St’át’imc – BC Hydro Agreement Review and Update: January 2012 1.
Review of recent development in hydro
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REVIEW OF RECENT DEVELOPMENT IN HYDRO
Heavy Ion Meeting 2013-10,Inha Univ., Korea, Nov.2, 2013
Tetsufumi HiranoSophia Univ.
K.Murase, TH, arXiv:1304.3243Y.Tachibana, TH, Nucl.Phys.A904-905(2013)1023c
Y.Tachibana, talk at Hard Probes 2013M.Hongo, Y.Hirono, TH, arXiv:1309.2823
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Outline Introduction Relativistic fluctuating hydrodynamics Medium response to jet propagation Anomalous hydrodynamics and chiral
magnetic effect Summary
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Introduction
0collision axis
time
3. QGP fluid
4. Hadron gas
1. QCD aspects of nuclear structure
2. Non-equilibriumstatistical physics
3. Relativistichydrodynamics
4. Relativistic kinetic theory
1. Color glass condensate
2. Glasma
“Form” of matterApproach
Hydrodynamics as a central framework of heavy ion collisions
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RELATIVISTIC FLUCTUATING HYDRODYNAMICS
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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
5See talks by Y.-S.Kim and M.-J. Kweon
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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)
• E-by-e hydro at work to describe higher harmonics (~ 2010)
𝑑≲5 fm
coarsegraining
size
initialprofile
𝑑≲1 fm
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Is fluctuation important in such a small system?
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Thermal Fluctuation • Conventional hydro describes space-time
evolution of (coarse-gained) thermodynamic quantities.
• Some of microscopic information must be lost through coarse-graining process.
• Does the lost information play an important role in dynamics on an e-by-e basis? Thermal (Hydrodynamic) fluctuation!
J.Kapusta, B.Muller, M.Stephanov, PRC85(2012)054906.J.Kapusta, J.M.Torres-Rincon, PRC86(2012)054911.P.Kovtun, G.D.Moore, P.Romatschke, PRD84(2011)025006.J.Peralta-Ramos, E.Calzetta, JHEP1202(2012)085.
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Causal Hydrodynamics
Π (𝑡 )=∫𝑑𝑡′𝐺𝑅 (𝑡 , 𝑡′ )𝐹 (𝑡 ′ )Linear response to thermodynamic force
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Dissipative current Themodynamic force Shear Shear stress tensor Gradient of flowBulk Bulk pressure Divergence of flow
Diffusion Diffusion current Gradient of chemical potential
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Causal Hydrodynamics (contd.)
Differential form
Π̇ (𝑡 )=− Π (𝑡 )−𝜅 𝐹 (𝑡 )𝜏 ,
Maxwell-Cattaneo Eq. (simplified Israel-Stewart Eq.)
𝑣 signal=√ 𝜅𝜏 <𝑐
2. Retarded Green function (causal, 2nd order hydro)
𝐺𝑅 (𝑡 ,𝑡 ′ )= 𝜅𝜏 exp(− 𝑡− 𝑡
′
𝜏 )𝜃 (𝑡−𝑡 ′)
1. Delta function (acausal, 1st order hydro)
𝐺𝑅 (𝑡 ,𝑡 ′ )=𝜅𝛿 (𝑡− 𝑡 ′ )→Π (𝑡 )=𝜅 𝐹 (𝑡 )
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Relativistic Fluctuating Hydrodynamics (RFH)
Π (𝑥 )=∫𝑑4 𝑥 ′𝐺𝑅 (𝑥 ,𝑥 ′ )𝐹 (𝑥′ )+𝛿Π (𝑥)⟨𝛿Π (𝑥 )𝛿Π (𝑥′ )⟩=𝑇𝐺∗(𝑥 , 𝑥′ )
Generalized Langevin Eq. for fields
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 function
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𝛿Π: Hydrodynamic fluctuation
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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 Correlation in Fourier space
Colored noise! As a consequence of causalityNote: white noise in differential form
K.Murase and TH, arXiv:1304.3243[nucl-th]
11cf.) C. Young, arXiv:1306.0472
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Fluctuation in Non-Linear Equation
Rayleigh-Taylor instability
Need numerical implementation of fluctuating hydro
Kelvin-Helmhortz instability
Figures from J.B.Bell et al., ESAIM 44-5 (2010)1085
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Ex.) Seeds for instabilities
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MEDIUM RESPONSE TOJET PROPAGATION
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Jet Quenching as Missing pT
Figures adapted from talkby C.Roland (CMS) at QM2011
Jet momentum balanced by low pT hadrons outside the jet cone! Medium response?
in-cone
out-of-cone
14See talk by Y.-S. Kim
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Momentum Balance Restored
Lost energy 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
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𝜕𝜇𝑇 fluid𝜇𝜈 = 𝐽 jet𝜈
Energy and Momentum Flowing into QGP• Jet quenching could affect QGP expansion
at LHC (and even at RHIC?)• Relativistic hydrodynamics with a source
term due to propagation of jets
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𝐽 jet0 (𝑥 )=−
𝑑𝑝0𝑑𝑡 𝛿(3 )(𝒙− 𝒙 (𝑡 ))
𝐽 jet𝑖 (𝑥 )=−
𝑑𝑝0𝑑𝑡
𝑝𝑖
𝑝0𝛿(3 )(𝒙− 𝒙 (𝑡 ))
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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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QGP Mach Cone Induced by a Jet
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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Central collisionat LHCJet momentum:
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Energy Balance in and out-ofJet Cone in Hydro + Jet Model
Energy balance Leading jet Soft particle
Qualitatively consistent with CMS results
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Another Channel to Constrain Shear Viscosity?• Jet propagation induces large flow velocity
along jet axis.• How much momentum diffuses perp. to
jet axis?
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Y.Tachibana and TH, Nucl.Phys.A904-905 (2013)1023c
Momentum diffusion
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ANOMALOUS HYDRODYNAMICS AND CHIRAL MAGNETIC EFFECT
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Strong EM fields in Heavy Ion Collisions
Figure taken from http://physics.aps.org/articles/v2/104
!!!
Electr
ic cu
rrent
Electr
ic cu
rrent
Positively charged heavy ion as a source ofelectro magnetic field Ampere’s law
Magnetic field
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Charge Asymmetry of Pion
𝐴±= 𝑁+¿−𝑁−
𝑁+¿+𝑁−¿¿
G.Wang (STAR), arXiv:1210.5498
Charge asymmetic results indicate existenceof electromagnetic effects of transport?
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Chiral Magnetic Wave
Slide taken from a seminar talk by M.Hongo at BNL (2013)
D.Kharzeev and H.U.Yee(2011);Y.Burnier et al.(2011)
For introduction of CME and CSE, see K.Fukushima, arXiv:1209.5064
chiralseparation
effect
chiralmagnetic
effect
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Anomalous HydroHydrodynamic eqs. under EM fields + anomaly
Constitutive eqs. incl. CME and CSE
D.T.Son and P.Surowka,PRL103(2009)191601 25
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Source of Axial Charge
𝜕𝜇 𝑗5𝜇=−12𝜋 2
𝐸𝜇𝐵𝜇
Quantum anomaly term
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Charge density and axial charge density in thetransverse plane at
Charge and Axial Charge in Expanding Case
𝑡=6 fm/c
M.Hongo, Y.Hirono, TH, arXiv:1309.2823
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Difference of btw. Positive and Negative pions
STAR data without anomalyIntercept is finite with anomaly
Need more quantitative analysis of anomalous hydro
M.Hongo, Y.Hirono, TH, arXiv:1309.2823
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SummaryHydrodynamic framework/simulation extended in three directions• Relativistic fluctuating hydrodynamics
• Source term from jet propagation
• Quantum anomalous effects
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Eccentricity Fluctuation in Small System
𝜀 part>𝜀stdPHOBOS, Phys. Rev. Lett. 98, 242302 (2007) 30
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Fluctuation in Initial Conditions
B.Alver et al., Phys. Rev. C 77, 014906 (2008)
𝜀𝑛=⟨𝑟2𝑒𝑖𝑛 𝜙 ⟩
⟨𝑟 2 ⟩
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Fluctuation Appears Everywhere
0collision axis
time Finite number of
hadrons
Propagation of jets
Hydrodynamic fluctuation
Initial fluctuation andparticle production 32
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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 34
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Coarse-Graining in Time
𝑡
𝐺𝑅
𝑡 ′
𝐺𝑅
𝐺𝑅=𝜅𝜏 𝑒
− 𝑡𝜏 𝐺𝑅≈ 𝜅𝛿 (𝑡 ′ )coarsegraining
???
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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Outlook for RFH
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• Numerical implementation in full hydro equation (not linearized one) and its consequences in observables• Langevin simulation in constrained system?• Dynamic critical phenomena
Need further formulation of RFH
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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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Outlook for Interplay between Soft and Hard• Alternative constraint for shear viscositythrough propagation of jets?
• Heat-up due to mini-jets propagation similar to neutrino reheating in Super Nova Explosion?
Jet propagationmomentum
transfer perp. tojet propagation
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Full 3D Anomalous Hydro Simulation
�⃗�
𝑛𝑅𝑛𝐿
𝑛=𝑛𝑅+𝑛𝐿
M.Hongo, Y.Hirono and TH, arXiv:1309.2823
𝑒𝐵𝑦=1.0GeV2
“Group velocity” of chiral magnetic wave CME under expanding background? Source of charge asymmetric ?
𝑇=0.5GeV
Backgroundtemperature
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