Download Hydrodynamic fluctuations

Collective Flows and Hydrodynamics
in High Energy Nuclear Collisions
Dec. 14-15, 2016,
Hefei, China
Relativistic Fluctuating
Hydrodynamics
Tetsufumi Hirano
Sophia Univ.
Outline
• Introduction
• Relativistic Fluctuating
Hydrodynamics
• Application to Heavy Ion Collisions
• Summary
time
Introduction
Finite number of
hadrons
Propagation of jets
Hydrodynamic
fluctuations
collision axis
0
Initial state fluctuation
Fluctuations appear everywhere!
Final hadronic observables  Whole history
Size of coarse-grained system
• Hydro at work to describe
elliptic flow (~ 2001)
𝑑 ≲ 5 fm
• E-by-e hydro at work to
describe higher harmonics
(~ 2010)
𝑑 ≲ 1 fm
• Hydro at work in p-p
and/or p-A??? (2012-)
𝑑 ≲ 1 fm?
Is fluctuation important
in such a small system?
Thermal fluctuation
• Conventional hydrodynamics
 Space-time evolution of (coarse-grained)
thermodynamic quantities
• Microscopic information
 Lost through coarse-graining process
• Does the lost information play an important role
in dynamics in small system and/or on an e-by-e
basis?
 Thermal (Hydrodynamic) fluctuation!
Calzetta, Kapusta, Muller, Stephanov, Young, Moore
Kovtun, Romatschke, Hirano, Murase,…
Green-Kubo formula
1
𝜂 = lim lim
𝜔→0𝑞→0 2𝜔
𝑑𝑡𝑑𝑥 𝑒 𝑖(𝜔𝑡−𝑞𝑥)
× 𝑇𝑥𝑦 𝑡, 𝑥 , 𝑇𝑥𝑦 (0,0)
Fluctuation of energy momentum tensor
Transport coefficients: shear viscosity,
bulk viscosity, diffusion coefficient, …
Relaxation and causality
Constitutive equations
at Navier-Stokes level
Instantaneous response
violates causality
 Critical issue in
relativistic theory
 Relaxation plays an
essential role
𝐹 or 𝑅/𝜅
𝜋 𝜇𝜈 = 2𝜂𝜕 <𝜇 𝑢𝜈> ,
Π = −𝜍𝜕𝜇 𝑢𝜇 ,
…
thermodynamics force
Realistic response
𝑡0
𝜏
𝑡
Linear Response relation
Π 𝑥 =
4 ′
′
𝑑 𝑥 𝐺 𝑥, 𝑥 𝐹 𝑥
′
Lorentz indices are omitted for simplicity.
Shear
Bulk
Diffusion
Dissipative current 𝛱
Shear stress tensor
Thermodynamic force 𝐹
Gradient of flow
Bulk pressure
Diffusion current
Divergence of flow
Gradient of 𝜇𝐵
 Constitutive equations
Causal constitutive equation
Retarded (Causal) response function
𝐺 𝑡, 𝑡 ′
𝜅
𝑡 − 𝑡′
= exp −
𝜃(𝑡 − 𝑡 ′ )
𝜏
𝜏
𝜅: transport coefficient
𝜏: relaxation time
𝛱 𝑡 − 𝜅𝐹 𝑡
𝛱 𝑡 =−
𝜏
Maxwell-Cattaneo Eq. (simplified Israel-Stewart Eq.)
Coarse-graining in time
Non-Markovian
Markovian
𝐺
𝐺
𝜅 −𝑡
𝐺= 𝑒 𝜏
𝜏
coarse
graining
𝐺 ≈ 𝜅𝛿 𝑡′
Navier Stokes
 causality
Maxwell-Cattaneo
𝑡
𝜆 → 𝑡′
𝑡
𝑡′
Relaxation and resolution
 Important in relativistic theory
𝑡′
Relativistic Fluctuating
Hydrodynamics
K. Murase and TH, arXiv:1304.3243[nucl-th]
K. Murase, Ph.D thesis, the U. of Tokyo (2015)
Brownian motion
𝑑𝑣(𝑡)
Q1) Solve the Langevin eq. 𝑚
= −𝛾𝑣(𝑡) + 𝑓(𝑡)
𝑑𝑡
𝑡
′
𝑓
𝑡
′
𝑑𝑡′
A1) 𝑣 𝑡 = 𝑣 0 𝑒 −𝛾𝑡 𝑚 + 𝑒 −𝛾 𝑡−𝑡 𝑚
𝑚
0
Q2) When the noise terms obey 𝑓(𝑡) = 0 and
𝑓 𝑡1 𝑓(𝑡2 ) = 2𝛾𝑘𝐵 𝑇𝛿(𝑡1 − 𝑡2 ), calculate the
ensemble average of kinetic energy at long-time limit.
A2) The equipartition of energy
𝑚 2
1
𝑣 (𝑡 → ∞) = 𝑘𝐵 𝑇
2
2
⋯ : ensemble average.
Fluctuation dissipation relation
Entropy
𝑑𝑣(𝑡)
𝑚
= −𝛾𝑣(𝑡) + 𝑓(𝑡)
𝑑𝑡
𝑓 𝑡1 𝑓(𝑡2 ) = 2𝛾𝑘𝐵 𝑇𝛿(𝑡1 − 𝑡2 )
Thermal equilibrium state
= Maximum entropy state
dissipation
fluctuation
𝑆 = 𝑆0 + 𝛿𝑆 + 𝛿 2 𝑆 + ⋯
<0
State
Balance between
fluctuation and
dissipation
 Stability of the
thermal equilibrium state
Relativistic Fluctuating Hydrodynamics
(RFH)
Generalized Langevin Eq. for dissipative currents
Π 𝑥 =
𝑑 4 𝑥 ′ 𝐺 𝑥, 𝑥 ′ 𝐹 𝑥 ′ + 𝛿Π(𝑥)
Fluctuation-Dissipation Relation (F.D.R.)
𝛿Π(𝑥)𝛿Π(𝑥 ′ ) = 𝑇𝐺 ∗ (𝑥, 𝑥 ′ )
𝐺 ∗ : Symmetrized correlation function
𝛿Π : Hydrodynamic fluctuation
K. Murase and TH, arXiv:1304.3243[nucl-th]
Colored Noise in relativistic system
𝐺 𝑡, 𝑡 ′
∗
𝛿Π𝜔,𝒌 𝛿Π𝜔′,𝒌′
𝜅
𝑡 − 𝑡′
= exp −
𝜃(𝑡 − 𝑡 ′ )
𝜏
𝜏
2𝜋 4 𝛿(𝜔 − 𝜔′ ) 𝛿 3 (𝒌 − 𝒌′ )
= 2𝜅
1 + 𝜔2𝜏 2
Colored noise!
 (Indirect) consequence of causality
K. Murase and TH, arXiv:1304.3243[nucl-th]
Integral form vs. differential form
Integral form
Π 𝑥 =
𝑑4 𝑥 ′ 𝐺 𝑥, 𝑥 ′ 𝐹 𝑥 ′ + 𝛿Π(𝑥)
Differential form
𝑑Π 𝑥
𝜏
+ Π 𝑥 = 𝜅𝐹 𝑥 + 𝜉 𝑥
𝑑𝑡
𝜉 = ℒ 𝑑 𝑑𝑡 𝛿Π: white noise
 No longer colored noise
 Practically convenient
K. Murase and TH, arXiv:1304.3243[nucl-th]
Fluid in a box
Ideal hydro
(1G hydro)
No dissipation
Wave propagation
Dissipative hydro
(2G hydro)
Towards global
equilibrium
Fluctuating hydro
(3G hydro)
Fluctuating
around mean
value
Application to
Heavy Ion Collisions
K. Murase, Ph.D thesis, the U. of Tokyo (2015)
K. Nagai et al., Nucl. Phys. A956, 781 (2016)
K. Murase and TH, Nucl. Phys. A956, 276 (2016)
Bjorken expansion with
viscosity and fluctuation
Equation of motion
𝑑𝑒
𝑒+𝑃 𝑒
=−
𝑑𝜏
𝜏
𝜋−𝛱
1−
𝑠𝑇
Stochastic constitutive equations
Shear:
Bulk:
𝑑𝜋
4𝜂
𝜏𝜋
+𝜋 =
+ 𝜉𝜋
𝑑𝜏
3𝜏
𝑑𝛱
𝜁
𝜏𝛱
+ 𝛱 = − + 𝜉𝛱
𝑑𝜏
𝜏
K. Nagai et al., Nucl. Phys. A956, 781 (2016)
𝑒 : Energy density
𝑃 : Pressure
𝑠 : Entropy density
𝜋 : Shear stress
𝛱 : Bulk pressure
𝜂 : Shear viscosity
𝜁 : Bulk viscosity
𝜏𝜋 , 𝜏𝛱 : Relaxation time
Hydrodynamic
noises!
Time evolution of dissipative current
Initial conditions: 𝜋 𝜏 = 𝜏0 = Π 𝜏 = 𝜏0 = 0
Dissipative currents fluctuating
around zero at long-time limit
K. Nagai et al., Nucl. Phys. A956, 781 (2016)
Entropy production rate
in Bjorken expansion
Israel-Stewart formalism
Fluctuating hydro
𝑑 𝑠𝜏
𝜏 3𝜋 2 𝛱2
=
+
≥0
𝑑𝜏
𝑇 4𝜂
𝜁
Constitutive equations
designed to obey the 2nd
law of thermodynamics
 Non-linear response
𝑑 𝑠𝜏
𝜋−𝛱
=
≶0
𝑑𝜏
𝑇
Entropy production
 Not positive definite
due to hydrodynamic noises
 The 2nd law of
thermodynamics on
(ensemble) average
K. Nagai et al., Nucl. Phys. A956, 781 (2016)
time
Full 3D fluctuating hydro +
hadronic cascade
Hadronic cascade JAM
Particlization 𝑇sw = 155 MeV
(3+1)-d relativistic
fluctuating hydro
EOS: s95p-v1.1 (Pasi’s 2nd lecture)
𝜂 𝑠 = 1 4𝜋
collision axis
0
MC-KLN
*MC-Glauber also availlable
For ideal hydro version, see TH et al., PPNP 70, 108 (2013)
𝜏𝑇
HF off
Without
initial
fluctuation
HF on
Courtesy of
K. Murase
𝜏𝜏
𝜏𝑇
HF off
With
initial
fluctuation
HF on
Courtesy of
K. Murase
𝜏𝜏
Effects of hydrodynamic
fluctuations on vn
Au+Au 𝑠NN = 200GeV
0-5% centrality
Analysis of vn
Event plane method
“eta-sub”
1.0 < 𝜂 < 2.8
Increase of anisotropy due to hydrodynamic fluctuations
K. Murase and TH, Nucl. Phys. A956, 276 (2016)
Flow fluctuations
Au+Au 𝑠NN = 200GeV
20-30% centrality
Analysis of vn
Multiparticle
cumulant method
𝑣 2 2 ~𝑣 2 + 𝜎 2
𝑣 4 2 , 𝑣 6 2 ~𝑣 2 − 𝜎 2
Flow fluctuations may contain information
about hydrodynamic fluctuations.
K. Murase and TH, Nucl. Phys. A956, 276 (2016)
Distribution of flow
Oversampling of hadrons
from each hydrodynamic
event
Flow distribution
broadened by
hydrodynamic fluctuations
−0.5 < 𝜂 < 0.5
K. Murase and TH, Nucl. Phys. A956, 276 (2016)
Summary
• Through fluctuation-dissipation relation,
hydrodynamic fluctuations should exist
during evolution of dissipative fluids.
• New channel to constrain transport
coefficients
• Flow fluctuations
• (Event-plane decorrelation)
• Systematic studies needed for analysis of
flow observables with hydrodynamic
fluctuations
Model for Equation of State
Effective degree of freedom
Effective d.o.f.
𝑑eff
𝑇 − 𝑇𝑐
𝑇 − 𝑇𝑐
1 − tanh
1 + tanh
𝑑
𝑑
= 𝑑𝐻
+ 𝑑𝑄
2
2
𝑑𝑄 = 37
4𝜋 2 4
𝑠 𝑇 = 𝑑eff
𝑇
90
𝑇
𝑑𝐻 = 3
𝑇𝑐 = 170 MeV
𝑑 = 𝑇𝑐 /50
𝑝 𝑇 =
𝑠 𝑇′ 𝑑𝑇′
0
𝑒 = 𝑇𝑠 − 𝑝
𝑇 (MeV)
Y.Asakawa, T.Hatsuda,Phys.Rev.D55, 4488 (1997)
Models for Transport Coefficients
Shear viscosity
𝜂
𝑠
𝜁
𝑠
𝜂
1
=
𝑠 4𝜋
P.Kovtun et al., PRL94, 111601 (2005)
Bulk viscosity
𝜁
1
= 15
− 𝑐𝑠2
𝑠
3
2
𝜂
𝑠
S.Weinberg, Astrophys.J.168, 175 (1971)
3𝜂
𝜏𝜋 = 𝜏Π =
Relaxation time
2𝑝
Caveat: Just for demonstration!
Some Details about Noises
F.D.R. for shear viscosity
𝜉𝜋 𝜇𝜈 (𝑥)𝜉𝜋 𝛼𝛽 (𝑥′) = 4𝑇𝜂Δ𝜇𝜈𝛼𝛽 𝛿 (4) 𝑥 − 𝑥 ′
Δ𝜇𝜈𝛼𝛽
1 𝜇𝛼 𝜈𝛽
1 𝜇𝜈 𝛼𝛽
𝜇𝛽
𝜈𝛼
= Δ Δ +Δ Δ
− Δ Δ
2
3
Δ𝜇𝜈 = 𝑔𝜇𝜈 − 𝑢𝜇 𝑢𝜈
Bjorken expansion case 𝑢𝜇 = (cosh 𝜂𝑠 , 0, 0, sinh 𝜂𝑠 )
𝜋 = 𝜋 00 − 𝜋 33 ⟹ 𝜉𝜋 = 𝜉𝜋00 − 𝜉𝜋33
Some Details about Noises (contd.)
Need F.D.R. for 𝜉𝜋 = 𝜉𝜋00 − 𝜉𝜋33
𝜉𝜋 𝑥 𝜉𝜋 𝑥′
= 𝜉𝜋00 𝜉𝜋00 − 2 𝜉𝜋00 𝜉𝜋33 + 𝜉𝜋33 𝜉𝜋33
Magnitude of fluctuation in discretized space-time
4𝑇𝜂 0000
8𝑇𝜂
0033
3333
⟹
Δ
− 2Δ
+Δ
=
Δ𝜏Δ𝑉
3Δ𝜏Δ𝑉
Width of Gaussian white noise  prop. to inverse volume
Δ𝑉 = 𝜏Δ𝜂𝑠 Δ𝑥Δ𝑦, Δ𝜂𝑠 = 1, Δ𝑥 = Δ𝑦 = 1 (fm)
Time Evolution of Temperature
Initial conditions
Fluctuating hydro
Viscous fluid
Perfect fluid
𝜏0 = 1 fm
𝑇0 = 0.22 GeV
𝜋0 = Π0 = 0
Time Evolution of Entropy
Fluctuating hydro
Viscous fluid
Perfect fluid
Fixed initial condition
 Final entropy fluctuation
due to hydrodynamic noises
Entropy Distribution
Fluctuating entropy around mean value
𝑣2 𝜂 from full 3D fluctuating
hydro + hadronic afterburner
v2 from two particle
correlation/cumulant
𝑣2 2
= 𝑣2
2
2
+ 𝛿𝑣2
2
𝜎: Coarse graining size
𝑣2 visc. < 𝑣2 ideal < 𝑣2 fluc. (!?)
Fluctuation of 𝑣2
 Information about thermal fluctuation