Hydrodynamization, hydrodynamic attractors and all that ...
Transcript of 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
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>
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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:
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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⌧⇡
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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
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⇡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↵⇡
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( )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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an<latexit sha1_base64="2t2imDR8eFJbWcNmECjDlODRnJI=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4Kkkt6LHgxWMF0xbaUDbbSbt0swm7G6GE/gYvHhTx6g/y5r9x2+agrQ8GHu/NMDMvTAXXxnW/nY3Nre2d3dJeef/g8Oi4cnLa1kmmGPosEYnqhlSj4BJ9w43AbqqQxqHATji5m/udJ1SaJ/LRTFMMYjqSPOKMGiv5dJDL2aBSdWvuAmSdeAWpQoHWoPLVHyYsi1EaJqjWPc9NTZBTZTgTOCv3M40pZRM6wp6lksaog3xx7IxcWmVIokTZkoYs1N8TOY21nsah7YypGetVby7+5/UyE90GOZdpZlCy5aIoE8QkZP45GXKFzIipJZQpbm8lbEwVZcbmU7YheKsvr5N2veZd1+oPjWqzUcRRgnO4gCvw4AaacA8t8IEBh2d4hTdHOi/Ou/OxbN1wipkz+APn8wcG5o7I</latexit>
an<latexit sha1_base64="2t2imDR8eFJbWcNmECjDlODRnJI=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4Kkkt6LHgxWMF0xbaUDbbSbt0swm7G6GE/gYvHhTx6g/y5r9x2+agrQ8GHu/NMDMvTAXXxnW/nY3Nre2d3dJeef/g8Oi4cnLa1kmmGPosEYnqhlSj4BJ9w43AbqqQxqHATji5m/udJ1SaJ/LRTFMMYjqSPOKMGiv5dJDL2aBSdWvuAmSdeAWpQoHWoPLVHyYsi1EaJqjWPc9NTZBTZTgTOCv3M40pZRM6wp6lksaog3xx7IxcWmVIokTZkoYs1N8TOY21nsah7YypGetVby7+5/UyE90GOZdpZlCy5aIoE8QkZP45GXKFzIipJZQpbm8lbEwVZcbmU7YheKsvr5N2veZd1+oPjWqzUcRRgnO4gCvw4AaacA8t8IEBh2d4hTdHOi/Ou/OxbN1wipkz+APn8wcG5o7I</latexit>
Divergent series: an ⇠ n!<latexit sha1_base64="SZrfiY1BJFfPvESuFiPmhW0xVAc=">AAACH3icdVBLSwMxGMz6rPVV9eglWgQPsuy21m3BQ8GLxwr2AdulZNO0Dc1mlyQrlGX/iRf/ihcPioi3/hvTx4KKDgSGmfmSL+NHjEplWVNjZXVtfWMzt5Xf3tnd2y8cHLZkGAtMmjhkoej4SBJGOWkqqhjpRIKgwGek7Y9vZn77gQhJQ36vJhHxAjTkdEAxUlrqFa6S7vwSVwx9L7FMu+Q4FfvCMmu1mlMpL4hVrqaol/AUdiUNID9Je4ViloVZFmZZaJvWHEWwRKNX+Oz2QxwHhCvMkJSubUXKS5BQFDOS5ruxJBHCYzQkrqYcBUR6yXy1FJ5ppQ8HodCHKzhXv08kKJByEvg6GSA1kr+9mfiX58ZqUPUSyqNYEY4XDw1iBlUIZ2XBPhUEKzbRBGFB9a4Qj5BAWOlK87qE7Kfwf9IqmXbZLN1dFuvXyzpy4BicgnNgAwfUwS1ogCbA4BE8g1fwZjwZL8a78bGIrhjLmSPwA8b0CzEAn10=</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
<latexit sha1_base64="OWYSSh3+UEs9UFTLd0pRIWZAtaY=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6EkiXjxGMA9IljA7mU3GzGOZmRXCkn/w4kERr/6PN//GSbIHTSxoKKq66e6KEs6M9f1vb2V1bX1js7BV3N7Z3dsvHRw2jUo1oQ2iuNLtCBvKmaQNyyyn7URTLCJOW9Hoduq3nqg2TMkHO05oKPBAspgRbJ3U7BLM0U2vVPYr/gxomQQ5KUOOeq/01e0rkgoqLeHYmE7gJzbMsLaMcDopdlNDE0xGeEA7jkosqAmz2bUTdOqUPoqVdiUtmqm/JzIsjBmLyHUKbIdm0ZuK/3md1MZXYcZkkloqyXxRnHJkFZq+jvpMU2L52BFMNHO3IjLEGhPrAiq6EILFl5dJs1oJzivV+4ty7TqPowDHcAJnEMAl1OAO6tAAAo/wDK/w5invxXv3PuatK14+cwR/4H3+ANnPjqM=</latexit>
A<latexit sha1_base64="OWYSSh3+UEs9UFTLd0pRIWZAtaY=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6EkiXjxGMA9IljA7mU3GzGOZmRXCkn/w4kERr/6PN//GSbIHTSxoKKq66e6KEs6M9f1vb2V1bX1js7BV3N7Z3dsvHRw2jUo1oQ2iuNLtCBvKmaQNyyyn7URTLCJOW9Hoduq3nqg2TMkHO05oKPBAspgRbJ3U7BLM0U2vVPYr/gxomQQ5KUOOeq/01e0rkgoqLeHYmE7gJzbMsLaMcDopdlNDE0xGeEA7jkosqAmz2bUTdOqUPoqVdiUtmqm/JzIsjBmLyHUKbIdm0ZuK/3md1MZXYcZkkloqyXxRnHJkFZq+jvpMU2L52BFMNHO3IjLEGhPrAiq6EILFl5dJs1oJzivV+4ty7TqPowDHcAJnEMAl1OAO6tAAAo/wDK/w5invxXv3PuatK14+cwR/4H3+ANnPjqM=</latexit>
an<latexit sha1_base64="2t2imDR8eFJbWcNmECjDlODRnJI=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4Kkkt6LHgxWMF0xbaUDbbSbt0swm7G6GE/gYvHhTx6g/y5r9x2+agrQ8GHu/NMDMvTAXXxnW/nY3Nre2d3dJeef/g8Oi4cnLa1kmmGPosEYnqhlSj4BJ9w43AbqqQxqHATji5m/udJ1SaJ/LRTFMMYjqSPOKMGiv5dJDL2aBSdWvuAmSdeAWpQoHWoPLVHyYsi1EaJqjWPc9NTZBTZTgTOCv3M40pZRM6wp6lksaog3xx7IxcWmVIokTZkoYs1N8TOY21nsah7YypGetVby7+5/UyE90GOZdpZlCy5aIoE8QkZP45GXKFzIipJZQpbm8lbEwVZcbmU7YheKsvr5N2veZd1+oPjWqzUcRRgnO4gCvw4AaacA8t8IEBh2d4hTdHOi/Ou/OxbN1wipkz+APn8wcG5o7I</latexit>
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.
A<latexit sha1_base64="OWYSSh3+UEs9UFTLd0pRIWZAtaY=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6EkiXjxGMA9IljA7mU3GzGOZmRXCkn/w4kERr/6PN//GSbIHTSxoKKq66e6KEs6M9f1vb2V1bX1js7BV3N7Z3dsvHRw2jUo1oQ2iuNLtCBvKmaQNyyyn7URTLCJOW9Hoduq3nqg2TMkHO05oKPBAspgRbJ3U7BLM0U2vVPYr/gxomQQ5KUOOeq/01e0rkgoqLeHYmE7gJzbMsLaMcDopdlNDE0xGeEA7jkosqAmz2bUTdOqUPoqVdiUtmqm/JzIsjBmLyHUKbIdm0ZuK/3md1MZXYcZkkloqyXxRnHJkFZq+jvpMU2L52BFMNHO3IjLEGhPrAiq6EILFl5dJs1oJzivV+4ty7TqPowDHcAJnEMAl1OAO6tAAAo/wDK/w5invxXv3PuatK14+cwR/4H3+ANnPjqM=</latexit>
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
(BRSSS) resummed hydrodynamics1503.07514 with Spalinski
Idea: resummed / far-from-equilibrium hydrodynamics = attractor solutions
A<latexit sha1_base64="OWYSSh3+UEs9UFTLd0pRIWZAtaY=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6EkiXjxGMA9IljA7mU3GzGOZmRXCkn/w4kERr/6PN//GSbIHTSxoKKq66e6KEs6M9f1vb2V1bX1js7BV3N7Z3dsvHRw2jUo1oQ2iuNLtCBvKmaQNyyyn7URTLCJOW9Hoduq3nqg2TMkHO05oKPBAspgRbJ3U7BLM0U2vVPYr/gxomQQ5KUOOeq/01e0rkgoqLeHYmE7gJzbMsLaMcDopdlNDE0xGeEA7jkosqAmz2bUTdOqUPoqVdiUtmqm/JzIsjBmLyHUKbIdm0ZuK/3md1MZXYcZkkloqyXxRnHJkFZq+jvpMU2L52BFMNHO3IjLEGhPrAiq6EILFl5dJs1oJzivV+4ty7TqPowDHcAJnEMAl1OAO6tAAAo/wDK/w5invxXv3PuatK14+cwR/4H3+ANnPjqM=</latexit>
A<latexit sha1_base64="OWYSSh3+UEs9UFTLd0pRIWZAtaY=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6EkiXjxGMA9IljA7mU3GzGOZmRXCkn/w4kERr/6PN//GSbIHTSxoKKq66e6KEs6M9f1vb2V1bX1js7BV3N7Z3dsvHRw2jUo1oQ2iuNLtCBvKmaQNyyyn7URTLCJOW9Hoduq3nqg2TMkHO05oKPBAspgRbJ3U7BLM0U2vVPYr/gxomQQ5KUOOeq/01e0rkgoqLeHYmE7gJzbMsLaMcDopdlNDE0xGeEA7jkosqAmz2bUTdOqUPoqVdiUtmqm/JzIsjBmLyHUKbIdm0ZuK/3md1MZXYcZkkloqyXxRnHJkFZq+jvpMU2L52BFMNHO3IjLEGhPrAiq6EILFl5dJs1oJzivV+4ty7TqPowDHcAJnEMAl1OAO6tAAAo/wDK/w5invxXv3PuatK14+cwR/4H3+ANnPjqM=</latexit>
A<latexit sha1_base64="OWYSSh3+UEs9UFTLd0pRIWZAtaY=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6EkiXjxGMA9IljA7mU3GzGOZmRXCkn/w4kERr/6PN//GSbIHTSxoKKq66e6KEs6M9f1vb2V1bX1js7BV3N7Z3dsvHRw2jUo1oQ2iuNLtCBvKmaQNyyyn7URTLCJOW9Hoduq3nqg2TMkHO05oKPBAspgRbJ3U7BLM0U2vVPYr/gxomQQ5KUOOeq/01e0rkgoqLeHYmE7gJzbMsLaMcDopdlNDE0xGeEA7jkosqAmz2bUTdOqUPoqVdiUtmqm/JzIsjBmLyHUKbIdm0ZuK/3md1MZXYcZkkloqyXxRnHJkFZq+jvpMU2L52BFMNHO3IjLEGhPrAiq6EILFl5dJs1oJzivV+4ty7TqPowDHcAJnEMAl1OAO6tAAAo/wDK/w5invxXv3PuatK14+cwR/4H3+ANnPjqM=</latexit>
A<latexit sha1_base64="OWYSSh3+UEs9UFTLd0pRIWZAtaY=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hd0o6EkiXjxGMA9IljA7mU3GzGOZmRXCkn/w4kERr/6PN//GSbIHTSxoKKq66e6KEs6M9f1vb2V1bX1js7BV3N7Z3dsvHRw2jUo1oQ2iuNLtCBvKmaQNyyyn7URTLCJOW9Hoduq3nqg2TMkHO05oKPBAspgRbJ3U7BLM0U2vVPYr/gxomQQ5KUOOeq/01e0rkgoqLeHYmE7gJzbMsLaMcDopdlNDE0xGeEA7jkosqAmz2bUTdOqUPoqVdiUtmqm/JzIsjBmLyHUKbIdm0ZuK/3md1MZXYcZkkloqyXxRnHJkFZq+jvpMU2L52BFMNHO3IjLEGhPrAiq6EILFl5dJs1oJzivV+4ty7TqPowDHcAJnEMAl1OAO6tAAAo/wDK/w5invxXv3PuatK14+cwR/4H3+ANnPjqM=</latexit>
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
A<latexit sha1_base64="Bqp9YfxlgaMyKJJ5pR0Fn+JG/YA=">AAAB73icbVA9SwNBEJ3zM8avqKXNYhCswl0UtLCI2FhGMB+QHGFvs5cs2ds7d+eEcORP2FgoYuvfsfPfuEmu0MQHA4/3ZpiZFyRSGHTdb2dldW19Y7OwVdze2d3bLx0cNk2casYbLJaxbgfUcCkUb6BAyduJ5jQKJG8Fo9up33ri2ohYPeA44X5EB0qEglG0UjvrMirJzaRXKrsVdwayTLyclCFHvVf66vZjlkZcIZPUmI7nJuhnVKNgkk+K3dTwhLIRHfCOpYpG3PjZ7N4JObVKn4SxtqWQzNTfExmNjBlHge2MKA7NojcV//M6KYZXfiZUkiJXbL4oTCXBmEyfJ32hOUM5toQyLeythA2ppgxtREUbgrf48jJpViveeaV6f1GuXedxFOAYTuAMPLiEGtxBHRrAQMIzvMKb8+i8OO/Ox7x1xclnjuAPnM8foFuPrQ==</latexit>
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)
✓=
2 9A
�4 3
◆
<latexit sha1_base64="Bnp4cDgn2DWX2RcEcUxWGLWwBv0=">AAACHHicbVDLSgNBEJz1GeNr1aOXwSDEg2E3CaigEPHiMYJ5QDaE2clsMmT2wUyvEJb9EC/+ihcPinjxIPg3TpI9aGJBQ1HVTXeXGwmuwLK+jaXlldW19dxGfnNre2fX3NtvqjCWlDVoKELZdoliggesARwEa0eSEd8VrOWObiZ+64FJxcPgHsYR6/pkEHCPUwJa6pkVRzAPivgKO54kNCmnyUWKE4cSga9TfJrJ1TSppNiRfDCEk55ZsErWFHiR2BkpoAz1nvnp9EMa+ywAKohSHduKoJsQCZwKluadWLGI0BEZsI6mAfGZ6ibT51J8rJU+9kKpKwA8VX9PJMRXauy7utMnMFTz3kT8z+vE4J13Ex5EMbCAzhZ5scAQ4klSuM8loyDGmhAqub4V0yHRaYDOM69DsOdfXiTNcsmulMp31ULtMosjhw7RESoiG52hGrpFddRAFD2iZ/SK3own48V4Nz5mrUtGNnOA/sD4+gEnhqAq</latexit>
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