-ion collisions, length and time scales ∼ 1 fm ∼ 1T , e ∼ p ∼ T 4,
s ∼ T 3: no mass scale exists to suppress the importance of fluctuations,only the smallness of η/s can.
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Noise and observables
Now Tµν = Tµνav. + δTµν :
Each solution for a given δTµν corresponds to one event at a heavy-ioncollider. The ensemble of δTµν is approximated by a data set. Mostobservables are averages of these sets.
〈δTµν〉 = 0 → hydrodynamical noise has no effect on one-particleobservables dN/dpT , dN/dy , averaged over many events.⟨δTµνδTαβ
⟩6= 0 → noise affects two-particle correlations even after
averaging over events.
δTµν 6= 0 → noise drives non-trivial variance of all observables, within acentrality class.
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The shape of second-order noise
The Israel-Stewart formalism of the hydrodynamical equations
∂µ(Tµνideal + ∂µW µν) = 0
∆µα∆ν
β(u · ∂)W αβ = − 1
τπ(W µν − Πµν),
where Πµν = η∆µuν + ∆νuµ − 23 (∂ · u)∆µν is the first-order viscous
energy-momentum and τπ is the relaxation time. What is the correlationfunction for thermal noise here?
Noise in energy-momentum: Tµν + Ξµν :Current conservation (∂µ(Tµν + Ξµν) = 0) and the fluctuation-dissipationrelation give the autocorrelation of thermal noise (for a fluid at rest):⟨
((τπ∂tΞij(x) + Ξij(x))(τπ∂tΞ
kl(x ′) + Ξkl(x ′))⟩
=[2ηT (δikδjl + δilδjk) + 2(ζ − 2η/3)T δijδkl
]δ4(x − x ′).
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Colored noise in numerical simulations
This makes the noise autocorrelation colored:⟨Ξij(x)Ξkl(x ′)
⟩=
[2ηT (δikδjl + δilδjk) + 2(ζ − 2η/3)T δijδkl
]× δ3(x − x ′)
exp(−|t − t ′|/τπ)
2τπ;
the noise decorrelates slowly in time. Numerically, it is easier to use thefirst equation to define a differential equation for Ξµν :
τπΞµν = −(Ξµν − ξµν),
where ξµν is now white noise.
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music for 3+1-dimensional viscous hydrodynamics
Noise breaks boost invariance→ (3+1)-dimensional simulationnecessary for K (∆η,∆φ).
Fluctuations related to dissipation→ viscous hydrodynamics necessary.
music (Schenke, Jeon, and Gale)solves the Israel-Stewart model ofviscous hydrodynamics in (τ, x , y , η),uses lattice EOS, determines3-dimensional freeze-out surface forhadron production.
=0.4 fm/c
-10 -5 0 5 10x [fm]
-10
-5
0
5
10
y [fm
]
0
100
200
300
400
500
600
[fm
-4]
=6.0 fm/c, ideal
-10 -5 0 5 10x [fm]
-10
-5
0
5
10
y [fm
]
0
2
4
6
8
10
12
[fm
-4]
=6.0 fm/c, /s=0.16
-10 -5 0 5 10x [fm]
-10
-5
0
5
10
y [fm
]
0
2
4
6
8
10
12
[fm
-4]
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Calculating linearized fluctuations numerically
The equations for δTµνid. and δW ′µν = W µν + Ξµν can be linearized:
∂tδT tνid. = −∂iδT iν
id. − ∂µδW ′µν ,
(u ·∂)δW ′µν = − 1
τπ(δW ′µν−δSµν−Ξµν)− 4
3(∂ ·δu)W µν− 4
3(∂ ·u)δW ′µν−(δu ·∂)W µν
−δuµ((u · ∂)uα)W αν − uµ((δu · ∂)uα)W αν + (u · ∂)δuα)W αν + (u · ∂)uα)δW ′αν)
−δuν((u · ∂)uα)W αµ − uν((δu · ∂)uα)W αµ + (u · ∂)δuα)W αµ + (u · ∂)uα)δW ′αµ).
The equations are discretized: ξ(x)→ ξi = 1∆V∆t
∫d4xξ(x).
〈ξ(x)ξ(x ′)〉 ∝ δ4(x − x ′) →⟨ξiξj
⟩∝ 1
∆V∆t :
hyperbolic equations with large gradients and sources.
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music with noise
δe(x , y , τ), in a√
sNN = 19.6 GeV Au+Au collision, b = 0.
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The variance of εp in very central collisions
εp = 〈T xx−T yy 〉〈T xx+T yy 〉 and 2〈T xy 〉
〈T xx+T yy 〉 added in quadrature:
0
5
10
15
20
25
30
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
dN/d
p
p
RHIC, b = 0 fm, /s = 0.08, including noise
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Conclusions
I Hydrodynamical noise in heavy-ion collisions produces a quiet butimportant correlation in heavy-ion collisions, capable of providingindependent measurements of viscosity.
I Thermal noise contributes to event-by-event variance of v2.
I Simulations with initial and freeze-out fluctuations necessary to findsignal of hydrodynamical noise in experimental observables.
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References I
L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Statistical PhysicsPart 2, Landau and Lifshitz Course of Theoretical Physics Volume 9,1980 Pergamon Press plc.
Joseph I. Kapusta and Charles Gale, Finite Temperature Field Theory:Principles and Applications, Cambridge University Press, 2006.
B. Schenke, S. Jeon and C. Gale, Phys. Rev. C 82, 014903 (2010)[arXiv:1004.1408 [hep-ph]].
D. T. Son and D. Teaney, JHEP 0907, 021 (2009) [arXiv:0901.2338[hep-th]].
J. I. Kapusta, B. Muller and M. Stephanov, Phys. Rev. C 85, 054906(2012) [arXiv:1112.6405 [nucl-th]].
W. Rumelin, SIAM Journal on Numerical Analysis 19, No. 3 (Jun.,1982), pp. 604-613
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References II
B. Schenke, S. Jeon and C. Gale, Phys. Rev. Lett. 106, 042301(2011) [arXiv:1009.3244 [hep-ph]].
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