indico.gsi.de€¦ · Thermalization At weak coupling the obvious deﬁnition would be to require thermal momentum distributions for quarks and gluons... At strong coupling, the picture

Thermalization of boost-invariant plasma at strongcoupling from AdS/CFT

Romuald A. Janik

Jagiellonian UniversityKraków

M. Heller, RJ, P. Witaszczyk, 1103.3452M. Heller, RJ, P. Witaszczyk, 1201.????

Romuald A. Janik (Kraków) Thermalization of expanding plasma from AdS/CFT 1 / 26

Outline

1 Key question

2 AdS/CFT, hydrodynamics and nonequilibrium processes

3 Boost-invariant flow

4 The AdS/CFT method

5 Main resultsNonequilibrium vs. hydrodynamic behaviourEntropyCharacteristics of thermalization

6 Conclusions

Key question:

Understand the features of (early)thermalization for an evolving (boost-invariant) plasma system

What do we mean by thermalization here?

Key question:

Thermalization

At weak coupling the obvious definition would be to require thermalmomentum distributions for quarks and gluons...

At strong coupling, the picture of a gas of gluons is not really valid— alternatively require that observables such as 2-point functions/spatial

Wilson loops/ entanglement entropy are the same as for a thermal system...explored in the AdS/CFT context

This is very good for studying relaxation processes where the final state issome uniform static plasma system — this is not so for the plasmaundergoing expansion

For an expanding plasma fireball we need local equilibrium — bilocal probesget contaminated by collective flow

We adopt an operational definition of thermalization — the point whenplasma starts being describable by (viscous) hydrodynamics.

Thermalization

AdS/CFT, hydrodynamics and nonequilibrium processes

Hydrodynamics isolates long wavelength effective degrees of freedom of atheoryThe energy-momentum tensor Tµν is expressed in terms of a localtemperature T and flow velocity uµ

Tµν is expressed as an expansion in the gradients of the flow velocities(shown here for N = 4 SYM)

Tµνrescaled = (πT )4(ηµν + 4uµuν)︸︷︷︸

perfect fluid

− 2(πT )3σµν︸︷︷︸viscosity

+ (πT 2)(

log 2Tµν2a + 2Tµν

2b + (2− log 2)

Tµν2c + Tµν

2d + Tµν2e

))︸︷︷︸

second order hydrodynamics

The coefficients of the various tensor structures are the transport coefficients.In a conformal theory these are pure numbers times powers of T .Full nonlinear hydrodynamic equations follow now from ∂µTµν = 0The above form of Tµν for N = 4 SYM at strong coupling is not anassumption but can be proven from AdS/CFT Minwalla et.al.

perfect fluid

+ (πT 2)(

2b + (2− log 2)

Tµν2c + Tµν

2d + Tµν2e

))︸︷︷︸

perfect fluid

+ (πT 2)(

2b + (2− log 2)

Tµν2c + Tµν

2d + Tµν2e

))︸︷︷︸

perfect fluid

+ (πT 2)(

2b + (2− log 2)

Tµν2c + Tµν

2d + Tµν2e

))︸︷︷︸

perfect fluid

+ (πT 2)(

2b + (2− log 2)

Tµν2c + Tµν

2d + Tµν2e

))︸︷︷︸

perfect fluid

+ (πT 2)(

2b + (2− log 2)

Tµν2c + Tµν

2d + Tµν2e

))︸︷︷︸

perfect fluid

+ (πT 2)(

2b + (2− log 2)

Tµν2c + Tµν

2d + Tµν2e

))︸︷︷︸

perfect fluid

+ (πT 2)(

2b + (2− log 2)

Tµν2c + Tµν

2d + Tµν2e

))︸︷︷︸

Linearized hydrodynamics

Look at small disturbances of the uniform static plasma. . .

If Tµν is described by (1st order viscous) hydrodynamics then one can derivedispersion relation of long wavelength modes from hydrodynamic equations:shear modes:

ωshear = −iη

E + pk2

sound modes:

ωsound =1√3

k − i23

E + pk2

If we were to include terms in Tµν with more derivatives (higher order viscoushydrodynamics), we would get terms with higher powers of k in thedispersion relations...

Hypothetical resummed all-order hydrodynamics would predict the fulldispersion relation for these modes ωshear (k), ωsound (k)

What happens in the AdS/CFT description?

ωshear = −iη

E + pk2

sound modes:

ωsound =1√3

k − i23

E + pk2

ωshear = −iη

E + pk2

sound modes:

ωsound =1√3

k − i23

E + pk2

ωshear = −iη

E + pk2

sound modes:

ωsound =1√3

k − i23

E + pk2

ωshear = −iη

E + pk2

sound modes:

ωsound =1√3

k − i23

E + pk2

ωshear = −iη

E + pk2

sound modes:

ωsound =1√3

k − i23

E + pk2

ωshear = −iη

E + pk2

sound modes:

ωsound =1√3

k − i23

E + pk2

The uniform static plasma system is described as a static planar black holeSmall disturbances of the uniform static plasma ≡ small perturbations of theblack hole metric (≡ quasinormal modes (QNM))

g5Dαβ = g5D,black holeαβ + δg5Dαβ (z)e−iωt+ikx

Dispersion relation fixed by linearized Einstein’s equations. Results for thesound channel

from Kovtun,Starinets hep-th/0506184This is equivalent to summing contributions from all-order viscoushydrodynamicsBut, in addition, there is an infinite set of higher QNM — effective degreesof freedom not contained in the hydrodynamic description at all!Romuald A. Janik (Kraków) Thermalization of expanding plasma from AdS/CFT 7 / 26

Einstein’s equations in AdS/CFT

contain all-order viscous hydrodynamic modes (with specific values of alltransport coefficients)

in addition contain the dynamics of genuine nonhydrodynamical modes

incorporate their interactions in a fully nonlinear (and unique) way

Consequence:Einstein’s equations can serve to study nonequilibrium processes in stronglycoupled N = 4 SYM and are an effective tool for exploring physics beyondhydrodynamics

Question:In the case of boost-invariant plasma expansion can we unambigously determinei) whether these nonhydrodynamical modes are really importantorii) whether it would be enough to consider just all-order viscous hydrodynamicmodes

Boost-invariant flow

Bjorken ’83Assume a flow that is invariant underlongitudinal boosts and does not dependon the transverse coordinates.

In a conformal theory, Tµµ = 0 and ∂µTµν = 0 determine, under the above

assumptions, the energy-momentum tensor completely in terms of a singlefunction ε(τ), the energy density at mid-rapidity.

The longitudinal and transverse pressures are then given by

pL = −ε− τ ddτε and pT = ε+

ddτε .

From AdS/CFT one can derive the large τ expansion of ε(τ) forN = 4 plasma

ddτε .

Large τ behaviour of ε(τ)

Current result for large τ : RJ,Peschanski;RJ;RJ,Heller;Heller

ε(τ) =1

212 334

+1 + 2 log 2

+−3 + 2π2 + 24 log 2− 24 log2 2

324 · 2 12 3 141

+. . .

Leading term — perfect fluid behavioursecond term — 1st order viscous hydrodynamicsthird term — 2nd order viscous hydrodynamicsfourth term — 3rd order viscous hydrodynamics...

As we decrease τ more and more dissipation will start to be important

Question: If we start from various initial conditions at τ = 0 when does theabove hydrodynamic form of ε(τ) starts being applicable?

ε(τ) =1

212 334

+1 + 2 log 2

+−3 + 2π2 + 24 log 2− 24 log2 2

324 · 2 12 3 141

+. . .

ε(τ) =1

212 334

+1 + 2 log 2

+−3 + 2π2 + 24 log 2− 24 log2 2

324 · 2 12 3 141

+. . .

ε(τ) =1

212 334

+1 + 2 log 2

+−3 + 2π2 + 24 log 2− 24 log2 2

324 · 2 12 3 141

+. . .

ε(τ) =1

212 334

+1 + 2 log 2

+−3 + 2π2 + 24 log 2− 24 log2 2

324 · 2 12 3 141

+. . .

ε(τ) =1

212 334

+1 + 2 log 2

+−3 + 2π2 + 24 log 2− 24 log2 2

324 · 2 12 3 141

+. . .

ε(τ) =1

212 334

+1 + 2 log 2

+−3 + 2π2 + 24 log 2− 24 log2 2

324 · 2 12 3 141

+. . .

Aim: Study the evolution of ε(τ) all the way from τ = 0 to large τ starting fromvarious initial conditions and investigate the transition to hydrodynamicbehaviour...

Method: Describe the time dependent evolving strongly coupled plasma systemthrough a dual 5D geometry — given e.g. by

ds2 =gµν(xρ, z)dxµdxν + dz2

z2≡ g5Dαβdxαdxβ

i) use Einstein’s equations for the time evolution

Rαβ −12

g5DαβR − 6 g5Dαβ = 0

ii) read off 〈Tµν(xρ)〉 from the numerical metric gµν(xρ, z)

gµν(xρ, z) = ηµν + z4g (4)µν (xρ) + . . . 〈Tµν(xρ)〉 =

N2c2π2· g (4)µν (xρ)

Different setup from [Chesler,Yaffe]: we need evolution from τ = 0, energy-momentumconservation and freedom to consider generic initial conditions

Rαβ −12

Results

We have considered 20+8 initial conditions, each given by a choice of themetric coefficient c(τ = 0, u).

We have chosen quite different looking profiles e.g.

c1(u) = cosh u

c3(u) = 1 +12

c7(u) = 1 +12u2

1 + 32u2

c10(u) = 1 +12

u2e− u2

c15(u) = 1 +12

c19(u) = 1 +12

tanh2(

Results

c1(u) = cosh u

c3(u) = 1 +12

c7(u) = 1 +12u2

1 + 32u2

c10(u) = 1 +12

u2e− u2

c15(u) = 1 +12

c19(u) = 1 +12

tanh2(

Results

c1(u) = cosh u

c3(u) = 1 +12

c7(u) = 1 +12u2

1 + 32u2

c10(u) = 1 +12

u2e− u2

c15(u) = 1 +12

c19(u) = 1 +12

tanh2(

Nonequilibrium vs. hydrodynamic behaviour

Introduce the dimensionless quantity w(τ) ≡ Teff (τ) · τViscous hydrodynamics (up to any order in the gradient expansion) leads toequations of motion of the form

dτw =

Fhydro(w)

where Fhydro(w) is a universal function completely determined in terms of thehydrodynamic transport coefficients (shear viscosity, relaxation time andhigher order ones). For strongly coupled N = 4 plasma it becomes

Fhydro(w)

1− log 227π2w2

+15− 2π2 − 45 log 2 + 24 log2 2

972π3w3+ . . .

Therefore if plasma dynamics would be given by viscous hydrodynamics (evento arbitrary high order) a plot of F (w) ≡ τ

dτ w as a function of w would bea single curve for all the initial conditions

Genuine nonequilibrium dynamics would, in contrast, lead to several curves...

dτw =

Fhydro(w)

1− log 227π2w2

+15− 2π2 − 45 log 2 + 24 log2 2

972π3w3+ . . .

dτw =

Fhydro(w)

1− log 227π2w2

+15− 2π2 − 45 log 2 + 24 log2 2

972π3w3+ . . .

dτw =

Fhydro(w)

1− log 227π2w2

+15− 2π2 − 45 log 2 + 24 log2 2

972π3w3+ . . .

dτw =

Fhydro(w)

1− log 227π2w2

+15− 2π2 − 45 log 2 + 24 log2 2

972π3w3+ . . .

dτw =

Fhydro(w)

1− log 227π2w2

+15− 2π2 − 45 log 2 + 24 log2 2

972π3w3+ . . .

A plot of F (w)/w versus w for various initial data

0 0.2 0.4 0.6 0.8w

F HwLw

An observable sensitive to the details of the dissipative dynamics (e.g.hydrodynamics) is the pressure anisotropy

∆pL ≡ 1− pL

ε/3= 12F (w)− 8

For a perfect fluid ∆pL ≡ 0. For a sample initial profile we get

For w = Teff · τ > 0.63 we get a very good agreement with viscoushydrodynamicsStill sizable deviation from isotropy which is nevertheless completely due toviscous flow.Romuald A. Janik (Kraków) Thermalization of expanding plasma from AdS/CFT 15 / 26

∆pL ≡ 1− pL

ε/3= 12F (w)− 8

∆pL ≡ 1− pL

ε/3= 12F (w)− 8

∆pL ≡ 1− pL

ε/3= 12F (w)− 8

∆pL ≡ 1− pL

ε/3= 12F (w)− 8

∆pL ≡ 1− pL

ε/3= 12F (w)− 8

∆pL ≡ 1− pL

ε/3= 12F (w)− 8

Entropy

The AdS/CFT prescription for 〈Tµν〉 is on a very solid ground in theframework of the AdS/CFT correspondence — in contrast entropy, especiallyfor nonequillibrium systems is much less understood

It is even not clear whether an exact local notion makes sense on the QFTside...

However, phenomenological notion of local entropy density is widely used in(dissipative) hydrodynamicsOn the AdS side entropy is obtained from the area element of a horizon butwe have to choose

the kind of horizon (currently: apparent horizon not event horizon)we have to map a point on the boundary to an appropriate point in the bulk(using null geodesics — but in general there are ambiguities)

For the boost-invariant setup fortunately the null geodesic ambiguities areabsent as well as ambiguities associated with defining the apparent horizon...

Entropy

We consider the entropy per unit rapidity and unit transverse area in units ofinitial temperature introducing a dimensionless entropy density s through

12N2c π2T 2eff (0)

Determine initial entropy from the area of a dynamical horizon at a pointwhere a null geodesic from τ = 0 intersects the horizon

Entropy

12N2c π2T 2eff (0)

Entropy

12N2c π2T 2eff (0)

Entropy

12N2c π2T 2eff (0)

Final entropy

For large proper-time, the dynamics is given by hydrodynamics, leading to thelarge τ expansion

Teff (τ)= Λ

(Λτ)1/3

{1− 16π(Λτ)2/3

+ −1+log 236π2(Λτ)4/3

+ −21+2π2+51 log 2−24 log2 21944π3(Λτ)2+...

We obtain the Λ parameter from a fit to the late time tail of our numericaldata.

Knowing Λ, we may use the standard perfect fluid expression for the entropyat τ =∞

sfinal =Λ2

T 2eff (0)

Final entropy

Teff (τ)= Λ

(Λτ)1/3

{1− 16π(Λτ)2/3

+ −1+log 236π2(Λτ)4/3

+ −21+2π2+51 log 2−24 log2 21944π3(Λτ)2+...

sfinal =Λ2

T 2eff (0)

Final entropy

Teff (τ)= Λ

(Λτ)1/3

{1− 16π(Λτ)2/3

+ −1+log 236π2(Λτ)4/3

+ −21+2π2+51 log 2−24 log2 21944π3(Λτ)2+...

sfinal =Λ2

T 2eff (0)

Final entropy

Teff (τ)= Λ

(Λτ)1/3

{1− 16π(Λτ)2/3

+ −1+log 236π2(Λτ)4/3

+ −21+2π2+51 log 2−24 log2 21944π3(Λτ)2+...

sfinal =Λ2

T 2eff (0)

Entropy production

Consider the entropy production sfinal − sinitial as a function of sinitial

Recall the complicated nonequilibrium dynamics...

Entropy production

0 0.2 0.4 0.6 0.8w

F HwLw

Entropy production

Yet the entropy production depends in surprisingly clean way on sinitial ...

à àà

0 0.15 0.3 0.45sinitial

sfinal-sinitial

The curve is a phenomenological fit

sfinal − sinitial ∼ 1.59 · s1.55initial

Entropy production

Yet the entropy production depends in surprisingly clean way on sinitial ...

à àà

0 0.15 0.3 0.45sinitial

sfinal-sinitial

The curve is a phenomenological fit

sfinal − sinitial ∼ 1.59 · s1.55initial

Observation:

The initial entropy turns out to be a key characterization of the initial state

There seems to be a lot of hidden regularity in the nonequilibrium dynamics

We will show below that the initial entropy also characterizes thecharacteristics of the transition to hydrodynamics (≡ thermalization)

The initial entropy is also strongly correlated with the position of theFefferman-Graham coordinate singularity corresponding to the initial data

Observation:

A numerical criterion for thermalization

We want to study systematically the properties of the plasma at the pointwhen the dynamics becomes describable by viscous hydrodynamics...

We adopted a numerical criterion for thermalization∥∥∥∥∥ τ ddτ w

F 3rd orderhydro (w)

∥∥∥∥∥ < 0.005

We looked at the following features of thermalization:1 the dimensionless quantity w = Teff · τ2 The thermalization time in units of initial temperature τth · Teff (0)3 The temperature at thermalization relative to the initial

temperature Tth/Teff (0)

∥∥∥∥∥ < 0.005

w = Teff · τ at thermalization

w at thermalization is approximately constant and for the initial profilesconsidered does not exceed w = 0.67. It seems to decrease for profiles withsmaller initial entropyN.B. sample initial conditions for hydrodynamics at RHIC (τ0 = 0.25 fm,T0 = 500 MeV ) assumed in [Broniowski, Chojnacki, Florkowski, Kisiel]correspond to w = 0.63The pressure anisotropy at thermalization is still sizable

∆pL ≡ 1− pL

ε/3= 12F (w)− 8 ' 12Fhydro(w)− 8 ∼ 0.72− 0.73

ààà

àà à

à àààà

0 0.15 0.3 0.45sinitial

∆pL ≡ 1− pL

ε/3= 12F (w)− 8 ' 12Fhydro(w)− 8 ∼ 0.72− 0.73

ààà

àà à

à àààà

0 0.15 0.3 0.45sinitial

∆pL ≡ 1− pL

ε/3= 12F (w)− 8 ' 12Fhydro(w)− 8 ∼ 0.72− 0.73

ààà

àà à

à àààà

0 0.15 0.3 0.45sinitial

∆pL ≡ 1− pL

ε/3= 12F (w)− 8 ' 12Fhydro(w)− 8 ∼ 0.72− 0.73

ààà

àà à

à àààà

0 0.15 0.3 0.45sinitial

∆pL ≡ 1− pL

ε/3= 12F (w)− 8 ' 12Fhydro(w)− 8 ∼ 0.72− 0.73

ààà

àà à

à àààà

0 0.15 0.3 0.45sinitial

∆pL ≡ 1− pL

ε/3= 12F (w)− 8 ' 12Fhydro(w)− 8 ∼ 0.72− 0.73

ààà

àà à

à àààà

0 0.15 0.3 0.45sinitial

∆pL ≡ 1− pL

ε/3= 12F (w)− 8 ' 12Fhydro(w)− 8 ∼ 0.72− 0.73

τth · Teff (0) at thermalization

Thermalization time in units of the initial effective temperature Teff (0)

Again we see a clean dependence on the initial entropy sinitial

0 0.15 0.3 0.45sinitial

Τth TeffH0L

0 0.15 0.3 0.45sinitial

Τth TeffH0L

Temperature at thermalization

It is interesting to consider the ratio of the temperature at thermalization tothe initial effective temperatureThis gives information on which part of the cooling process occurs in the farfrom equilibrium regime and which part occurs during the hydrodynamicevolution

Note: for initial profiles with large sinitial , the energy density initially rises andonly then falls −→ even for Tth/Teff (0) ∼ 1 there is still sizablenonequilibrium evolutionFor profiles with small initial entropy most of the cooling is ofa nonequilibrium nature.Romuald A. Janik (Kraków) Thermalization of expanding plasma from AdS/CFT 25 / 26

ààà

0 0.15 0.3 0.45sinitial

Tth � TeffH0L

ààà

0 0.15 0.3 0.45sinitial

Tth � TeffH0L

ààà

0 0.15 0.3 0.45sinitial

Tth � TeffH0L

ààà

0 0.15 0.3 0.45sinitial

Tth � TeffH0L

Conclusions

AdS/CFT provides a very general framework for studying time-dependentdynamical processes

The AdS/CFT methods do not presuppose hydrodynamics so are applicableeven to very out-of-equilibrium configurations

Even though genuine nonequilibrium dynamics is very complicated, weobserved surprising regularities

Initial entropy seems to be a key physical characterization of the initial statedetermining the total entropy production and thermalization time andtemperature

For w = Tth · τth > 0.7 we observe hydrodynamic behaviour but with sizeablepressure anisotropy (described wholly by viscous hydrodynamics)

Conclusions

indico.gsi.de€¦ · Thermalization At weak coupling the obvious deﬁnition would be to require thermal momentum distributions for quarks and gluons... At strong coupling, the picture

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