Hydrodynamic fluctuations in integrated dynamical model Koichi Murase

Hydrodynamic fluctuations in integrated dynamical model Koichi Murase Sophia University Workshop on Partonic and Hadronic Transport Approaches for Relativistic Heavy Ion Collisions May 11, 2019, Hi Chance Science Technology Center, Dalian, China 2019/5/11 1

INTRODUCTION 2019/5/11 2

Fluctuations in heavy-ion collisions Part. X (1/1) – flow coefficients vn, etc. Matter response Eo. S, η, ζ, τR, etc. Additional fluctuations + hydro fluctuations + jets/mini-jets, etc • Initial state fluctuation time • Final observables Relativistic hydrodynamics (～QGP) collision axis 0 – nucleon distribution, – gluon/color fluctuations, etc. 2019/5/11 3

Part. X (1/1) QCD critical point search Search of QCD critical point and 1 st order phase transition Dynamical models for high-energy collisions (Hydro + cascade + …) Needed extensions • Eo. S modeling • critical fluctuations • dynamical initialization • dynamical core-corona separation Schematic phase diagram of QCD [taken from the 2007 NSAC Long Range Plan] 2019/5/11 Dynamical models for lower-energy collisions? 4

Part. X (1/1) Hydrodynamic fluctuations Thermal fluctuations of fluid fields spontaneous field fluctuations of fluid fields such as πμν, Π, etc. at each t and each x πμν, Π x c. f. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (1959) Fluctuation-dissipation relation (FDR) Magnitude of fluctuations δπ, etc. is determined by dissipation η, etc. (and temperature T) 2019/5/11 η≠ 0 δπ ≠ 0 5

Part. X (1/1) Hydrodynamic fluctuations FDR: 〈δπ2〉 ≃ 2 Tη / Δt Δx 3 Water in a glass (Δt, Δx: typical length/time scale) QGP in heavy-ion collisions η ≃ 10 -3 [Pa sec] (≃ 50 s) T ≃ 300 [K] (≃ 3× 10 -9 [Me. V]) Δx ≃ 10 -3 [m] Δt ≃ 10 -1 [sec] η ≃ 0. 1 s ≃ 0. 2 [fm-3] T ≃ 300 [Me. V] Δx ≃ 1 [fm] Δt ≃ 1 [fm] δπ ≃ 3× 10 -8 [Pa] P ≃ 105 [Pa] δπ ≃ 1. 5× 102 [Me. V/fm 3] P ≃ 4× 103 [Me. V/fm 3] δπ / P = 3× 10 -13 δπ / P = 0. 04 Hydrodynamic fluctuations are not negligible in QGP 2019/5/11 6

HYDRODYNAMIC FLUCTUATIONS IN INTEGRATED DYNAMICAL MODELS 2019/5/11 7

Part. X (1/1) Integrated dynamical model 4. Hadronic cascades (JAM) time 5. Observables 3. Particlization at Tsw = 155 Me. V Cooper-Frye formula: f 0 + δf 2. (3+1)-dim. Relativistic Fluctuating Hydrodynamics Eo. S: lattice QCD & HRG, η/s = 1/4π collision axis 0 1. Initial condition MC-KLN, MC-Glauber/modified BGK 2019/5/11 T. Hirano, P. Huovinen, KM, Y. Nara, Prog. Part. Nucl. Phys. 70 (2013) 108; KM, T. Hirano, Nucl. Phys. A 956 (2016) 276 8

Numerical Hydrodynamic Simulation: evolution Evolution conventional 2 nd-order viscous hydro τT ττ x [fm] without HF y [fm] Part. X (1/1) ηs 2 nd-order fluctuating hydro 2019/5/11 x [fm] with HF y [fm] x [fm] ηs 9

Part. X (1/1) p. T-spectra (pions) Au. Au 200 Ge. V For fixed I. C. local flows by hydro fluctuations • High-pt particles increase with HF accelerated by local flows 2019/5/11 10

Part. X (1/1) vn{EP} vs p. T EP: η-sub, |η| = 1. 0 -2. 8 Au. Au 200 Ge. V, MC-KLN 20 -30% • v 2: (IS Fluct. ) + (HF) + (Collision geometry) • v , v : (IS Fluct. ) + (HF) 3 4 2019/5/11 11

Part. X (1/1) vn{2 m} vs p. T v{2}2 ＝ v 2 ＋ σ 2, v{4}2 , v{6}2 ～ v 2 － σ 2, ～ flow fluct. Au. Au 200 Ge. V, MC-KLN 2019/5/11 20 -30% • v{4}2 ＝ v{6}2 ＜ v{2}2 • flow fluctuations σ: larger with HF 12

Part. X (1/1) E-by-E distribution of v 2 v{2}2 ＝ v 2 ＋ σ 2, v{4}2 , v{6}2 ～ v 2 － σ 2, Au. Au 200 Ge. V, MC-KLN 20 -30% Broader distribution of v 2 2019/5/11 13

Part. X (1/1) by A. Sakai, KM, T. Hirano Hydrodynamic fluctuations Longitudinal decorrelation Pb+Pb 2. 76 Te. V MC-Glauber 0 -10%: cutoff λ = 1. 0 10 -50%: cutoff λ = 1. 5 Centrality dependence cannot be reproduced only by hydro fluctuations 2019/5/11 14

Part. X (1/1) by A. Sakai, KM, T. Hirano Hydrodynamic fluctuations + Initial longitudinal fluctuations (PYTHIA×Modified BGK model) Pb+Pb 2. 76 Te. V MC-Glauber Reproduce centrality dependence of data Both fluctuations are important to understand factorization ratio 2019/5/11 15

DISCUSSION 2019/5/11 16

FDR under background evolution Part. X (1/1) When background is non-static / inhomogeneous, FDR has corrections : KM, ar. Xiv: 1904. 11217 Otherwise, Fluctuation theorem (FT) from nonequilibrium statistical mechanics is broken in 2 ndorder fluctuating hydrodynamics T. Hirano, R. Kurita, KM, Nucl. Phys. A 984, 44 (2019), KM, in preparation 2019/5/11 17

Part. X (1/1) Renormalization of Eo. S/viscosity λ-dependence of Eo. S and transport coefficients • e. g. Decomposition of energy density in equilibrium 〈T 00〉 = e = 〈eλ〉 + 〈(eλ+Pλ)uλ 2 + πλ 00〉 internal energy in ordinary sense Global equilibrium in Conventional hydrodynamics 2019/5/11 “internal” energy “kinetic” energy Global equilibrium in Fluctuating hydrodynamics 18

Part. X (1/1) Renormalization of Eo. S/viscosity λ-dependence of Eo. S and transport coefficients • e. g. Decomposition of energy density in equilibrium 〈T 00〉 = e internal energy in ordinary sense = 〈eλ〉 + 〈(eλ+Pλ)uλ 2 + πλ 00〉 “internal” energy “kinetic” energy • Every “macroscopic” quantities (eλ, uλ, etc. ) are redefined for each cutoff λ. • Macroscopic relations such as viscosity, Eo. S, etc. should be modified not to change the bulk properties • Additional terms in hydrodynamic eqs: “long-time tails” 2019/5/11 P. Kovtun, et al. , Phys. Rev. D 68, 025007 (2003); P. Kovtun, et al. , Phys. Rev. D 84, 025006 (2011); P. Kovtun, J. Phys. A 45, 473001 (2012); Y. Akamatsu, et al. , Phys. Rev. C 95, 014909 (2017); Y. Akamatsu, et al. , Phys. Rev. C 97, 024902 (2018) 19

Other renormalization Part. X (1/1) • Cooper-Frye formula : How to sample hadrons from (eλ, uλμ, πλμν, Πλ)? (eλ, uλ) ? • Initialization model (from partons, etc. ) : Width and shape of smearing kernel should match with those in fluctuating hydrodynamics? Larger λ Smaller λ Renormalization? 2019/5/11 larger internal energy eλ smaller initial flow uλ smaller internal energy eλ larger initial flow uλ 20

SUMMARY 2019/5/11 21

Part. X (1/1) Summary • Hydrodynamic fluctuations are thermal fluctuations of fluid fields whose power is determined by FDR • Integrated dynamical model with (3+1)-dim relativistic fluctuating hydrodynamics Non-negligible effects on flow observables, in particular, longitudinal dynamics • Further development for physical consistency: FDR corrections, FT, Renormalization, non-Gaussian statistics, etc. 2019/5/11 22

BACKUP 2019/5/11 23

Fluctuation-dissipation relation Part. X (1/1) log (Probability) Thermal distribution of fluid fields (in 2 nd order hydro) dissipation fluctuations FDR = 2019/5/11 P + Π, πμν Balance of fluctuations and dissipation to maintain thermal distribution 24

FDR in higher-order constitutive eqs Part. X (1/1) Constitutive equation for 1 + (higher order terms) finite-order in derivatives. (D: time derivative) 1 st order term hydrodynamic fluctuations FDR, causality, etc. white noise K. Murase and T. Hirano, ar. Xiv: 1304. 3243; K. Murase, Ph. D. Thesis (The University of Tokyo), Sec. 4. 4 (2015) 2019/5/11 25

Smeared fluctuating hydrodynamics Part. X (1/1) Noise terms with autocorrelations ～ δ(4)(x-x’) Hydrodynamic eqs are non-linear Continuum limit is non-trivial Regularization : cutoff length λ (or cutoff momentum Λ=1/λ) ～ coarse-graining scale > microscopic scale Noise terms w(x) are smeared by, e. g. , Gaussian of width σ = λ 2019/5/11 26

Part. X (1/1) Stochasitic Integrals Structure of 2 nd order fluctuating hydrodynamics noise τ: proper time No difference between Ito/Stratonovich SDE Stratonovich product dτ 0 Ito product 2019/5/11 27

Conservation in curved coordinate Part. X (1/1) Conservation in curved coordinate Tμν=AμαAνβTαβ Conservation for “mixed” tensor Tμν=AμαTαν is conserved 2019/5/11 28

Conservation in curved coordinate Part. X (1/1) time evolution with √(-g)Tτν ⇔ time evolution with √(-g)Tτν with discretized Γμαβ discretized version of Γ analytic expression of Γ Yasuki Tachibana, KM In a conservative scheme , Γμαβ should be discretized along with other derivatives of fields 2019/5/11 29

Conservation and dynamical variables Part. X (1/1) Conserved quantity Dynamical variables (representation of fields) • . This LRF: dissipative currents in local rest frame 2019/5/11 30

Part. X (1/1) Constitutive equation in LRF In ordinary coordinate In the local rest frame Rotation 2019/5/11 31