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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