Physics | Hydrodynamics » Relativistic Hydrodynamics for heavy-ion collisions

Relativistic Hydrodynamics for heavy-ion collisions 严 力 复旦大学现代物理研究所 高能核物理暑期学校@山东大学,青岛, 2023.07 Outline 1. Some theoretical aspects of relativistic hydrodynamics 2. Hydrodynamic modeling of heavy-ion collisions 3. Evolution towards hydro in heavy-ion collisions: emergence of attractor 4. Hydrodynamic fluctuations 2 Evolution stages in heavy-ion collisions Hydrodynamic modeling 3 Collectivity in QGP • Long-range correlation in small and large colliding systems: collectivity 4 Collective flow • Decompose hadron spectrum into Fourier harmonics • Vn is complex, with magnitude and phase. 5 Collective flow and hydrodynamics • Collectivite flow vs. centrality, pT, particle species, correlations, 6 Hydro modeling is unreasonably successful • Hydrodynamics successfully characterizes “all” flow observables in HIC, • New flow observables from v0-vn correlations, . 7 Non-relativistic hydro

• Conservation of mass: equation of continuity • Conservation of momentum: Euler equation • In particular, Navier-Stokes equation 8 Relativistic hydrodynamics • Conservation of energy-momentum and (charge: baryon, strangeness, etc.) • Energy-momentum tensor: • Energy density: • Momentum dentisy: • Energy flux: • Pressure tensor: 9 Formulation of energy-momentum tensor • Fluid velocity: four vector (natural unit with c = 1) 1. Normalization: 2. Projection operator: 10 Formulation of energy-momentum tensor • Fluid local rest frame: • Fluid cell with velocity: • Ideal fluid at rest: • Ideal fluid: Lorentz boost • Dissipative hydro: Below we focus on the (-,+,+,+) convention! 11 Relativistic hydrodynamics • Conservation of energy-momentum and (charge: baryon, strangeness, etc.) • Constitutive equations: • Hydro fields (variables): energy density, pressure, flow velocity, stress tensor. 12 Relativistic hydrodynamics EoM •

Continuity equation, • Euler equation, • Note that and in particular Exercise: using this decomposition of derivative to derive Eular and continuity equations. 13 Theoretical formualtion of hydrodynamics • As an EFT, theory of hydrodynamics emerges w.rt the dominance of long wavelength formulation and small frequency modes -- hydro modes • In practice, long-wavelength limit allows for gradient expansion: up to : nth order viscous hydro. 14 Navier-Stokes hydrodynamics • Leading order in gradient is well understood -- NS hydro 1. This is the most general decomposition at 1st order of gradient of 2. where stands for symmetric, traceless and transverse to 3. There are two unknown scalar parameters (shear and bulk viscosities) 15 Entropy production • Entropy production satisfies, Exercise: prove the result using hydro EoM and the thermodynamic relation de=Tds 1. Ideal fluid corresponds to equilbrium, entropy conserved 2. It is one way to identify the

unkown form of from the condition of the production of entropy. [Landau&Lifshitz “Fluid dynamics”] 3. Positive entropy production applies to generalized hydro formulation with magnetic fields dof. (magnetohydrodynamics), spin dof (spin-hydro), etc 16 Constraints on shear and bulk viscosities 17 Hydro modes in NS hydro (neglect bulk) • Consider static fluid with perturbations and and , • Sound mode propogation, • Shear mode propogation, (large k modes are more damped) 18 Causal Israel-Stwart hydrodynamics • Acausal propogation, [W. Israel, Annl Phys 100 (1976), W Israel and J. Stewart, Annl Phys 118 (1979)] • UV regulation of acausality, • Israed-Stewart hydro: 19 IS hydrodynamics • Stress tensor expands to 2nd order in gradient, with a new transport coefficient: (shear) relaxation time • For a N=4 SYM system, 20 Hydro and non-hydro modes in IS hydro • Sound mode propogation, • Hydro modes go to 0 in long wavelength limit. •

One therefore finds a non-hydro mode. 21 Hydro and non-hydro modes in IS hydro • Shear mode propogation, • Non-hydro mode evolution: hydro pole X X non-hydro pole • These pole structure arises also in retarded Green funcion 22 Hydro and non-hydro modes in other theories • AdS/CFT: strongly coupled system without quasi-particle excitation • Kinetic theory: weakly coupled system with quasi-particle excitation [A. Kurkela et al, EPJC 79, 965(2019), P. Romatschke, EPJC 76:352 (2016)] 23 Hydro with more gradient corrections • BRSSS hydro (2nd order conformal viscous hydro) [Baier et al., JHEP04(2008)100] Note the nonlinear couplings of gradients and new transport coefficients. 24 Hydro with more gradient corrections • Third order hydro [A. Jaiswal, 13053480] • Note the nonlinear couplings of gradients and new transport coefficients. • Formulation becomes much more complicated: n! growth? 25 Brief summary (1) • Ideal relativistic hydro:

local equilibrium + local rest frame + Lorentz boost • 1st ordor vscous hydro: Navier-Stokes hydro, with shear and bulk viscous corrections. Well determined, but acausal • 2nd order viscous hydro: with new transport coefficients (relaxation time, etc.), causal, but not well determined Non-hydro mode introduced • Higher orders can be derived (in principle) order by order. 26 Hydro modeling of system evolution 1. Initial condition from effective models: IP-Glasma, MC-Glauber, 2. Assuming onset of hydro at , after which system expansion is captured by hydrodynamics with respect to a proper equation of state, Hydro EoM (2nd viscous hydro) + Lattice QCD EoS 3. Convert to particles after freeze-out, and particle re-scatterings, resonance decay, etc. (UrQMD) Crucial condition: system (close to) local equilibrium, hydro applicability? 27 Initial condition • Spatial configuration of the nucleas-nucleus collision • Determine energy deposition from AA collisions from

effective models: Glauber, IP-Glasma, Trento, . 28 Initial condition -- e.g, Glauber model • Npart and Ncoll determined by NN inelastic scatterings + nuclear geometry • Assuming energy (entropy) deposition according to Npart and Ncoll 29 Analytical solution to NS hydro: Bjorken flow [J. Bjorken, PRD27 (1983)140 ] • 0+1 D Bjorken expansion: Boost invariant symmetry along space-time rapidity and translational invariant symmetry in transverse plane. • Applies to very early stages of very high energy nuclear collisions 1. dominated by longitudinal expansion 2. expansion in transverse plane negligible 3. boost invariance approximated x 30 Analytical solution to NS hydro: Bjorken flow • Bjorken symmetry fixes flow velocity: , ∞ + = t. s n co = =− ∞ • In Milne coordinates: • Hydro EoM becomes: (continuity eq.) Euler eq. is trival 31 Analytical solution to NS hydro: Bjorken flow • Consider conformal fluid: and • Reparameterize shear

viscosity: 1. AdS/CFT: 2. Kinetic theory of massless particles: 32 Analytical solution to NS hydro: Gubser flow • 1+1 D Gubser expansion: Boost invariant symmetry along space-time rapidity and rotational symmetry in transverse plane. • Applies to ultra-central high energy nuclear collisions 1. isotropic expansion in transverse plane 2. boost invariance approximated [S. Gubser, 10060006, S Gubser and A. Yoram, 10121314 ] x 33 Analytical solution to NS hydro: Gubser flow • Symmtry is obvious in new coordinates: • Flow velocity is fixed by the symmetry, • NS hydro EoM reduces to • For conformal flow, 34 Cooper-Frye freeze-out formalism • Freeze-out hyper surface: hydro description stops to work particles fluid Limitation: heavy particles not applicable. 35 Cooper-Frye freeze-out formalism • Duality between hydro and kinetic theory • Derive Cooper-Frye formula 36 Cooper-Frye freeze-out formalism • Determine freeze out distribution, 

Ideal hydro (local equlibrium):  Viscous hydro: • Resampling w.rt Cooper-Frye freeze-out 37 Brief summary (2) • Hydro modeling for heavy-ion collisions consists of 1. Effective description of initial condition 2. Solving hydro EoM (coupled with EoS) 3. Convert hydro fields to particles (hadrons) 4. In realistic cases, hadron interaction after hydro evolution wrt models, UrQMD, SMASH, . 38 Outline 1. Some theoretical aspects of relativistic hydrodynamics 2. Hydrodynamic modeling of heavy-ion collisions 3. Evolution towards hydro in heavy-ion collisions: emergence of attractor 4. Hydrodynamic fluctuations 39 Evolution stages in heavy-ion collisions What happens here? 40 Onset of hydro from far-from-equilibrium • Pre-equilbrium expansion -- simple picture: hydro starts at later times what happens in small systems, where ? 41 Collective flow in small systems • Multi-particle correlations and collectivite flow in small colliding systems, Hydro even

works in small systems! [CMS, 1904.11519] 42 Theoretical aspects of thermalization [Bottom-up thermalization, A. Muller, D Son, et al] • Thermalization of weakly coupled (QCD): • Thermalization of strongly coupled (SYM): [Chesler, Yaffe, Shuryak] can be small. • Isotropization can be achieved faster: PL = PT + viscous corrections • Hydrodynamization: Hydro emerges as d.of reduced 43 Nature of hydro gradient expansion • (Global) gradient expansion of stress tensor w.rt Knudsen number, • Knudsen number is a dimensionless parameter, 44 Nature of hydro gradient expansion • Kn must be small so that hydro applies, namely, when Kn<<1, system is close to equilibrium, hydro can be truncated • E.g, space shuttle entering atmosphere at 20 km experiences Kn~10-8 • However, it is known that expansion in Kn is asymptotic. [H. Grad, Phys Fluids 6 (1963) 147, S Groot and W Leeuwen and C van Weert, G Denicol and J. Noronha, 160807869, J Blaizot and LY,

170310694 ] 45 Example, solving BRSSS hydro in Bjorken flow • BRSSS hydro becomes coupled ODE in Milne coordinates, with • To solve the equation, one may reparameterize the transport coefficients with the dimensionless parameters determined via underlying theory. Eg, kinetic theory: 46 • The coupled ODE can be recast into nonlinear ODE (*) where we define and • Note that corresponds to ideal fluid. 47 Solution to eq.(*) • Numerical solution. • Semi-analytical solution w.rt expansion in Kn, ideal hydro 1st order viscous hydro 48 Solution to eq.(*) • Hydro gradients summation does not improve solution. • Numerical solution insensitive to initial condition -- attractor 2nd order 1st order 50th order • In the mathematical field of dynamical systems, an attractor is a set of numerical values toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values

remain close even if slightly disturbed. – Wikipedia 49 Attractor from different theories [G. Denicol et al, 170906644] 50 Attractor from different theories Muller-Israel-Stewart Hydro [M. Heller and M Splinski, PRL 115, 072501 (2015)] [G. Denicol et al, 180404771] attractor in Gubser flow (attractor in Hubble flow [Z.Du et al 210412534]) 51 Attractor from different theories [P. Romatschke, 107408699 ] 52 Hydro gradient expansion in Kn is aymptotic • One can solve the expansion coefficient order by order, that • S and beta depend on transport coefficients. • n! leads to zero radius of convergence, similar to perturbative expansion in QFT. 53 Borel resum of asymptotic series For asymptotic series, Borel resummation technique can be applied. 1. Borel transform of the hydro gradient expansion leads to a convergent series, 2. Borel resum (Laplace transform) 3. Borel resum effective represents the asymptotic series, if it is Borel summable, i.e, there

is no singularity on R+ of 54 Hydro gradient is not Borel summable • There exists singularity of the Borel transform of hydro gradient expansion, [Basar and Dunne, PRD92,125011 ] Pade approximation of gives a branch cut on the real axis. The leading pole corresponds to S, i.e, the radis of convergence of Standard procedure to determine radius of convergence of a series, e.g, exp of LQCD wrt 55 Imaginary ambiguity from Borel resum • To avoid branch cut in the resum, one needs analytic continuation to complex plane, with a complex ambiguity arises w.rt integration contour, accordingly, Borel sum leads to 56 Extends to trans-series and resurgence • Borel resum must give rise to real solution, which implies trans-series solution, • Each is an asymptotic series, to be resummed. • Especially, • The complex constant is to be fixed via i.c and resurgence relations: very commonly, this has to be done numerically. 57 Brief summary (3a) • Hydro gradient

expansion w.rt Kn diverges • Borel resum applies to the divergent hydro gradient expansion. • After resum, hydro series expansion extends to trans-series, • Note that the factor emerges naturally along with i.c: initial condition dependent evolution decays! -- attractor behavior • This is how attractor emerges mathematically. [M. Heller and M Splinski, PRL 115, 072501 (2015), J. Blaizot and LY, 2006.08815] 58 Observed attractor [A.Kurkela et al, PRL124 (102301) ] • IS hydro power-law decay • RTA kinetic theory Exponential decay power-law decay Exponential decay 59 Attractor and fixed point analysis hydro fixed point at late times free streaming fixed point at early times 60 Attractor and fixed point analysis hydro fixed point at late times free streaming fixed point at early times 61 Fixed points in the pre-equilibrium (e.g, IS hdyro) • EoM of IS hydro w.rt Bjorken flow can be written in a matrix form, free streaming collision where, • In

late time limit, EoM reduces to hdyro fixed point solution: 62 Fixed points in the pre-equilibrium (e.g, IS hydro) • At early times, eigenvalues of the free-streaming matrix detemine freestreaming fixed points, which implies decay of energy density in free streaming, [J. Blaizot and LY, 171203856] 63 Fixed points in the pre-equilibrium (e.g, IS hydro) • Alternatively, one may derive the beta function defined with respect to the evolution of g(w), fixed points can be solved accordingly via [J. Blaizot and LY, 190408677] 64 Slow mode evolution and attractor (e.g, IS hydro) • Solving the coupled EoM of IS hydro effectively via the time-dependent eigenvalue problem of matrix, in analogy to QM, arbitrary evolution should be dominated by the slowest mode, [J. Brewer et al, 191000021] 65 Slow mode evolution and attractor (e.g, IS hydro) • Gap of eigenvalues: explains early-time power-law decay and late-time exponential decay. 66 Onset of hydro from

far-from-equilbrium • Pre-equilbrium expansion -- attractor picture: hydro starts much earlier 67 Brief summary (3b) • Hydro attractor solution can be as well understood in terms of 1. Fixed point analysis: Free-streaming fixed point corresponds to 1D expansion. Hydro fixed point corresponds to collisions among excitations. Therefore, emergence of hydro attractor due to competing effects of expasion and collision. 2. Slow mode (adiabatic) evolution: Slow mode dominates system evolution throughout all stages, if the gap is perserved. In particular, the fast mode plays the role of non-hydro mode, which at late times, behavior as 68 Hydrodynamic fluctuations in heavy-ion collisions 69 Fluctuating hydrodynamics • Hydro fluctuation: thermal fluctuation in fluids, universal in nature. • Classical hydro (w/o fluc.) ==> hydro with thermal fluctuations where characterizes random noise, and perturbations accordingly. are induced thermal • Here classical hydro

consists of classical hydro fields -- hydro fields without correction from thermal fluctuations, (what we have discussed so far) 70 Fluctuating hydrodynamics • Fluctuation-dissipation relation, (define {.} emsemble average) [Landan and Lifshtz, “Fluid dynamics”] • Similarly for charge current: ( charge conductivity) [J. Kapusta, B Muller and M Stephanov, PRC85, 054906 ] 71 Formulation of fluctuating hydro (conformal) • Recall that entropy production rate in viscous hydro satisfies: where for Navier-Stokes hydro, and the projection operator 72 Formulation of fluctuating hydro (conformal) • EoM of xa with random thermal noise, (e.g Langevin equation) • Auto correlation of thermal noise: fluctuation-dissipation relation. 1. necessary for system to achieve equilibrium 2. time scale of noise much smaller than time scale of background system 73 Formulation of fluctuating hydro (conformal) • For viscous hydro, drag can be determined by identifying and

more importantly, so that 74 Properties of hydrodynamic fluctuations • Qualitatively different from quantum fluctuations: initial state fluctuations e.g: event-by-event fluctuations of initial energy density X ( , ⊥) • Hydro fluctuations are strong in dissipative and small systems. • Hydro fluctuations are substantial near the critical region. 75 Hydrodynamic fluctuations in heavy-ion collisions • Numerical simulations of fluctuating hydrodynamics. • Solving stochastic partial differential equations. • Numerical realization of Dirac delta function on grid. • Needs average for obserbles. [B. Schenke et al, 200500621, see also C Young, PRC89, 024913, A. Sakai et al, 2111.08963, A De et al, 220302134] 76 Evolution of n-point correlation: EFT • Fluctuating hydro implies deterministic equation for n-point correlations. • Tree-level: classical hydro EoM • One-point level: { } = 0 • At two-point level (one-loop): { } ==> hydro-kinetic equation,

where • N-point cumulants. [XAn et al, 200910742] 77 Renormalization from hydro fluctuations • Taking into account of hydro fluctuations, averaged emergy-momentum tensor receives corrections from n-point correlations. • For the Bjorken flow, for instance, for energy density, one has loop contains to k-integral 78 Renormalization from hydro fluctuations • Renormalized shear viscosity, [P. Kovtun, G Moore and P Romatschke, PRD84, 025006(2016)] • Renormalized pressure, [Y.Aakamatsu et al, 160607742] • Long-time tails 79 Brief summary (4) • Hydro fluctuations: 1. Formulation based on Landau-Lifshitz approach Flutuating hydro can be solved via stochastic ODE. 2. Formulatoin with respect to n-point function as EFT: hydro-kinetic Fluctuating hydro can be solved in deterministic way. 80 QFT, EFT Condensed matter hydro Heavy-Ion Cosmology You are very welcome to join the journey to the future of hydrodynamics! 81