Green functions of T during weak coupling hydrodynamization Aleksi - - PowerPoint PPT Presentation

Green functions of T µν during weak coupling hydrodynamization

Aleksi Kurkela

AK, Mazeliauskas, Paquet, Schlichting, Teaney, in progress Keegan, AK, Mazeliauskas, Teaney JHEP 1608 (2016) 171 AK, Zhu PRL 115 (2015) 18, 182301; AK, Lu PRL 113 (2014) 18, 182301 AK, Moore JHEP 1111 (2011) 120 AK, Moore JHEP 1112 (2011) 044

Oxford, March 2017

Motivation?

Pre thermal plasma Locally thermalised plasma Lorentz contracted nuclei

Soft physics of HIC described by relativistic hydrodynamics ∂µT µν = 0 Gradient expansion around local thermal equilibrium T µν = T µν

• eq. − η2∇<µuν> + . . .

Motivation?

Anisotropy: PL/PT

Time: τ

+1 H y d r

• τi

At early times pre-equilibrium evolution Hydro simulations start at intialization time τi

Motivation:

Anisotropy: PL/PT

Time: τ

+1 H y d r

• τi

P r e

• e

q . If prethermal evolution converges smoothly to hydro, independence of unphysical τi

Motivation:

Anisotropy: PL/PT

Time: τ

+1 H y d r

• τi

P r e

• e

q . If prethermal evolution converges smoothly to hydro, independence of unphysical τi In most current pheno: either free streaming, or nothing at all

Motivation:

In AA collisions: pre-equilibrium evolution ∼ 10% of the evolution

Pre-equilibrium evolution major uncertainty affects η/s, etc

Motivation:

In AA collisions: pre-equilibrium evolution ∼ 10% of the evolution

Pre-equilibrium evolution major uncertainty affects η/s, etc

Motivation:

In AA collisions: pre-equilibrium evolution ∼ 10% of the evolution

Pre-equilibrium evolution major uncertainty affects η/s, etc

In pA collisions: currently no quantitative description

even if the system becomes hydrodynamical, ”pre-equilibrium” evolution O(1) of the evolution

Motivation:

In AA collisions: pre-equilibrium evolution ∼ 10% of the evolution

Pre-equilibrium evolution major uncertainty affects η/s, etc

In pA collisions: currently no quantitative description

even if the system becomes hydrodynamical, ”pre-equilibrium” evolution O(1) of the evolution

pp collisions: ?????

Hydrodynamization in weak coupling

Anisotropy: PT / PL

Occupancy: f

Overoccupied Underoccupied Thermal Initial f~1 f~α f~α−1

Color Glass Condensate: Initial condition overoccupied

McLerran, Venugopalan PRD49 (1994) , PRD49 (1994); Gelis et. al Int.J.Mod.Phys. E16 (2007), Ann.Rev.Nucl.Part.Sci. 60 (2010)

f(Qs) ∼ 1/αs, Qs ∼ 2GeV Expansion makes system underoccupied before thermalizing

Baier et al PLB502 (2001)

f(Qs) ≪ 1

Hydrodynamization in weak coupling

Anisotropy: PT / PL

Occupancy: f

Thermal Kinetic theory Classical YM Initial f~1 f~α f~α−1 Both

Degrees of freedom:

f ≫ 1: Classical Yang-Mills theory (CYM) f ≪ 1/αs: (Semi-)classical particles, Eff. Kinetic Theory (EKT)

Hydrodynamization in weak coupling

Anisotropy: PT / PL

Occupancy: f

Thermal Kinetic theory Classical YM Initial f~1 f~α f~α−1 Both

Transmutation of fields to particles: Field-particle duality

Son, Mueller PLB582 (2004) 279-287; Jeon PRC72 (2005) 014907; Mathieu et al EPJ. C74 (2014) 2873; AK et al PRD89 (2014) 7, 074036

1 ≪ f ≪ 1/αs ”Bottom-up thermalization” of underoccupied system

Strategy at weak coupling

Anisotropy: PL/PT

Time: τ

+1 H y d r

• τi~1fm/c

CYM EKT

τEKT~0.1 fm/c

Strategy: Switch from CYM to EKT at τEKT , 1 ≪ f ≪ 1/αs From EKT to hydro at τi, PL/PT ∼ 1

Early times 0 < Qsτ 1: classical evolution

Time: Qsτ

• 1

PT/ε PL/ε +1

Epelbaum & Gelis, PRL. 111 (2013) 23230

Melting of the coherent boost invariant CGC fields

Initial condition from CGC: MV-model, JIMWLK

After τ ∼ 1/Qs, fields decohere, PL > 0

Later times Qsτ > 1: classical evolution

Berges et al. Phys.Rev. D89 (2014) 7, 074011

Anisotropy: PT / PL

Occupancy: f

Overoccupied Underoccupied Thermal Initial f~1 f~α f~α−1

Numerical demonstration of overoccupied part of the diagram Classical theory never thermalizes or isotropizes Before f ∼ 1, must switch to kinetic theory

Outline

Effective kinetic theory Hydrodynamization and thermalization at weak coupling in effective kinetic theory Apples to apples comparison of weak and strong coupling hydrodynamization Green functions of T µν in during hydrodynamization and phenomenological application to HIC

Effective kinetic theory of Arnold, Moore, Yaffe

JHEP 0301 (2003) 030

Soft and collinear divergences lead to nontrivial matrix elements

soft: screening, Hard-loop; collinear: LPM, ladder resum

= Re

                      ∗                      

No free parameters; LO accurate in the αs → 0, αsf → 0 limit, for ∆t ∼ ω−1 > Typical scattering time ∼ 1/(α2T) , Caveat: in anistropic systems screening complicated. Here with isotropic screening. Also no fermions here

plasma instabilities, . . .

Why kinetic theory needed?

LO spectral function in unresummed pert-theory ρφ2φ2(ω, k) ∼

• d4k

(2π)4

• 1 + n(−k0 + ω)
• (1+n(k0))ρ(k, −k0+ω)ρ(k, k0)

K

ω

Jeon PRD47 (1993)

Why kinetic theory needed?

LO spectral function in unresummed pert-theory ρφ2φ2(ω, k) ∼

• d4k

(2π)4

• 1 + n(−k0 + ω)
• (1+n(k0))ρ(k, −k0+ω)ρ(k, k0)

Free spectral function ρfree = sign(k0)2πδ(−(k0)2 + k2 + m2) No overlap if ω < 2m

• Ek

Ek+ω

k0 = −

• k2 + m2,

k0 =

• k2 + m2 + ω

Jeon PRD47 (1993)

Why kinetic theory needed?

In interactive theory ρ(k0, k) ≈ 4k0Γk

• (k0)2 − E2

k

2 + 4(k0Γk)2 Smooth limit lim

ω→0

ρ(0, ω) ω ∼

• d4k

(2π)4 n(Ek)(1 + n(Ek)) 1 E2

kΓk

• Ek Ek+ω

In weak coupling Γk ∼ α2T coupling in the denominator → resummation needed

Why kinetic theory needed?

ω

1/α2Τ

Lifetime: Frequency of scattering: 1/α2Τ

Physical reason: Both lines long lived (α2T)−1, of the order or scattering time Diagrammatic resummation (in λφ4 )

Jeon PRD52 (1995)

Interpretation of the diagrammatic resummation in terms of effective kinetic theory

Jeon, Yaffe PRD53 (1996)

Generalization to gauge theories through power counting

Arnold et al. JHEP 0301 (2003) 030

Outline

Effective kinetic theory Hydrodynamization and thermalization at weak coupling in effective kinetic theory Apples to apples comparison of weak and strong coupling hydrodynamization Green functions of T µν in during hydrodynamization and phenomenological application to HIC

Anisotropy: PT / PL

Occupancy: f

Overoccupied Underoccupied Thermal Initial f~1 f~α f~α−1

Isotropic overoccupied: Transmutation of d.o.f’s Isotropic underoccupied: Radiative break-up Effect of longitudinal expansion: Hydrodynamization

Overoccupied cascade

AK, Moore JHEP 1112 (2011) 044

What happens if you have too many soft gluons, f ∼ 1/αs.

ln(p) ln(f) Thermal f ~ 1 Initial condition (eβp-1)-1

1/α

Q

Overoccupied cascade

AK, Moore JHEP 1112 (2011) 044

What happens if you have too many soft gluons, f ∼ 1/αs. No longitudinal expansion.

ln(p) ln(f) Thermal f ~ 1 Initial condition (eβp-1)-1 Self-similar cascade pmax ~ t1/7 f(pmax)~ t-4/7

1/α

Q

τinit ∼ [σn(1 + f)]−1 ∼ Q T 7 1 α2

sT ≪

1 α2

sT ∼ τthem.

c.f. Bokuslawski’s talk

Overoccupied cascade

AK, Lu, Moore, PRD89 (2014) 7, 074036

0.01 0.1 1 10 Momentum p

~ = p/Q (Qt)

• 1/7

1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10 100 1000 Rescaled occupancy: f

~= λ f (Qt) 4/7

Lattice (continuum extrap.) Lattice (large-volume) Lattice and Kinetic Thy. Compared

Form of cascade from classical lattice simulation, 1 ≪ f 1/αs

Large-volume: (Qa)=0.2, (QL)=51.2, Cont. extr.: down to (Qa)=0.1, (QL)=25.6, Qt=2000, ˜ m = 0.08

Overoccupied cascade

AK, Lu, Moore, PRD89 (2014) 7, 074036

0.01 0.1 1 10 Momentum p

~ = p/Q (Qt)

• 1/7

1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1 10 100 1000 Rescaled occupancy: f

~= λ f (Qt) 4/7

Kinetic thy (discrete-p) Lattice (continuum extrap.) Lattice (large-volume) Lattice and Kinetic Thy. Compared

Same system, very different degrees of freedom 1 f ≪ 1/αs

Numerical demonstration of field-particle duality

Anisotropy: PT / PL

Occupancy: f

Overoccupied Underoccupied Thermal Initial f~1 f~α f~α−1

Isotropic overoccupied: Transmutation of d.o.f’s Isotropic underoccupied: Bottom-up thermalization Effect of longitudinal expansion: Hydrodynamization

Bottom-up thermalization

Hard particles emit soft radiation: creation of a soft thermal bath

Soft bath starts to dominate dynamics (screening, scattering, etc.)

Hard particles undergo radiative break-up

System thermalizes in a time scale it takes to quench a jet of momentum Q

AK, Moore 1107.5050

teq ∼ Q T 1/2 1 λT 2

Q t (Q)

split

t (Q/2)

split

t (Q/4) . . . . . . . . . . . . T Q/4 Q/2

split

Bottom-up thermalization

AK, Lu, PRL 113 (2014) 18, 182301

0.1 1 10 100 Momentum: p/T 1e-12 1e-10 1e-08 1e-06 0.0001 0.01 1 Occupancy: f Initial condition Q/T = 404.9 λ = 0.1

Start with an underoccupied initial condition p ∼ Q after a very short time, an IR bath is created

(1↔ 2–processes)

Bottom-up thermalization

AK, Lu, PRL 113 (2014) 18, 182301

0.1 1 10 100 Momentum: p/T 1e-12 1e-10 1e-08 1e-06 0.0001 0.01 1 Occupancy: f λ

2Tt(T/Q) 1/2=100

50 25 12.5 6.25 Q/T = 404.9 λ = 0.1 teq, 147

More energy flows to the IR, temperature increases, “Bottom-up” When “bottom” reaches final T, “up” is quenched

AK, Moore JHEP 1112 (2011) 044

teq ∼ (Q/T)1/2 1 α2

sT

Bottom-up thermalization

AK, Lu, PRL 113 (2014) 18, 182301

0.1 1 10 100 Momentum: p/T 1e-12 1e-10 1e-08 1e-06 0.0001 0.01 1 Occupancy: f λ

2Tt(T/Q) 1/2= 200

250 300 Q/T = 404.9 λ = 0.1 teq, 147

Hardest scales reach equilibrium last.

Anisotropy: PT / PL

Occupancy: f

Overoccupied Underoccupied Thermal Initial f~1 f~α f~α−1

Isotropic overoccupied: Transmutation of d.o.f’s Isotropic underoccupied: Radiative break-up Application to HIC: effect of longitudinal expansion

Route to equilibrium in EKT

AK, Zhu, PRL 115 (2015) 18, 182301

1 10 100 Rescaled occupancy: <pαsf>/<p> 1 10 100 1000 Anisotropy: PT/PL αs=0 αs=0.03 αs=0.15 αs=0.3 Classical YM Bottom-Up αs=0.015 Realistic coupling

Anisotropy: PT / PL

Occupancy: f

Overoccupied Underoccupied Thermal Initial f~1 f~α f~α−1

Initial condition (f ∼ 1/αs) from classical field thy calculation

Lappi PLB703 (2011) 325-330

In the classical limit (αs → 0, αsf fixed), no thermalization At small values of couplings, clear Bottom-Up behaviour Features become less defined as αs grows

Bottom-up thermalization

p2f(p⊥, pz) pz −pz p⊥

1 10 100 0.1 1 10 anisotropy

• ccupancy

Bottom-up thermalization

p2f(p⊥, pz) pz −pz p⊥

1 10 100 0.1 1 10 anisotropy

• ccupancy

Bottom-up thermalization

p2f(p⊥, pz) pz −pz p⊥

1 10 100 0.1 1 10 anisotropy

• ccupancy

Bottom-up thermalization

p2f(p⊥, pz) pz −pz p⊥

1 10 100 0.1 1 10 anisotropy

• ccupancy

Smooth appreach to hydrodynamics

0.5 1 1.5 2 2.5 3 3.5 4 1 10 20 30 40 50 (e + T zz)(τQs)4/3/e(τo) Qsτ kinetic theory 2nd order free streaming

AK, Zhu, PRL 115 (2015) 18, 182301

Kinetic theory smoothly and automatically goes to hydrodynamics

Outline

Effective kinetic theory Hydrodynamization and thermalization at weak coupling in effective kinetic theory Apples to apples comparison of weak and strong coupling hydrodynamization Green functions of T µν in during hydrodynamization and phenomenological application to HIC

Weak and strong coupling hydrodynamization compared

Question: To what extent are the strong coupling and weak coupling hydrodynamizations similar or different? Challenge: How to setup similar initial condition in theories with different microscopic physics?

Weak and strong coupling hydrodynamization compared

Start with thermal equilibrium Ti perform same macroscopic deformation on both ds2 = −dt2 + dx2 + dy2 + g(t)dL2

g(t → −∞) = 1, Minkowski g(t → ∞) = t2, Milne

Weak and strong coupling hydrodynamization compared

0.01 0.1 1 5 10 15 20 25 30 ε(t)/εinitial t Ti Energy density Ideal hydro λ= ∞ λ=10 λ= 5 Free-streaming

Keegan et al JHEP 1604 (2016)

λ = ∞: N=4 SUSY, λ = 5, 10: pure gauge

Weak and strong coupling hydrodynamization compared

0.2 0.4 0.6 0.8 1 1.2 1.4 10 20 30 40 50 60 PL/PT t Ti Pressure anisotropy λ=∞ λ=10 λ= 5 NS BRSSS

Large quantitative difference due to different η/s

Weak and strong coupling hydrodynamization compared

0.2 0.4 0.6 0.8 1 1.2 10 20 30 40 50 60 70 80 PL/PT (t Ti)2/3/(η/\s)(χ)1/3 λ=Inf λ=10 λ=5 λ=3 λ=1 1st order hydro

Keegan et al JHEP 1604 (2016)

PL PT = 1 − 8 3 (η/s) (tTi)2/3χ1/3 , χ = Seq(t → ∞) Seq(t → −∞) All hydrodynamize at very large anisotropy!

Spectrum of non-hydro modes in weak coupling

x x x x x x x x x x x x x x k

• k

x

G00,00(w, k) = −6(ǫ + P)

• 1 − ω

4k log ω − k + iǫ ω + k + iǫ

• = −6(ǫ + P)

    1 − ω 2 dΩ 4π 1 ω − v · k + iǫ

• δ(x−vt)

    

Spectrum of non-hydro modes in weak coupling

x x x x x x x x x x x x x x x x k

• k

x x x

G00,00(w, k) = −6(ǫ + P)

• 1 − ω

4k log ω − k + iǫ ω + k + iǫ

• = −6(ǫ + P)

    1 − ω 2 dΩ 4π 1 ω − v · k + iǫ

• δ(x−vt)

    

Spectrum of non-hydro modes in weak coupling

x x x x x x x x x x x x x x x x k

• k

τπ

−1

x i) x x

τπ

pν∂νf(t, x, p) = p0 τΠ (f − feq) RTA δ′(x − vt) → δ′(x − vt) exp(−τ/τπ) log ω − k ω + k

• → log

ω − k + i/τπ ω + k + i/τπ

• Bit more complicated than that... Romatschke, EPJC76 (2016)

Spectrum of non-hydro modes in weak coupling

x x x x x x x x x x x x x x x x k

• k

τπ

−1

x x

ωs(k)

x x

Enter hydro pole: k ≪ 1/τπ ωs(k) = ±csk − i 2 τπ 5

• η/(P+ǫ)

k2 + . . .

Spectrum of non-hydro modes compared

x x x x x x x x x x x x x x x x k

• k

τπ

−1

x x

ωs(k)

x x x x x x x k

• k

τπ

−1

x x

ωs(k)

x x x x x x x

No singularity at the complex infinity → cut may be deformed log ω − k + i/τπ ω + k + i/τπ

Hydrodynamization through decay of non-hydro modes

In both, holography and kinetic theory the hydrodynamical gradient expansion in divergent R ≡ PL − PT P =

∞

• n=1

rn(Tτ)−n ≈ 8Cη Tτ + 16Cη(CτΠ − Cλ) 3(Tτ)2 + O( 1 (Tτ)3 )

here τΠ = T −1

RTA

AK, Heller, Spalinski al. 1609.04803

Holography

Heller et al. PRL 110 (2013)

Hydrodynamization through decay of non-hydro modes

Divergence signals that powerlaw form is not sufficient Needs to be supplemented with terms ∼ e−ξ0Tτ × (constants of integration) No surprise? In kinetic theory, need f(p) as an initial condition. In gradient expansion, the only boundary condition T at t → ∞

Hydrodynamization through decay of non-hydro modes

Find ξ0 through analytical properties of Borel transform RB(ξ) =

∞

• n=1

rn Γ(n + b)ξn, RI−B(Tτ) = 1 Tτ ∞ dξe−ξ/TτξbRB(ξ) Exponential decay is governed by the lowest non-hydro mode

Also in IS hydro

e−ωnh

• T(τ)dτ ∼ e−3/2ωnhTτ = e−ξ0Tτ

RTA

AK, Heller, Spalinski al. 1609.04803

Holography

Heller et al. PRL 110 (2013)

Outline

Effective kinetic theory Hydrodynamization and thermalization at weak coupling in effective kinetic theory Apples to apples comparison of weak and strong coupling hydrodynamization Green functions of T µν in during hydrodynamization and phenomenological application to HIC

Transverse dynamics and preflow

Nuclear radius R ≪ cτi ∼ Nucleon radius Rp ≪ 1/Qs Transverse structure small perturbation within the causal horizon Linear response theory for the transverse structures

Transverse dynamics and preflow

Green functions on top of non-equilibrium background

Linearized perturbations in EKT

Keegan et al. 1605.04287

Transverse perturbations characterized by wavenumber k f(x⊥, p) = ¯ f(p) + exp(ix · k)δf(p)

• ∂τ − pz

τ ∂pz

• f = C[f]
• ∂τ − pz

τ ∂pz + ik · p

• f = C[ ¯

f, f] For thermal ¯ f: large wavelenght pert. described by hydro

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 k/T 0.55 0.6 0.65 0.7 0.75 Re[ ω(k)/k ] Dispersion relation λ=10 2nd order hydro Ideal hydro cs

2=1/3

EKT

ω(k) k = c2

s + 4

3 η e + p

• csτπ − 2

3cs η e + p

• k2

For larger k, c2

s → 1, with polynomial

decay

no plot unfortunately...

Hydrodynamization of perturbations

Keegan et al.JHEP 1608 (2016)

δT xx = δe

e

• 1

3e + 1 3ητπk2 + η 2τ − 2(λ1−ητπ) 9τ 2

• − ikδT 0x

e

• η − 1

τ

• η2

2e + ητπ 2 − 2 3λ1

• k ∼ 1/Rnucleus

−0.6 −0.4 −0.2 0.2 0.4 0.6 1 10 100 1000 k/Teff

k = 0.01Qs

δT xx/T 00 τQs kinetic theory 2nd order hydro 1st order hydro ideal hydro

k ∼ 1/Rproton

−0.6 −0.4 −0.2 0.2 0.4 0.6 1 10 100 1000

k = 0.1Qs

δT xx/T 00 τQs

Perturbations hydrodynamize also at Qτ ∼ {10, 20}.

Hydrodynamization of perturbations

Keegan et al.JHEP 1608 (2016)

δT xx = δe

e

• 1

3e + 1 3ητπk2 + η 2τ − 2(λ1−ητπ) 9τ 2

• − ikδT 0x

e

• η − 1

τ

• η2

2e + ητπ 2 − 2 3λ1

• k ∼ 1/0.5Rnucleus

−0.6 −0.4 −0.2 0.2 0.4 0.6 1 10 100 1000

k = 0.2Qs

δT xx/T 00 τQs

k ∼ 1/0.25Rnucleus

−0.6 −0.4 −0.2 0.2 0.4 0.6 1 10 100 1000

k = 0.4Qs

k/Teff > 0.6 δT xx/T 00 τQs

No hydrodynamics for the large-k modes

Green function in coordinate space

−1.0 −0.5 0.0 0.5 1.0 1.5 2.0 0.2 0.4 0.6 0.8 1 1.2 1.4 δe response to δe perturbation Qsτ = 10 τ/τπ = 0.99 τ 2E(r) r/(τ − τ0) free streaming kinetic theory −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 0.2 0.4 0.6 0.8 1 1.2 1.4 δe response to δe perturbation Qsτ = 20 τ/τπ = 1.6 τ 2E(r) r/(τ − τ0) free streaming kinetic theory

Nanscent formation of dip in the origin hall mark of hydro

Green function in coordinate space

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0 0.2 0.4 0.6 0.8 1 1.2 1.4 δe response to δe perturbation Qsτ = 50 τ/τπ = 3.1

2nd hydro

• τ 2E(r)

r/(τ − τ0) kinetic theory Qsτ = 10 Qsτ = 20 −3.0 −2.0 −1.0 0.0 1.0 2.0 3.0 4.0 0.2 0.4 0.6 0.8 1 1.2 1.4 δe response to δe perturbation Qsτ = 500 τ/τπ = 15

2nd hydro

• τ 2E(r)

r/(τ − τ0) kinetic theory Qsτ = 10 Qsτ = 20

Evolution after Qτi > {10, 20}, evolution described by hydro

Transverse dynamics and preflow

With free streaming pre-equilibrium evolution:

20 40 60 80 100 eτ 4/3/[norm] First hydro starts Second hydro starts Third hydro starts 20 40 60 80 100

• 3 -2 -1 0

1 2 3 eτ 4/3/[norm] x fm

• 3 -2 -1 0

1 2 3 x fm

• 3 -2 -1 0

1 2 3 x fm τ = 0.4 fm τ = 0.8 fm τ = 1.2 fm τinit = 0.4 fm τinit = 0.8 fm τinit = 1.2 fm τ = 1.5 fm τ = 2.0 fm τ = 3.0 fm

AK, Mazeliauskas, Paquet, Schlichting, Teaney, in progress

Strong dependence on initialization time!

Transverse dynamics and preflow

With full EKT pre-equilibrium evolution:

20 40 60 80 100 eτ 4/3/[norm] First hydro starts Second hydro starts Third hydro starts 20 40 60 80 100

• 3 -2 -1 0

1 2 3 eτ 4/3/[norm] x fm

• 3 -2 -1 0

1 2 3 x fm

• 3 -2 -1 0

1 2 3 x fm τ = 0.4 fm τ = 0.8 fm τ = 1.2 fm τinit = 0.4 fm τinit = 0.8 fm τinit = 1.2 fm τ = 1.5 fm τ = 2.0 fm τ = 3.0 fm

AK, Mazeliauskas, Paquet, Schlichting, Teaney, in progress

Initialization time removed

Summary

Weak coupling hydrodynamization quantitatively and qualitatively understood, with some caveats Push towards phenomenologically useful pre-equilibrium description Some similarities and differences between weak and strong coupling

Big quantitative difference in η/s − → time scales very different Non-hydro modes near equilibrium:

Imaginary parts: T in strong, τΠ in weak coupling Real parts T in strong, and k in weak coupling

Similar divergent hydrodynamic series and hydrodynamization through decay of non-hydro modes

Weak and strong coupling hydrodynamization compared

0.1 1 10 100 η/s 1 2 4 8 χ = S/Si kinetic AdS (1+7.0η/s)

1/3

2.0 (η/s)

1/3

Total entropy production

Strong coupling quite close to where weak coupling goes haywire?

Weak coupling param. estimate for entropy production: Titeq ∼

Ti λ2T (teq) , before free streaming: T 4(teq) ∼ T 4 i /(Titeq) then teq ∼ λ−8/3 ∼ (η/s)4/3