Hydrodynamization, hydrodynamic attractors and all that ...

Michal P. Heller

aei.mpg.de/GQFI

Hydrodynamization, hydrodynamic attractors and all that — an overview of the field

1610.02023: lecture notes emphasizing the holographic component1707.02282: review emphasizing the hydrodynamic component

state of the field: www.birs.ca/events/2019/5-day-workshops/19w5048

http://aei.mpg.de/GQFI

http://www.birs.ca/events/2019/5-day-workshops/19w5048

Introduction & motivation

// in this lecture the vast majority of results concern conformally-invarianttheories without a set of dim-less scales (~ strong coupling) //

Motivation: hydrodynamic simulations for HICGeneric energy density at the moment hydrodynamics simulation starts:

1009.3244 by Schenke, Jeon & Gale

Simulations like this match the data, however, they apply hydrodynamics in the

regime of large gradients: . Does it even make sense?�E�d

⇥

Eav( 1Eav/10

)1/4

!�1

⇡ 1<latexit sha1_base64="/KzwrWT0jfSIRfpwUYUXV88oEqI=">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</latexit>

1/13

Hydrodynamization

plot from 1609.04803v2 with Kurkela, Spalinski & Svensson

Ab initio studies in holography and later studies in other models show that viscous hydro can work even when deviations from local equilibrium are large:

0906.4426, 1011.3562 by Chesler & Yaffe; 1103.3452 with Janik & Witaszczyk

N=4 SYMEKT with ⌘/s = 0.624

RTA with ⌘/s = 0.624

sample n-eq states in:

�PE/3 =

2

⇡w�1

viscous hydro prediction:

2/13

Preview: hydrodynamic attractors

plot from 1609.04803v2 with Kurkela, Spalinski & Svensson

Viscous hydrodynamics works* despite huge gradients in the system:0906.4426, 1011.3562 by Chesler & Yaffe; 1103.3452 with Janik & Witaszczyk

hydrodynamics

“attractor”=???

3/13

Relativistic hydrodynamics - textbook definitionan EFT of the slow (?) evolution of conserved currents in collective media close to equilibrium (?)hydrodynamics is

DOFs: always local energy density and local flow velocity ( )EOMs: conservation eqns for expanded in gradients

✏ uµ u⌫u⌫ = �1

rµhTµ⌫i = 0 hTµ⌫i

Tµ⌫ = ✏uµu⌫ + P (✏){ gµ⌫ + uµu⌫ }� ⌘(✏)�µ⌫ � ⇣(✏){ gµ⌫ + uµu⌫ }(r · u) + . . .hTµ⌫i

shear viscositycontribution

bulk viscosity(vanishes for CFTs)

microscopicinput:

EoSP (✏) =

1

3✏( for CFTs)

�PE/3 =

2

⇡w�1

⇡µ⌫<latexit sha1_base64="xc8fxwU0zR5vc1P2BBMD9mu3AuM=">AAAB9HicbVDLSgNBEOyNrxhfUY9eBoPgKexGQU8S8OIxgnlAdg2zk9lkyOzsOo9AWPIdXjwo4tWP8ebfOEn2oIkFDUVVN91dYcqZ0q777RTW1jc2t4rbpZ3dvf2D8uFRSyVGEtokCU9kJ8SKciZoUzPNaSeVFMchp+1wdDvz22MqFUvEg56kNIjxQLCIEaytFPgpe8z82CBfmGmvXHGr7hxolXg5qUCORq/85fcTYmIqNOFYqa7npjrIsNSMcDot+UbRFJMRHtCupQLHVAXZ/OgpOrNKH0WJtCU0mqu/JzIcKzWJQ9sZYz1Uy95M/M/rGh1dBxkTqdFUkMWiyHCkEzRLAPWZpETziSWYSGZvRWSIJSba5lSyIXjLL6+SVq3qXVRr95eV+k0eRxFO4BTOwYMrqMMdNKAJBJ7gGV7hzRk7L86787FoLTj5zDH8gfP5A9tukiQ=</latexit>

4/13

Here we assume Landau frame, c.f. 1907.08191 by Kovtun / 1708.06255 by Bemfica et al.

Hydrodynamic & transient modes

Theories of viscous hydrodynamics

Tµ⌫ = ✏uµu⌫ + P (✏){ gµ⌫ + uµu⌫ }� ⌘(✏)�µ⌫ � ⇣(✏){ gµ⌫ + uµu⌫ }(r · u) + . . .rµ

⇣+ . . .

⌘= 0The crucial subtlety: does not

have a well-posed initial value problem hydrodynamic theories

Overall idea: make obey an independent PDE ensuring its to�⌘ �µ⌫⇡µ⌫

Müller 1967, Israel 1976, Israel & Stewart 1976decay timescale(⌧⇡ u

↵D↵ + 1) [⇡µ⌫ � (�⌘ �µ⌫)] = 0 ⇡µ⌫ = �⌘ �µ⌫ � ⌧⇡ u↵D↵ ⇡µ⌫ � ⌧⇡ u

↵D↵ (⌘ �µ⌫)

New incarnation: 0712.2451 by Baier, Romatschke, Son, Starinets & Stephanov

⇡µ⌫ = �⌘�µ⌫ � ⌧⇡ u↵D↵⇡

µ⌫ + �1⇡hµ

↵�⌫i↵ + �2⇡

hµ↵⌦

⌫i↵ + �3⌦hµ

↵⌦⌫i↵⇡

5/13

(there is a brand new approach that deals with the problem somewhat differently, see1907.08191 by Kovtun and 1708.06255 by Bemfica, Disconzi & Noronha)

Modes in BRSSS theoryMode = solution of linearized equations of finite-T state without any sources

Technical issue: tensor perturbs. channels (here and later sound channel):

Assuming momentum along x3 direction : , &�T �u3 �⇡33e�i! x0+i k x3

⇡µ⌫ = �⌘�µ⌫ � ⌧⇡ u↵D↵⇡

µ⌫ + �1⇡hµ

↵�⌫i↵ + �2⇡

hµ↵⌦

⌫i↵ + �3⌦hµ

↵⌦⌫i↵

conservation+

●●

●

■ ■

■

◆◆

◆

-6 -4 -2 0 2 4 6-5

-4

-3

-2

-1

0

Re(ω) / T

Im(ω

)/T

k = 0.1 Tk = 1.0 Tk = 10 T

two modes:

!3 + (. . .)!2 + (. . .)! + (. . .) = 0

hydro (sound wave)

transient (pure decay):!��k=0

= �i1⌧⇡

<latexit sha1_base64="1cvwFfTpUr7yKkg+0rIEzPaseBU=">AAACRHicdVBNTxsxFPQCLSEtJW2PXJ6IKvUA0e4SETgURXDhCBIBpPVq5XW8ixXvh2xvpcjsj+PCD+DGL+iFQyvEtao3JFJbtU+yNZp5b/w8cSm40q774Cwtr7x6vdpaa795u/5uo/P+w4UqKknZiBaikFcxUUzwnI0014JdlZKRLBbsMp4cN/rlVyYVL/JzPS1ZmJE05wmnRFsq6gQGz0wCmcahcXsH+3t+f2/b7bnuwPO9BviD/m6/xkXGUgL4iKc3kZnAF3Bre+0AB7wNOJGEGq82WJMqMrjkdV1Hne7CEBaGsDAEzzJNddG8TqPOPR4XtMpYrqkgSgWeW+rQEKk5Faxu40qxktAJSVlgYU4ypkIz27+GT5YZQ1JIe3INM/b3CUMypaZZbDszoq/V31pD/ksLKp3sh4bnZaVZTl8eSioBuoAmURhzyagWUwsIldzuCvSa2Dy0zb1tQ1j8FP4PLvyet9vzz/rd4eE8jhbaRFvoM/LQAA3RCTpFI0TRLfqGvqMfzp3z6Dw5zy+tS8585iP6o5yfvwBGMq0R</latexit>

6/13

Modes in Einstein-Hilbert holography = QNMs

ds2 =L2

u2

n�2dx0du� (1� ⇡4T 4u4)

�x0

�2+ d~x2

o+ �gab(u) e

�i! x0+i k x3

●

●

●

●

●

●

●

●

●●

■

■

■

■

■

■

■

■

■■

◆

◆

◆

◆

◆

◆

◆

◆

◆ ◆

-40 -20 0 20 40-30

-25

-20

-15

-10

-5

0

Re(ω) / T

Im(ω

)/T

!/T ⇡ ± 1p3k/T � i

2

3

1

4⇡(k/T )2 ± 3� 2 log 2

24p3⇡2

(k/T )3

hep-th/0506184 by Kovtun & Starinets

ingoing (regular) at the horizon

vanishing at the boundary

7/13

here focus on 1503.07514 with Spalinskisee also 1302.0697 with Janik & Witaszczyk for an earlier work

Hydrodynamics at large orders

Boost-invariant flowBjorken 1982

x1x1

⌧ = 0

⌧ > 0 x0const slice:

Boost-invariance: in coords no -dep(⌧ ⌘q

x20 � x2

1, y ⌘ arctanhx1

x0, x2, x3) y

In a CFT: hTµ⌫i = diag

⇢�E(⌧),�E � ⌧ E , E +

1

2⌧ E , E +

1

2⌧ E

�

and via scale-invariance is a function of

hT 22i � hT y

yi

w ⌘ ⌧ T

✓E(⌧)

38⇡

2N2c

◆1/4

⌘�PE/3 ⌘ R

⌘A

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Gradient expansion: series in 1w 1103.3452 with Janik & Witaszczyk

8/13

Large order gradient expansion: BRSSS1503.07514 with Spalinski

conservation (always the same) ⌧

w

dw

d⌧=

2

3+

1

18RA


⇡µ⌫ = �⌘�µ⌫ � ⌧⇡ u↵D↵⇡

µ⌫ + �1⇡hµ

↵�⌫i↵ + �2⇡

hµ↵⌦

⌫i↵ + �3⌦hµ

↵⌦⌫i↵

⇡µ⌫ = �⌘�µ⌫ � ⌧⇡ u↵D↵⇡

µ⌫ + �1⇡hµ

↵�⌫i↵ + �2⇡

hµ↵⌦

⌫i↵ + �3⌦hµ

↵⌦⌫i↵⇡

+<latexit sha1_base64="DxFm1e9y3l1ZKrzv3I486B431AY=">AAAB6HicbVDLSgNBEOyNrxhfUY9eBoMgCGE3CnqSgBePCZgHJEuYnfQmY2Znl5lZIYR8gRcPinj1k7z5N06SPWhiQUNR1U13V5AIro3rfju5tfWNza38dmFnd2//oHh41NRxqhg2WCxi1Q6oRsElNgw3AtuJQhoFAlvB6G7mt55QaR7LBzNO0I/oQPKQM2qsVL/oFUtu2Z2DrBIvIyXIUOsVv7r9mKURSsME1brjuYnxJ1QZzgROC91UY0LZiA6wY6mkEWp/Mj90Ss6s0idhrGxJQ+bq74kJjbQeR4HtjKgZ6mVvJv7ndVIT3vgTLpPUoGSLRWEqiInJ7GvS5wqZEWNLKFPc3krYkCrKjM2mYEPwll9eJc1K2bssV+pXpeptFkceTuAUzsGDa6jCPdSgAQwQnuEV3pxH58V5dz4WrTknmzmGP3A+fwBxn4yv</latexit>

A<latexit sha1_base64="OWYSSh3+UEs9UFTLd0pRIWZAtaY=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6EkiXjxGMA9IljA7mU3GzGOZmRXCkn/w4kERr/6PN//GSbIHTSxoKKq66e6KEs6M9f1vb2V1bX1js7BV3N7Z3dsvHRw2jUo1oQ2iuNLtCBvKmaQNyyyn7URTLCJOW9Hoduq3nqg2TMkHO05oKPBAspgRbJ3U7BLM0U2vVPYr/gxomQQ5KUOOeq/01e0rkgoqLeHYmE7gJzbMsLaMcDopdlNDE0xGeEA7jkosqAmz2bUTdOqUPoqVdiUtmqm/JzIsjBmLyHUKbIdm0ZuK/3md1MZXYcZkkloqyXxRnHJkFZq+jvpMU2L52BFMNHO3IjLEGhPrAiq6EILFl5dJs1oJzivV+4ty7TqPowDHcAJnEMAl1OAO6tAAAo/wDK/w5invxXv3PuatK14+cwR/4H3+ANnPjqM=</latexit>




( )1

4⇡=

2� log 2

2⇡=

1

2⇡=

a n+1/a

n<latexit sha1_base64="BT2XpVD7CrGgy1v+GXgS2uAQ3lo=">AAAB9HicdVDLSsNAFJ3UV62vqks3g0UQhJg0rXUlBTcuK9gHtKFMppN26GQSZyaFEvIdblwo4taPceffOGkjqOiByz2ccy9z53gRo1JZ1odRWFldW98obpa2tnd298r7Bx0ZxgKTNg5ZKHoekoRRTtqKKkZ6kSAo8BjpetPrzO/OiJA05HdqHhE3QGNOfYqR0pKLhgk/s9PzrKfDcsUy67Wq07DgktSdnFw40DatBSogR2tYfh+MQhwHhCvMkJR924qUmyChKGYkLQ1iSSKEp2hM+ppyFBDpJoujU3iilRH0Q6GLK7hQv28kKJByHnh6MkBqIn97mfiX14+Vf+kmlEexIhwvH/JjBlUIswTgiAqCFZtrgrCg+laIJ0ggrHROJR3C10/h/6RTNW3HrN7WKs2rPI4iOALH4BTYoAGa4Aa0QBtgcA8ewBN4NmbGo/FivC5HC0a+cwh+wHj7BL1jkhU=</latexit>

R(w) ⇡1X

n=1

rnwn

= + . . .

(note that ’s do not depend on ini. cond.)


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Divergent series: an ⇠ n!<latexit sha1_base64="SZrfiY1BJFfPvESuFiPmhW0xVAc=">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</latexit>

9/13

Transseries and resurgence1503.07514 with Spalinski

R(w) ⇡1X

n=1

rnwn

BR(⇠) =1X

n=1

rnn!

⇠n ⇡ a0 ++ . . .+ a100⇠100

b0 ++ . . .+ b100⇠100Borel trafo.A




an<latexit sha1_base64="2t2imDR8eFJbWcNmECjDlODRnJI=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4Kkkt6LHgxWMF0xbaUDbbSbt0swm7G6GE/gYvHhTx6g/y5r9x2+agrQ8GHu/NMDMvTAXXxnW/nY3Nre2d3dJeef/g8Oi4cnLa1kmmGPosEYnqhlSj4BJ9w43AbqqQxqHATji5m/udJ1SaJ/LRTFMMYjqSPOKMGiv5dJDL2aBSdWvuAmSdeAWpQoHWoPLVHyYsi1EaJqjWPc9NTZBTZTgTOCv3M40pZRM6wp6lksaog3xx7IxcWmVIokTZkoYs1N8TOY21nsah7YypGetVby7+5/UyE90GOZdpZlCy5aIoE8QkZP45GXKFzIipJZQpbm8lbEwVZcbmU7YheKsvr5N2veZd1+oPjWqzUcRRgnO4gCvw4AaacA8t8IEBh2d4hTdHOi/Ou/OxbN1wipkz+APn8wcG5o7I</latexit> ⇡ b0 + . . .+ b100 ⇠100

c0 + . . .+ c100 ⇠100<latexit sha1_base64="SrTo+TH+Rj8o3IOLMtHsStqzQqo=">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</latexit>

Borel (re)summation:✓Z

C1

d⇠ �Z

C2

d⇠

◆⇥w e�w ⇠BR(⇠)

⇤

⇠ e� 3

21

C⌧⇡ww

C⌘�2C�1C⌧⇡ . . .

⌘�

⌘A

A<latexit sha1_base64="qVw6/IQYfQuKenh1VKji6n4SJzc=">AAAB73icdVBNS8NAEJ3Ur1q/qh69LBbBU0haSxvwUPHisYK1hTaUzXbbLt1s4u5GKKF/wosHRbz6d7z5b9y0FVT0wcDjvRlm5gUxZ0o7zoeVW1ldW9/Ibxa2tnd294r7B7cqSiShLRLxSHYCrChngrY005x2YklxGHDaDiaXmd++p1KxSNzoaUz9EI8EGzKCtZE6aY9gji5m/WLJsb16tVqpI8d2HK/s1QzxPM+tucg1SoYSLNHsF997g4gkIRWacKxU13Vi7adYakY4nRV6iaIxJhM8ol1DBQ6p8tP5vTN0YpQBGkbSlNBorn6fSHGo1DQMTGeI9Vj99jLxL6+b6GHdT5mIE00FWSwaJhzpCGXPowGTlGg+NQQTycytiIyxxESbiAomhK9P0f/ktmy7Fbt8fVZqnC/jyMMRHMMpuFCDBlxBE1pAgMMDPMGzdWc9Wi/W66I1Zy1nDuEHrLdPMrKQEg==</latexit>

A =3

2i

✓�i

1

C⌧⇡

◆

C1

C2Ambiguity in resummation ~ transient mode

R(w) =1X

j=0

�j e�j Aw wj � �(j)(w)Transseries:~ resum. ambig. + ini. cond.

~ 1/w expansions

nonlinear effects

Resurgence: transseries yields an unambiguous answer up to 1 real int. const.


10/13

Attractors

1503.07514 with Spalinski1704.08699 by Romatschke

1907.08101 by Kurkela, van der Schee, Wiedemann & Wu… ~15 other distinct studies of attractors …

1912.xxxxx with Jefferson, Spalinski & Svensson, see https://bit.ly/2DCt0bl

https://bit.ly/2DCt0bl

(BRSSS) resummed hydrodynamics1503.07514 with Spalinski

Idea: resummed / far-from-equilibrium hydrodynamics = attractor solutions





BRSSS:

„slow roll” approximation reveals an attractor solution

Re(�) ⇡ 0.875

0.4 0.6 0.8 1.0 1.2 1.4

0.5

1.0

1.5

2.0

2.5

w = T τ

R=P T

-P L

POne can also approx. resum transseries:

R(w) =1X

j=0

�j e�j Aw wj � �(j)(w)2

⇡

Requires 3 Borel summations

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11/13

Anything fundamentalabout it?

Attractor in kinetic theory and holography

Idea: use the slow roll approximation to generate attractors in other theories1704.08699 by Romatschke (figure imported from the arXiv ver)

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Note: centre and right are projections from infinitely-dimensional phase space12/13

BRSSS (old) RTA (new) holography (new)

Longitudinal expansion important for the initial memory loss of solutions1907.08101 by Kurkela, van der Schee, Wiedemann & Wu

Executive summary

Executive summaryWhat seems to control the applicability of hydrodynamics is not the gradient expansion itself, but what comes on top of it — expansion & transients

transient oscillationin N = 4 SYM

Hydrodynamic attractors = intermediate-to-late time universalities13/13

As a result, applying hydro to HIC early on is not a priori crazy

Hydrodynamization, hydrodynamic attractors and all that ...

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