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Feb 24, 2016
重イオン衝突における散逸と揺らぎ
RHIC-LHC 高エネルギー原子核反応の物理研究会2013年 6月 22-23日
平野哲文(上智大理工)共同研究者 : 村瀬功一 , 橘保貴
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Outline1. はじめに2. Initial fluctuation3. Relativistic fluctuating
hydrodynamics4. Jet propagation in expanding
medium5. Summary and outlook
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はじめに発見のステージ 精密科学に向けて 新奇な現象の発見
重イオン衝突における QGP の物理
より詳細な QGP ダイナミクスの記述に向けた試み3
INITIAL FLUCTUATION
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Eccentricity Fluctuation in Small System
𝜀 part>𝜀stdPHOBOS, Phys. Rev. Lett. 98, 242302 (2007) 5
Fluctuation in Initial Conditions
B.Alver et al., Phys. Rev. C 77, 014906 (2008)
𝜀𝑛=⟨𝑟2𝑒𝑖𝑛 𝜙 ⟩
⟨𝑟 2 ⟩
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Higher Harmonics is Finite!
Figures adapted from talkby J.Jia (ATLAS) at QM2011
Two particle correlation function is composed solely of higher harmonics
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Impact of Finite Higher Harmonics• Most of the people
did not believe hydro description of the QGP (~ 1995)
• Hydro at work to describe elliptic flow (~ 2001)
• Hydro at work (?) to describe higher harmonics (~ 2010)
𝑑≲5 fm
coarsegraining
sizeinitialprofile
𝑑≲1 fm8
Open Questions• Can system respond
hydrodynamically to such a fine structure?
• Why is local thermalization achieved at such a short length/time scale (~1 fm)?
• Concept of hydrodynamics? Is ensemble average taken?
• Thermal fluctuation on an event-by-event basis ? 9
RELATIVISTIC FLUCTUATING HYDRODYNAMICS
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Fluctuation Appears Everywhere
0collision axis
time Finite number of
hadrons
Propagation of jets
Hydrodynamic fluctuation
Initial fluctuation andparticle production
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Green-Kubo Formula
Slow dynamics How slow?Macroscopic time scale ~ Microscopic time scale ~
cf.) Long tail problem (liquid in 2D, glassy system, super-cooling, etc. )
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Relaxation and CausalityConstitutive equations at Navier-Stokes level
…
𝑡
thermodynamics force
𝑡 0
𝐹∨𝑅
/𝜅
Realistic response
Instantaneous response violates causality Critical issue in
relativistic theory Relaxation plays an
essential role
𝜏 13
Causal Hydrodynamics
Π (𝑡 )=∫𝑑𝑡′𝐺𝑅 (𝑡 , 𝑡′ ) 𝐹 (𝑡 ′ )Linear response to thermodynamic force
Retarded Green function𝐺𝑅 (𝑡 ,𝑡 ′ )= 𝜅
𝜏 exp(− 𝑡− 𝑡′𝜏 )𝜃 (𝑡−𝑡 ′)
Differential formΠ̇ (𝑡 )=− Π (𝑡 )−𝜅 𝐹 (𝑡 )
𝜏 ,
Maxwell-Cattaneo Eq. (simplified Israel-Stewart Eq.)
𝑣 signal=√ 𝜅𝜏 <𝑐
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Relativistic Fluctuating Hydrodynamics (RFH)
)
⟨ 𝛿Π (𝑥 )𝛿Π (𝑥′ )⟩=𝑇𝐺∗(𝑥 , 𝑥′ )
Generalized Langevin Eq.dissipative current
hydrodynamicfluctuationthermodynamic force
For non-relativistic case, see Landau-Lifshitz, Fluid Mechanics
Fluctuation-Dissipation Relation (FDR)
K.Murase and TH, arXiv:1304.3243[nucl-th]
𝐺∗: Symmetrized correlation function15
Coarse-Graining in Time
𝑡
𝐺𝑅
𝑡 ′
𝐺𝑅
𝐺𝑅=𝜅𝜏 𝑒
− 𝑡𝜏 𝐺𝑅≈ 𝜅𝛿 (𝑡 ′ )coarse
graining???
Existence of upper bound in coarse-graining time (or lower bound of frequency) in relativistic theory???
Navier Stokes causality
Non-Markovian Markovian
Maxwell-Cattaneo
𝑡→ 𝑡 ′
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Colored Noise in Relativistic System
⟨𝛿Π 𝜔 ,𝒌∗ 𝛿Π 𝜔 ′ ,𝒌 ′ ⟩=2𝜅 (2𝜋 )4 𝛿(𝜔−𝜔 ′)𝛿(3 )(𝒌−𝒌 ′)
1+𝜔2𝜏2
𝐺𝑅 (𝑡 ,𝑡 ′ )= 𝜅𝜏 exp(− 𝑡− 𝑡′
𝜏 )𝜃 (𝑡−𝑡 ′)
: Extention to t<t’Correlation in Fourier space
Colored noise! (Indirect) consequence of causalityNote: white noise in differential form
K.Murase and TH, arXiv:1304.3243[nucl-th]
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A Comment on Heat Conductivity
Phenomenological constitutive eqs.,
for Gluonic system () ???𝜏 𝐷𝑞𝜇
𝐷𝑡 +𝑞𝜇=𝜅𝑇𝑋 𝜇
1st order:
2nd order:
𝜏 𝐷𝑞𝜇
𝐷𝑡 +𝑞𝜇=𝜅𝑇𝑋 𝜇+𝜉𝜇
Getting meaningful solely in RFH Fluctuation of ? 18
RFH: ,
Need Fluctuation?
Rayleigh-Taylor instabilityNon-linearity, instability, dynamic critical phenomena,…
Kelvin-Helmhortz instability
Figures from J.B.Bell et al., ESAIM 44-5 (2010)1085
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Seeds for instabilities
Outlook for RFH
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• Numerical implementation and its consequences in observables• Langevin simulation in constrained
system?• Dynamic critical
phenomenaSeparation of systematic motion from noise?• RFH from quantum Langevin eq.?• Heat conductivity in gluonic system
from Hosoya-Kajantie lattice QCD? AdS/CFT?
JET PROPAGATION INEXPANDING MEDIUM
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Fluctuation Appears Everywhere
0collision axis
time Finite number of
hadrons
Propagation of jets
Hydrodynamic fluctuation
Initial fluctuation andparticle production
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Jet Quenching as Missing pT
Figures adapted from talk
by C.Roland (CMS) at QM2011
Jet momentum balanced by low pT hadrons out of jet cone!
in-coneout-of-cone
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Momentum Balance Restored
Lost energy goes to low pT particles at large angle
Figure adapted from talk by C.Roland (CMS) at QM2011
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Horizontal: Jet asymmetryVertical: Transverse momenta (anti-)parallel to jet axis
𝜕𝜇𝑇 𝜇𝜈= 𝐽𝜈
Energy-Momentum Flow from Jets• Jet quenching could affect QGP
expansion at LHC (and even at RHIC)
• Many jets could disturb fluid evolution (or even heat them up?)
• Beyond linearized hydro
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𝐽 0 (𝑥 )= 𝐽 1 (𝑥 )=−𝑑𝑝0𝑑𝑡 𝛿(3 )(𝒙− 𝒙 (𝑡 ))
𝐽 2 (𝑥 )= 𝐽 3 (𝑥 )=0in the case that jet propagates parallel to x-axis
QGP Wake by Jet
A 50 GeV jet traverses a background with T=0.5 GeV Mach-cone like structure
Vortex ring behind a jet
Y.Tachibana and TH, Nucl.Phys.A904-905(2013)1023c
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April 13, 2013 at Mt. Etna
http://photoblog.nbcnews.com/_news/2013/04/12/17720274-mount-etna-blows-smoke-ring-during-volcanic-eruptions
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QGP Mach Cone Induced by a Jet
First fully non-linear and fully 3D hydro simulation of Mach cone
Y.Tachibana and TH, Nucl.Phys.A904-905(2013)1023c
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Large Angle Emission from Expanding Medium
Jet pair created atoff central position Enhancement of low
momentum particles at large angle from jet axis
Δ ( 𝑑𝑝∥
𝑑cos𝜃 )= 𝑑𝑝∥w . jet
𝑑 cos𝜃 −𝑑𝑝∥
w .o . jet
𝑑cos𝜃
out-of-coneqcos𝜃=−1cos𝜃=1
Y.Tachibana and TH, Nucl.Phys.A904-905 (2013)1023c
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Momentum Balance
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preliminaryin cone out of cone
Pb+Pb central collisions at sqrt(s_NN) = 2.76 TeV
Outlook for Interplay between Soft and Hard• Alternative constraint for shear
viscositythrough propagation of jets?
• Heat-up from mini-jets propagation similar to neutrino reheating in SNE?
Jet propagationmomentumtransfer
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Summary and Outlook• Fluctuation appears everywhere• Hydrodynamic fluctuation• Fluctuation induced by jet
propagation• Development of a more
sophisticated dynamical model towards precision/new heavy-ion physics
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BACKUPS
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HYDRO-BASED EVENT GENERATOR
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Current Status: E-by-E H2C
Hadronic cascade
3D ideal hydro
Monte Carlo I.C.(MC-KLN or MC-Glauber)0
collision axis
time
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Initial Density Fluctuation
1𝑁 ∑ initial condition single initial condition
conventionalhydro
event-by-eventhydro 36
Eccentricity, Triangularity, …𝜀𝑛 , part=
|⟨𝑟 2𝑒𝑖𝑛 𝜙 ⟩|⟨𝑟2 ⟩ 𝑛𝜓𝑛=arg ⟨𝑟2𝑒𝑖𝑛 𝜙 ⟩
40-50%
Almost boost invariant37
v2{***}v2{EP}, v2{2}, v2{4}, v2{6}, v2{LYZ}, …
Hydro-based event generator Analysis of the outputs almost in the sameway as experimental people do.
Demonstration of vn analysis according toevent plane method by ATLAS setup.E.g.) Centrality cut using ET in FCal region
ATLAS, arXiv:1108.601838
Resolution of Event Plane
𝑅𝑛=√ ⟨cos [𝑛 (𝛹 𝑛❑1−𝛹 𝑛
2 ) ]⟩
Reaction (Event) plane is not known experimentally nor in outputs from E-by-E H2C. Event plane method
Event plane resolution using two subevents
𝑛𝛹𝑛=tan−1(∑ 𝐸𝑇 ,𝑖 sin𝑛𝜙𝑖
∑ 𝐸𝑇 ,𝑖cos𝑛𝜙𝑖)
c.f.) A.M.Poskanzer and S.Voloshin, Phys. Rev. C 58, 1671 (1998)39
Resolution of Event Plane from E-by-E H2C
ATLAS, arXiv:1108.6018
Even Harmonics Odd Harmonics
# of events: 80000 (Remember full 3D hydro+cascade!)Subevent “N”: charged, -4.8< h < -3.2 (FCal in ATLAS)Subevent “P”: charged, 3.2< h < 4.8 (FCal in ATLAS)
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Vn{EP}(h)
Even Harmonics Odd Harmonics
Not boost inv. almost boost inv. for epsilon
40-50%40-50%
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vn{EP} vs. vn{RP}
Even Harmonics Odd Harmonics40-50% 40-50%
vodd{RP}~0veven{EP}>veven{RP}due to fluctuation
vn{RP}: vn w.r.t. reaction plane known in theory
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