1. Koichi Murase, The University of Tokyo, Sophia University Relativistic fluctuating hydrodynamics 2016/2/18, ATHIC 2016, New Delhi, India 2016/2/181

2. PartX (1/1) Fluctuating hydrodynamics Subject: Thermal fluctuations of fluids Two keywords: Hydrodynamic fluctuations (HF) = thermal fluctuations of fluid fields Fluctuating hydrodynamics = dissipative hydrodynamics with hydrodynamic fluctuations as noise terms  Hydrodynamic eqs. become Stochastic Partial Differential Equations (SPDEs) like the Langevin equation 2016/2/182 c.f. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (1959)

3. PartX (1/1) Final observables – flow harmonics v n, etc. Initial-state fluctuations –nucleon distribution –quantum fluctuations Fluctuations in heavy-ion collisions 2016/2/183 Response of Matter EoS, η, ζ, τ R, λ, … 0 collision axis time

4. PartX (1/1) Final observables – flow harmonics v n, etc. Initial state fluctuation –nucleon distribution –quantum fluctuations Fluctuations in heavy-ion collisions 2016/2/184 Other fluctuations + hydro fluctuations + jets/mini-jets + critical phenomena + … 0 collision axis time Relativistic hydrodynamics ( ～ QGP)

5. PartX (1/1) Hydrodynamics Hydro = Basically, conservation laws “Hydrodynamics” works iff… fluxes T ij ( ～ pressure, stress, etc.) are uniquely determined by conserved densities T 0μ ( ～ e, u μ ). 2016/1/215 e.g. e.g. EoS & Constitutive eqs. near equilibrium Pressure Probability ～ V -1/2 thermodynamic limit V  ∞ w/ fixed E/V unique! P(e)

6. PartX (1/1) Hydrodynamic Fluctuations HF: deviation from the equilibrium value 2016/2/186 ensemble/event averaged thermal fluctuations at each spacetime x e.g. V: 3+1 dim. volume decrease anisotropy HF are important in e-by-e description of HIC Balance Fluctuation-Dissipation Relation FDR increase anisotropy HF

7. HF in causal dissipative hydrodynamics 2016/2/187

8. PartX (1/1) Causal hydrodynamics 1 st -order constitutive equation (Navier-Stokes) – instability, and acausal modes 2 nd -order constitutive equation 2016/2/188 How is the FDR modified? … J. I. Kapusta, et.al., PRC85, 054906 (2012) + δπ μν FDR

9. PartX (1/1) Constitutive equation (CE) Constitutive eq. in linear response regime Integral form: Differential form: 2015/10/269 1 st order term memory function e.g. Γ + (higher order terms) 1 st order finite-order in derivatives

10. PartX (1/1) Fluctuation-dissipation relation Hydro fluctuations appearing in constitutive eq. Fluctuation-dissipation relation (FDR) G is not delta function in general: “colored noise” 2016/2/1810 hydrodynamic fluctuations

11. PartX (1/1) In differential form Fluctuation in the differential form FDR 2014/02/0411 1 st order term hydro fluctuation colored noise ? white/colored? Properties of memory function G(x)

12. PartX (1/1) General properties of memory function G A)Retarded G(x) = 0 if t<0 B)Relaxing in a finite time G(x)  0 as t  ∞ C)Timelike G(x) = 0 if x 2 <0 D)Positive-semidefinite (from FDR) 〈 δΓ x δΓ x’ 〉 = TG’ xx’ : variance-covariance matrix of δΓ 2014/02/0412

13. PartX (1/1) In differential form Fluctuations in the differential form are white 2014/02/0413 L is expanded to finite order in derivatives retarded: G=0 if t<0 relaxation: G  0 as t  ∞ timelike: G=0 if x 2 <0 Covariance matrix is positive- semidefinite G(τ)G(τ) τ prob. δΓδΓ

14. Event-by-event calculations 2016/2/1814

15. PartX (1/1) Setup: Target Collision system Au+Au, √s NN = 200 GeV Fluctuations Initial-state fluctuations from nucleon positions Hydrodynamic fluctuations  Comparison between the effects of both fluctuations 2016/2/1815 Au

16. PartX (1/1) Setup: IS Fluctuations and HF Initial condition Centrality: Minimum bias Model: MC-KLN (CGC) (3+1)-dim Relativistic Fluctuating Hydrodynamics: 2016/2/1816 x-y planeη s -x plane Typeη/sη/sHF (Gaussian noise)HydroCascades Viscous1/4πnone100 k events 10 M (100k ☓ 100) Fluctuating1/4πλ = 1.0 (fm)100 k events 10 M (100k ☓ 100) λ : HF cutoff length scale (Gaussian-smearing width)

17. PartX (1/1) Numerical Simulation: Evolution 2016/2/1817 Hydrodynamic evolution ηsηs x [fm] ηsηs y [fm] x [fm] without HF with HF conventional 2 nd -order viscous hydro 2 nd -order fluctuating hydro τT ττ

18. PartX (1/1) v n {EP} vs p T 2016/2/1818 EP: η-sub, |η| = 1.0-2.8 0-5% HF increase anisotropies larger effects in a higher order  HF ∝ V -1/2

19. Cutoff scale of HF 2016/2/1819

20. PartX (1/1) Cutoff dependence of “macroscopic” quantities cutoff λ: lower bound of considered HF  coarse-graining scale of the system  λ-dependence of EoS and transport coefficients e.g. Decomposition of energy density in equilibrium 2016/2/1820 “internal” energy “kinetic” energy 〈 T 00 〉 = e = 〈 e λ 〉 + 〈 (e λ +P λ )u λ 2 + π λ 00 〉 internal energy in ordinary sense Global equilibrium in Conventional hydrodynamics Global equilibrium in Fluctuating hydrodynamics

21. PartX (1/1) Cutoff dependence of “macroscopic” quantities cutoff λ: lower bound of considered HF  coarse-graining scale of the system  λ-dependence of EoS and transport coefficients e.g. Decomposition of energy density in equilibrium  Every “macroscopic” quantities (e λ, u λ, etc.) are redefined for each cutoff λ.  Macroscopic relations such as viscosity, EoS, etc. should be renormalized not to change the global equilibrium properties (Future tasks) 2016/2/1821 “internal” energy “kinetic” energy 〈 T 00 〉 = e = 〈 e λ 〉 + 〈 (e λ +P λ )u λ 2 + π λ 00 〉 internal energy in ordinary sense c.f. P. Kovtun, J. Phys. A: Math. Theor. 45 (2012) 473001

22. Summary 2016/2/1822

23. PartX (1/1) Summary Hydrodynamic fluctuations (HF) ＝ thermal fluctuations of hydrodynamics Fluctuating hydrodynamics ＝ dissipative hydrodynamics with HF as noise terms Properties of HF in causal hydrodynamics: – Colored/white in integral/differential form Effects on observables from event-by-event simulations HF is Important in extracting the transport properties Cutoff scale of HF: – Coarse-graining scale is needed in non-linear eqs. – ”Renomalization” of η(λ), EoS, f(p, λ), etc. by cutoff λ. 2016/2/1823

24. Backup 2014/08/0624

25. PartX (1/1) White noise in the differential form 2014/02/0425 2T Re G(ω) ω 〈 ξ(τ)ξ(0) 〉 τ always delta fn. ～ δ(τ) 2T Re G(ω) ω fluctuation of dissipative currents δΠ : colored noise 〈 δΠ(τ)δΠ(0) 〉 τ always exponential ～ e － τ/τ r (k) noise term in the differential form ξ=LδΠ : white noise

26. PartX (1/1) Integrated Dynamical Model 2016/2/1826 0 collision axis time 2. (3+1)-dim. Relativistic Fluctuating Hydrodynamics EoS: lattice QCD&HRG, η/s = 1/4π 1. Initial condition MC-KLN 5. Analysis of hadron distribution 3. Particlization at T sw = 155 MeV Cooper-Frye formula: f + δf 4. Hadronic cascades (JAM) Updated version of T. Hirano, P. Huovinen, KM, Y. Nara, PPNP 70, 108 (2012) [arXiv:1204.5814]

27. PartX (1/1) v n {EP} vs p T 2016/2/1827 EP: η-sub, |η| = 1.0-2.8 Central: (IS Fluct.) + (HF) Non-central: (IS Fluct.) + (HF) + (Collision geometry) 20-30%

28. PartX (1/1) v n {2m} vs p T 2016/2/1828 v{4} 2 ＝ v{6} 2 ＞ v{2} 2 flow fluctuations σ: larger with HF ～ flow fluct. v{2} 2 ＝ v 2 ＋ σ 2, v{4} 2, v{6} 2 ～ v 2 － σ 2, 20-30%

29. PartX (1/1) E-by-E distribution of v 2 2016/2/1829 v{2} 2 ＝ v 2 ＋ σ 2, v{4} 2, v{6} 2 ～ v 2 － σ 2, Broader distribution of v2 20-30%

30. PartX (1/1) Event-plane fluctuations by HF 2016/2/1830 Event-plane decorrelation by HF 20-30% CMS, arXiv:1503.01692 J. Jia, and P. Huo, Phys. Rev. C 90, 034905 (2014),

31. Setup A: HF only 2015/7/2731

32. PartX (1/1) Setup A: HF only Initial condition b = 6.45 fm ( ～ Centrality 20%) Averaged MC-KLN (CGC) (3+1)-dim Relativistic Fluctuating Hydrodynamics: 2015/7/2732 x-y planeη s -x plane Typeη/sη/sHFHydroCascades Ideal0none1 event10 4 events Viscous1/4πnone1 event10 4 events Fluctuating1/4πσ=0.8, 1.0, 1.2 (fm) ＊ 10 4 events ☓ 3 σ: HF cutoff length scale

33. PartX (1/1) Numerical Simulation: Evolution 2015/7/2233 A: Hydrodynamic evolution ηsηs x [fm] ηsηs y [fm] x [fm] without HF with HF conventional 2 nd -order viscous hydro 2 nd -order fluctuating hydro τT ττ

34. PartX (1/1) A: dN ch /dη Increase  entropy production by HF Larger effect with a shorter cutoff length σ 2015/7/2234

35. PartX (1/1) A: p T -spectra (pions) High-pt particles increase with HF  accelerated by local flows 2015/7/2235 local flows by HF

36. PartX (1/1) A: Elliptic flow v 2 {2} decreased by viscosity, unchanged by HF  local flows do not change the global flow profile 2015/7/2236 local flows by HF

37. PartX (1/1) A: Elliptic flow v 2 {2}(η) Decrease with viscosity, increase with HF  increase of high-p T particles 2015/7/2237 v2(p T ) in the last slide High-p T particles v2v2 η

38. PartX (1/1) B: p T -spectra (pions) 2015/7/2238 Points: PHENIX PRC69 (2004) 034909 Black lines: ideal hydrodynamics Blue lines: viscous hydrodynamics Red lines: fluctuating hydrodynamics 0-4%, 5-10%, 10-15%, 15-20%, 20-30%, 30-40%, 40-50%, 50-60%, 60-70%, 70-80% from top to bottom (multiplied by 10 4 -10 -5 ) Increase of high-p T pions: larger in peripheral collisions  larger thermal fluctuations in smaller systems Correction of distribution in Cooper-Frye formula by HF?

39. Setup B: IS fluctuations + HF 2015/7/2739

40. PartX (1/1) Elliptic flow v 2 (η) v2(fluctuating) > v2(ideal) in central collisions Central: (IS Fluct.) + (HF) Non-central: (IS Fluct.) + (HF) + (Collision geometry) Same order with IS fluctuations 2015/9/2940 ηpηp ηpηp 10-15% 0-5%

41. PartX (1/1) v 2 {2}(p T ), v 3 {2}(p T ) v 2 increase with HF in central collisions Similar behavior for v 3 2015/9/2941 v2v2 v3v3

42. PartX (1/1) v 2 {2} vs p T Calculations: η/s = 1/4pi 2015/9/2942