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Physics of Relativistic Heavy Ion Collisions At QUARK MATTER 2014 Darmstadt, Germany, 19-24 May, 2014 Tetsufumi Hirano Sophia Univ. In collaboration with R. Kurita, K. Murase and K. Nagai 1 Introduction Fluctuating Hydrodynamics in Bjorken Expansion Fluctuation theorem Summary 2 • 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, Moore, Murase, Young, … 3 Thermal equilibrium state Entropy = Maximum entropy state 4 dissipation Thermal equilibrium state Entropy = Maximum entropy state Balance between fluctuation and dissipation fluctuation Stability of thermal equilibrium state 5 dissipation Thermal equilibrium state Entropy = Maximum entropy state Balance between fluctuation and dissipation fluctuation Stability of thermal equilibrium state 훿Π(푥)훿Π(푥′) = 푇퐺∗(푥, 푥′) 퐺∗ : Symmetrized correlation function 훿Π : Thermal (hydrodynamic) fluctuation 6 7 Equation of motion 푒 : Energy density 푑푒 푒 + 푃 푒 휋 − Π 푃 : Pressure = − 1 − 푑 푠푇 푠 : Entropy density 휋 Constitutive equations : Shear stress Π : Bulk pressure 푑휋 4휂 + 휋 = + 휉 Shear: 휋 푑 3 휋 휂 : Shear viscosity 휁 : Bulk viscosity 푑Π 휁 Bulk: Π + Π = − + 휉Π 휋, Π : Relaxation time 푑 8 Equation of motion 푒 : Energy density 푑푒 푒 + 푃 푒 휋 − Π 푃 : Pressure = − 1 − 푑 푠푇 푠 : Entropy density 휋 Constitutive equations : Shear stress Π : Bulk pressure 푑휋 4휂 + 휋 = + 휉 Shear: 휋 푑 3 휋 휂 : Shear viscosity 휁 : Bulk viscosity 푑Π 휁 Bulk: Π + Π = − + 휉Π 휋, Π : Relaxation time 푑 휉: Gaussian white noise* Hydrodynamic *K. Murase, TH, arXiv:1304.3243; noises! K. Murase, Poster H-21 9 푇 − 푇 푇 − 푇 1 − tanh 푐 1 + tanh 푐 푑 = 푑 푑 + 푑 푑 eff 퐻 2 푄 2 . 2 푑푄 = 37 4휋 3 d.o.f 푠 푇 = 푑 푇 eff 90 푇 푇푐 = 170 MeV 푝 푇 = 푠 푇′ 푑푇′ Effective Effective 푑퐻 = 3 0 푑 = 푇푐/50 푒 = 푇푠 − 푝 푇 (MeV) D.H.Rischke, nucl-th/9809044 10 Shear viscosity 휂 1 휂 = 푠 4휋 푠 P.Kovtun et al., PRL94, 111601 (2005) 휁 Bulk viscosity 2 푠 휁 1 휂 = 15 − 푐2 푠 3 푠 푠 S.Weinberg, Astrophys.J.168, 175 (1971) 3휂 Relaxation time = = 휋 Π 2푝 Caveat: Just for demonstration! 11 Initial conditions Fluctuating hydro 0 = 1 fm 푇 = 0.22 GeV Viscous fluid 0 휋0 = Π0 = 0 Perfect fluid 12 Fluctuating hydro Viscous fluid Perfect fluid Fixed initial condition Final entropy fluctuation due to hydrodynamic noises 13 Perfect fluid Viscous fluid The number The number of events Fluctuating entropy around mean value 14 15 D.J.Evans et al., PRL71,2401(1993); G.Gallavotti and E.G.D.Cohen, PRL74, 2694(1995); D.J.Evans and D.J.Searles, PRE52, 5839(1995). 푃 = 퐴 = 푒퐴푡 푃 : Probability distribution 푃 = −퐴 of entropy production rate 1 푡 푡 = 푠 푑푠 Entropy production rate 푡 0 averaged over 0 < 푠 < 푡 푃 Probability of negative entropy production 푃 = 퐴 푃 = −퐴 Quantified through fluctuation theorem! −퐴 0 퐴 *Milestone in non-equilibrium statistical physics since linear response theory 16 Consequence of fluctuation theorem in Bjorken expansion* ∆푆 2 ∆ 푠 1 fin = 푆fin 0푠0 + ∆ 푠 ∆휂푠Δ푥Δ푦 Δ 푠 : Average entropy production per volume of local thermal system Δ푉 = Δ휂푠Δ푥Δ푦: Volume of local thermal system *For brief derivation, see backup slide 17 The number of independent local thermal system: 푆 푏 Δ푥 푁 = Δ푥Δ푦 Δ푦 Theorem: Upper bound of total entropy fluctuation in Bjorken expansion Δ푆 1 ∆푆 2 ∆ 푠 1 tot = fin = 푆 푆 푠 + ∆ 푠 tot 푁 fin 0 0 ∆휂푠푆 푏 1 1 1 ≤ = 20푠0 ∆휂푠푆 푏 2푆ini 18 The number of independent local thermal system: 푆 푏 Δ푥 푁 = Δ푥Δ푦 Δ푦 Theorem: Upper bound of total entropy fluctuation in Bjorken expansion Δ푆 1 ∆푆 2 ∆ 푠 1 tot = fin = 푆 푆 푠 + ∆ 푠 tot 푁 fin 0 0 ∆휂푠푆 푏 1 1 1 ≤ = 20푠0 ∆휂푠푆 푏 2푆ini 19 푆 푏 p-p collision in = 61 mb at LHC ) 푅 = 6 fm = 42 mb at RHIC 1 in − fm 1 ( ~0.4 − 0.5 푆 in 1 Enhancement of 푏 (fm) fluctuation effects in small system 20 • Hydrodynamic fluctuation in 1D Bjorken expansion • Final entropy (multiplicity) fluctuation due to hydro noises • Fluctuation theorem in 1D Bjorken expansion Upper bound of fluctuation • More quantitative analysis needed • Fluctuation of flow velocity? • Higher harmonics in 2D expansion? 21 Finite number of hadrons time Propagation of jets Hydrodynamic fluctuation collision axis 0 Initial state fluctuation Fluctuations appear everywhere! 22 • Hydro at work to describe elliptic flow (~ 2001) 푑 ≲ 5 fm • E-by-e hydro at work to describe higher harmonics 푑 ≲ 1 fm (~ 2010) • Hydro at work in p-p 푑 ≲ 1 fm? and/or p-A??? (2012-) Is fluctuation important in such a small system? 23 1 휂 = lim lim 푑푡푑푥 푒푖(휔푡−푞푥) 휔→0푞→0 2휔 × 푇푥푦 푡, 푥 , 푇푥푦(0,0) Slow dynamics How slow? Macroscopic time scale ~ 1 휔 푡"macro" Microscopic time scale ~ cf.) Long tail problem (liquid in 2D, glassy system, super-cooling, etc. ) 24 Constitutive equations Instantaneous response at Navier-Stokes level violates causality Critical issue in 휋휇휈 = 2휂휕<휇푢휈>, relativistic theory 휇 Π = −휕휇푢 , Relaxation plays an … essential role 휅 thermodynamics force / 푅 or Realistic response 퐹 푡 푡0 25 Linear response to thermodynamic force ′ ′ ′ Π 푡 = 푑푡 퐺푅 푡, 푡 퐹 푡 Retarded Green function (as an example) 휅 푡 − 푡′ 퐺 푡, 푡′ = exp − 휃(푡 − 푡′) 푅 Differential form Π 푡 − 휅퐹 푡 휅 Π 푡 = − , 푣 = < 푐 signal Maxwell-Cattaneo Eq. (simplified Israel-Stewart Eq.) 26 퐺 퐺 푅 Non-Markovian 푅 Markovian 푡 휅 − coarse 퐺 = 푒 휏 퐺푅 ≈ 휅훿 푡′ 푅 graining Maxwell-Cattaneo ??? Navier Stokes causality 푡 → 푡′ 푡 푡′ Existence of upper bound in coarse-graining time (or lower bound of frequency) in relativistic theory??? 27 Figures from J.B.Bell et al., ESAIM 44-5 (2010)1085 Ex.) Seeds for instabilities Rayleigh-Taylor instability Kelvin-Helmhortz instability Non-linearity, instability, dynamic critical phenomena,… 28 29 Linear response to thermodynamic force ′ ′ ′ Π 푡 = 푑푡 퐺푅 푡, 푡 퐹 푡 Dissipative current 횷 Thermodynamic force 푭 Shear Shear stress tensor Gradient of flow Bulk Bulk pressure Divergence of flow Diffusion Diffusion current Gradient of chemical potential 퐺푅 Transport coefficient, relaxation time, … 30 • Generalized Langevin Eq. for currents 4 ′ ′ ′ Π 푥 = 푑 푥 퐺푅 푥, 푥 퐹 푥 + 훿Π(푥) • Fluctuation-Dissipation Relation (F.D.R.) 훿Π(푥)훿Π(푥′) = 푇퐺∗(푥, 푥′) 퐺∗: Symmetrized correlation function 훿Π : Hydrodynamic fluctuation K. Murase and TH, arXiv:1304.3243[nucl-th] 31 휅 푡 − 푡′ 퐺 푡, 푡′ = exp − 휃(푡 − 푡′) 푅 2휋 4훿(휔 − 휔′) 훿 3 (풌 − 풌′) 훿Π∗ 훿Π = 2휅 휔,풌 휔′,풌′ 1 + 휔22 Colored noise! (Indirect) consequence of causality K. Murase and TH, arXiv:1304.3243[nucl-th] 32 Integral form 4 ′ ′ ′ Π 푥 = 푑 푥 퐺푅 푥, 푥 퐹 푥 + 훿Π(푥) Differential form 푑Π 푥 + Π 푥 = 퐹 푥 + 휉 푥 푑푡 휉 = ℒ 푑 푑푡 훿Π: white noise No memory effect nor colored noise Practically convenient K. Murase and TH, arXiv:1304.3243[nucl-th] 33 푑휋 4휂 + 휋 = + 휉 휋 푑 3 휋 Discretized 휋 4휂 휉 휋 + Δ = 휋 + − + + 휋 Δ 휋 3휋 휋 34 F.D.R. for shear viscosity 휇휈 훼훽 휇휈훼훽 (4) ′ 휉휋 (푥)휉휋 (푥′) = 4푇휂Δ 훿 푥 − 푥 1 1 Δ휇휈훼훽 = Δ휇훼Δ휈훽 + Δ휇훽Δ휈훼 − Δ휇휈Δ훼훽 2 3 Δ휇휈 = 푔휇휈 − 푢휇푢휈 휇 Bjorken expansion case 푢 = (cosh 휂푠 , 0, 0, sinh 휂푠) 00 33 00 33 휋 = 휋 − 휋 ⟹ 휉휋 = 휉휋 − 휉휋 35 00 33 Need F.D.R. for 휉휋 = 휉휋 − 휉휋 00 00 00 33 33 33 휉휋 푥 휉휋 푥′ = 휉휋 휉휋 − 2 휉휋 휉휋 + 휉휋 휉휋 Magnitude of fluctuation in discretized space-time 4푇휂 8푇휂 ⟹ Δ0000 − 2Δ0033 + Δ3333 = ΔΔ푉 3ΔΔ푉 Width of Gaussian white noise Δ푉 = Δ휂푠Δ푥Δ푦, Δ휂푠 = 1, Δ푥 = Δ푦 = 1 (fm) 36 Our approach Conventional method 푑 푠 휋 − Π 푑 푠 3휋2 Π2 = ≶ 0 = + ≥ 0 푑 푇 푑 푇 4휂 휁 Entropy production Not positive definite Constitutive equations due to hydrodynamic noises designed to obey 2nd law 2nd law of thermodynamics of thermodynamics on (ensemble) average 37 Suppose entropy production rate obeys Gaussian 1 − 2 푃 = exp − 2휋푎2 2푎2 From fluctuation theorem, 2 = 푡 ⟹ 푡푎 = 2 푡 푎2 2 푆fin − 푆fin 푃 푆fin ∝ exp − 2 2 2 푡 38 39.
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