流体揺らぎの運動論に基づく
体積粘性係数のくりこみ赤松幸尚（大阪大学）

2017 年 8 月 28–30 日@熱場の量子論とその応用

Collaborators:
Aleksas Mazeliauskas (Heidelberg) & Derek Teaney (Stony Brook)
Refs: 1606.07742, 1708.05657
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1. Introduction

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Ultra-relativistic heavy-ion collisions and the Bjorken expansion

Heavy-ion collision is an overwhelmingly complicated system! Assumption that hydrodynamics is applicable in the collision center
Bjorken expansion is a simple and characteristic hydrodynamic solution! An approximate boost invariance along the beam axis
uz =zt(|z| ≤ t)

Bjorken expansion is a one-dimensional Hubble expansion
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Hydrodynamics with noise
Kovtun-Yaﬀe (03), Gavin-AbdelAziz (06), Kovtun-Moore-Romatschke (11), Kapusta-Muller-Stephanov (12),Young-Kapusa-Gale-Jeon-Schenke (15), Murase-Hirano (16), Yan-Grönqvist (16), Gavin-Moschelli-Zin (16),YA-Mazeliauskas-Teaney (17), · · ·Thermal fluctuations – for example Landau-Lifshitz
Ngg (t, k) ≡ ⟨g i (t, k)g j∗ (t, k)⟩ = (e + p)T δ ij!
"#$ !
"#$momentum, g i ≡ T 0iequilibrium! Conceptually important (required by the FDT)

! Larger in smaller systems: Nparticle ∼ 10000 in the heavy-ions! Essential near a critical point

How do thermal fluctuations evolve during a Bjorken expansion?
How do thermal fluctuations change the Bjorken expansion?

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Kinetic regime of hydrodynamic fluctuations – a new scale k∗
1. For hydrodynamic fluctuations with wavenumber k:!
!

Equilibration rate ∼ γη k 2Expansion rate ω ∼ 1/τ for a Bjorken expansion(γη ≡ η/(e + p))

2. Compete at a critical scale:k∗ ∼%ωγη

3. Derivative expansion controlled by ϵ ≡ γη ω ≪ 1
ωω≪ k∗ ∼ √csc ϵ! "# s $k∗ is hard !

We derive an eﬀective “Hydro-kinetic theory” for the kinetic regime k∗

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Renormalization in hydrodynamics with noise
1. Modes above the cutoﬀ contribute to local parameters, e.g. p0 , ζ0
2. Modes below the cutoﬀ are dynamical
3. “Usual” p and ζ are defined in the thermodynamic limit a → ∞!
We will show how the cut-oﬀ dependence arises in Hydro-kinetic theory
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Brief review of our results for conformal fluid (c2s = 1/3, ζ = 0)
1. Renormalization based on Hydro-kinetic theory!

Reproduced previous diagrammatic calculation on a trivial backgroundKovtun-Yaﬀe (03), Kovtun-Moore-Romatschke (11)p = p0 (Λ) +

Λ3 T
,
6π 2

e = e0 (Λ) +Λ3 T,2π 2η = η0 (Λ) +17ΛT e0 + p0
120π 2 η0

2. Fractional order correction to the Bjorken expansion (long-time tail)&'3/24η18zz
T = p −+ 1.08318 T+ (λ1 − ητπ ) 2 + · · ·!"#$3τ
4πγη τ!"#$"# 9τ $!
"#$ !
ideal
1st order

1.5th order (long-time tail)Hard Hydro Thermal Loops (HHTLs)

Hard ⇠ k⇤ =hxy (!)

2nd order

Kovtun, Yaffe; Kovtun, Moore, Romatschkeq!

⌘T xy = (e + p)ux uySoft ⇠ !
sound propagator
Evaluate the “Hard Hydro Thermal Loop”

What about
nonconformal
fluidfrom Hydro-kinetic
theory?
✓ ◆3/2
3/2
hi
hT xy (!)i = pi!⌘+(7 + (3/2))

T!

+O(! 2 ) hxy (!)

7 / 25Renormalization for nonconformal fluid
1. Bulk viscosity renormalization:!

2Renormalization proportional to conformal breaking Cζ,ηTΛ
ζ(T ) = ζ0 (T ; Λ) +18π 2
Cζ (T ) ≡ 1 +(2Cζ0Cη2+4 0γζ02γη0

3T dc2s− 3c2s ,2 dT

2. Pressure renormalization:)Cη (T ) ≡ 1 − 3c2s&

T dc2s0p(T ) = p0 (T ; Λ) + 1 +2 dT'T Λ36π 2

3. Energy density renormalization:e(T ) = e0 (T ; Λ) +T Λ32π 2Let us see how these results are obtained by Hydro-kinetic theory
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2. Hydrodynamic fluctuations in the equilibrium

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Hydro-kinetic theory: An analogy with Brownian motion

1. Langevin equation
dp= −γp + ξ ,!"#$ !"#$dtdrag⟨ξ(t)ξ(t′ )⟩ = 2T M γδ(t − t′ )noise

2. Calculate how ⟨p2 (t)⟩ evolves through Langevin process*+d 2⟨p ⟩ = −2γ ⟨p2 ⟩ − M Tdt!
"#$equilibration

Follow the same steps for hydrodynamic fluctuations

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Hydro-Langevin equation in equilibrium

1. Linearized fluctuations: e = e0 + δe, ⃗g = (e0 + p0 )⃗v
φ(t, ⃗k) ≡ (cs δe, ⃗g )2. Hydro-Langevin equation for φ(t, ⃗k)−φ̇(t, ⃗k) =iLφ!"#$ideal ∼ ics k

3. Fluctuation-dissipation theorem+Dφ!"#$

viscous ∼ γk2+ ξ!"#$noise⟨ξ(t, ⃗k)ξ(t′ , ⃗k ′ )⟩ = 2T (e0 + p0 )Dδ(⃗k − ⃗k ′ )δ(t − t′ )The Hydro-Langevin equation is similar to the Brownian motion
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The matrices in the equilibrium

L=,0cs kics kj0γζ0 =-,D = k2

ζ0 + 43 η0,e 0 + p0,00L
T0 γζ0 δij+ γη0 δij

γη0 =

η0
e 0 + p0

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Hydro-kinetic equations for the hydro fluctuations
1. Four eigenmodes of L: φ+ , φ− , φT1 , φT2left moving sound φ−!
"#$λ− = −cs kright moving sound φ+!
"#$λ+ = cs k

2. Analyze the two-point functions – e.g.

transverse modes φT!
"#$λT = 0N++ (t, ⃗k) ≡ ⟨φ+ (t, ⃗k)φ∗+ (t, ⃗k)⟩!

Neglect oﬀ-diagonal components (rotating wave approximation)

3. Hydro-kinetic equations for NAA*+Ṅ++/−− = −γζ0 k 2 N++/−− − T (e0 + p0 )*+ṄT1 T1 /T2 T2 = −2γη0 k 2 NT1 T1 /T2 T2 − T (e0 + p0 )!
"#$equilibration

The Hydro-kinetic equations describe equilibration (FDT)

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Nonlinear contribution to energy-momentum tensoreq
1. Equilibrium fluctuations – using NAA= T (e0 + p0 )⟨g i (⃗x)g j (⃗y )⟩eq = T (e0 + p0 )δ ij δ(⃗x − ⃗y )2. Nonlinear contributions (for conformal fluid)⟨T xx ⟩eq = p0 + (e0 + p0 )⟨ux ux ⟩eq = p0 +⟨g x g x ⟩eq
e 0 + p0

3. Renormalization (for conformal fluid). Λ

p = p0 (Λ) + T!
"#d3 k(2π)3$≡ pphys , independent of Λ

Background pressure and energy density need to be renormalized
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3. Hydrodynamic fluctuations under an external forcing

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Expanding systems

1. Analyze in a comoving frame v0 = 0
2. Examples:!

Tensor/scalar metric perturbations (h(t) = he−iωt , |h| ≪ 1)ds2 = −dt2 + (1 + h(t))dx2 + (1 + h(t))dy 2 + (1 − 2h(t))dz 2"ds2 = −dt2 + (1 + h(t))dx2 + (1 + h(t))dy 2 + (1 + h(t))dz 2!

Bjorken expansion
ds2 = −dτ 2 + dx2⊥ + τ 2 dη 2What is the eﬀect of expansions for hydrodynamic fluctuations?
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Hydro-Langevin equation in an expanding system

1. Hydro-Langevin equation for φ = (cs δe, ⃗g )−φ̇(t, ⃗k) = iLφ +!"#$ideal

Dφ + ξ! "# $viscous + noise+Pφ!"#$expansion1⟨ξ(t, ⃗k)ξ(t′ , ⃗k ′ )⟩ = 2T0 (e0 + p0 )Dδ(⃗k − ⃗k ′ )δ(t − t′ ) √g& &''23T0 dcs0
2
P = ḣ diag
1 + cs0 +, 2, 2, 2
22 dT02. Repeat the same steps with the equilibrium case
2.1 Find 4 eigenmodes of L: φ+ , φ− , φT1 , φT2
2.2 Kinetic equation for each mode NAA = V −1 ⟨φA φ∗A ⟩2.3 Nonlinear fluctuations in T µνThe expanding background perturbs the fluctuations by P
17 / 25Hydro-kinetic equation in a scalar metric perturbation
1. Uniform background solution is time-dependente0 (t) = ē0 −3h(t)(ē0 + p̄0 )2

2. Hydro-kinetic equations for NL = N++/−− and NT = NT1 T1 /T2 T2!

The scalar gravitational field does not distinguish T1 and T2/0ḣ
3T0 dc2s02ṄL = −γζ0 k [NL − Neq (t)] −3cs0 ++ 7 NL
22 dT02ṄT = −2γη0 k 2 [NT − Neq (t)] −!
"#$equilibration

Neq (t) =T0 (e0 + p0 )√g4ḣNT! "# $external forcing

Equilibration and disturbance balance at k ∼ k∗ ∼1ω/γ18 / 25

Nonlinear contribution to T µν
1. Solve the kinetic equation in the linear order of h:iωh(ω)−iω + γ0 k 2
2. Compute fluctuation contribution to T µν in the linear order of h:
N = Neq (t) + δN (t),δN (ω) ∼⟨⃗g 2 ⟩e0 + p0/091−h
3T0 dc2s0 2 2ii2
T = 3(1 − h)p0 − ζ0 ḣ +⟨⃗g ⟩ +⟨c δe ⟩ +O(h2 )! "# $ !2"# $ e0 + p02 dT0 s0!
"#$idealT tt =e0+viscousii
fluctuations Tfluct

3. Contribution from the fluctuations:. Λ√2⟨⃗g ⟩ =gd3 k [NL + 2NT ]!
"#$!

Neq ∝ Λ3 ↔ p0 ,!

,√⟨c2s0 δe2 ⟩ = g. Λd3 k N L

δN ∝ ḣΛ ↔ ζ0 ?

Nonlinear fluctuations require renormalization of bulk viscosity and ...
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Absorb divergences in T tt – Temperature shift
1. T tt does not have a counter term to absorb a divergence ∝ ḣΛT tt = e0 (T0 ; Λ) +T0 Λ3+ O(ḣΛ) + · · ·2π 2

2. Temperature depends on the cutoﬀ!

To account for oﬀ-equilibrium fluctuations near the cutoﬀ

T0 (t; Λ) = T (t) + ∆T (t; Λ)
3. Express T tt in terms of T :T Λ3 de0T tt ≃ e0 (T ; Λ) ++∆T + O(ḣΛ) + · · ·2!
"# 2π $ !dT"#$≡ ephys (T )=0, by which ∆T ∝ ḣΛ fixed

T tt is made finite by the temperature shift ∆T ∝ Λ(∂ · u)

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Absorb divergences in T ii – Renormalization
1. Temperature shift in p0 :p0 (T0 ; Λ) ≃ p0 (T ; Λ) +dp0
∆TdT0! "# $≡ ∆p ∝ ḣΛ

2. Compute isotropic stress:
9T ii = 3(1 − h)p0 (T0 ) − ζ0 ḣ +!
"#$ !2"# $ideal

viscousii
Tfluct,! "# $fluctuationsii
Tfluct=TNiieq!"#$∝ (1 − h)Λ39ii= 3(1 − h)p0 (T ) + TNiieq − ζ0 ḣ + 3(1 − h)∆p + TδN!
"#$! 2"#$∼ (1 − h)(p0 + #Λ3 )ii+ TδN!"#$∝ ḣΛ∼ ḣ(ζ0 + #Λ) + long-time tailTemperature shift aﬀects how bulk viscosity is renormalized

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Renormalization of bulk viscosity, pressure, and energy density
1. Bulk viscosity renormalization:!

2Renormalization proportional to conformal breaking Cζ,ηc.f. Diagrammatic computation for cold Fermi gas [Martinez-Schaefer (17)]TΛζ(T ) = ζ0 (T ; Λ) +18π 2(2Cζ0Cη2+4 0γζ02γη0)

3T dc2s− 3c2s , Cη (T ) ≡ 1 − 3c2s
2 dT
2. Pressure renormalization:&'T dc2s0 T Λ3p(T ) = p0 (T ; Λ) + 1 +2 dT
6π 2Cζ (T ) ≡ 1 +3. Energy density renormalization:
T Λ32π 2
Bulk viscosity renormalization is derived for the first time!
e(T ) = e0 (T ; Λ) +22 / 25

Bulk viscosity from thermal fluctuations
1. Bulk viscosity is finite even if ζ0 (Λ∗ ) = 0 at some scale Λ∗( &)'2ζs Λ∗ /T
1
3
3T dc2s22 2= ··1+
− 3cs + 2(1 − 3cs )sη s/T 3 18π 2 42 dT
2. Lattice EoS by Hot QCD Collaboration
0.08Λ = 2T − 4T
η/s = 1/(4π)

0.07
0.06

ζ/s

0.05
0.04
0.03
0.02
0.01
0
150

200

250

300

350

400

T MeV

Bulk viscosity is enhanced near transition temperature T ≈ 150MeV

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4. Summary & Outlook

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! Hydro-kinetic equation for k∗ , advantageous in expanding systems! Universal renormalization of energy density, pressure, and viscosities! Background-dependent long-time tails ∝ ω 3/2 , τ −3/2

! Bulk viscosity enhanced due to scale symmetry breaking

ζ(T ) = ζ0 (T ; Λ)⎡&⎤
'23T dc2s0
e 0 + p021+− 3cs0⎥TΛ ⎢⎢2 dT
ζ0 + 43 η0 ⎥+⎢⎥
18π 2 ⎣⎦
562 2 e0 + p0+ 4 1 − 3cs02η0! Application to the critical dynamics [YA-Teaney-Yan-Yin, in preparation]!

Kibble-Zurek scaling for critical fluctuations in a Bjorken expansion

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