The role of plasma instabilities in thermalization fileThe role of plasma instabilities in...

The role of plasma instabilities in thermalization

Aleksi Kurkelawith Guy Moore

Arxiv:

1107.5050thermalization in generic setup,

complete treatment of plasma instabilities

1108.4684 specialized to heavy ion collisions

869 order of magnitude estimates in total

Motivation

In: Out:

Pb+Pb @ few TeV per nucleonAnisotropic yield of ∼ 104

hadrons (”v2”)

Z

⇒Z

Motivation

Model:Hydro flow of nearly thermal fluid:

Collective flow turns spatial anisotropy to momentum anisotropyZ

Motivation

Model:Hydro flow of nearly thermal fluid:

Collective flow turns spatial anisotropy to momentum anisotropyZ

Assumes: Local thermal equilibrium T r .f .µν ≈ diag(e,p,p,p)

Inputs:

Equation of state, viscosities. . .Initial geometryThermalization time τ0 ∼ 0.4 . . . 1.2fm/c

MotivationObjective:

Why is τ0 . 0.4 . . . 1.2fm/c ???What happens before? Maybe observable in experiment

Method:

Extremely big: Nnucl → ∞Extremely high energy:

√s → ∞

weak coupling: αs ≪ 1⇒ Separation of scales: Kinetic theory, Hard loops, Vlasov equations,. . .Might be still non-perturbative (αf ≷ 1)

Purely parametric: counting powers of αs

Warning: all scales logarthmic

MotivationObjective:

Why is τ0 . 0.4 . . . 1.2fm/c ???What happens before? Maybe observable in experiment

Method:

Extremely big: Nnucl → ∞Extremely high energy:

√s → ∞

weak coupling: αs ≪ 1⇒ Separation of scales: Kinetic theory, Hard loops, Vlasov equations,. . .Might be still non-perturbative (αf ≷ 1)

Purely parametric: counting powers of αs

Objective of this talk: What are the relevant physical process that lead tothermalization in a HIC?

Warning: all scales logarthmic

Initial condition: t ∼ Q−1s

At weak coupling, well understood: Color Glass Condensate

Characteristic, saturation scale: Qs

High occupancy: f (p < Qs) ∼ 1/α (need something to cancel α in σlarge angle!)





Anisotropic: pz ≡ δpt ≪ pt

Initial energy density: ε(t ∼ Q−1s ) ∼ α−1Q4

s







s

Distribution of gluons: f (p) ∼ α−cθ(Qs − p)θ( αd︸︷︷︸

δ

Qs − pz)







s

Distribution of gluons: f (p) ∼ α−cθ(Qs − p)θ( αd︸︷︷︸

δ

Qs − pz)

t

Q

δQ

f(p)

z

p

p

-1 1

0

1

Initial condition

d=ln(δ)/ln(α)

c=-ln(f)/ln(α)f~α−1

High occupancyLow occupancy

High anisotropy

Low anisotropy

f~1f~α

Out of equilibrium systems: descriptors

-1 0 1

-1

0

1 d

c

f~α−1

High anisotropy:

f (p) ∼ α−cθ(Qs−p)θ(αdQs−pz)

Small anisotropy:

f (p) ∼ α−c(1 + α−dF (p))

Longitudinal expansionSpatial expansion translates into redshift in pz ∼ δQs

Changes only pz , Qs stays constant

Changes δ = αd

If pz ≪ pt , ε(t) ∼ α−1Q4s /(Qs t)

t

Q

δQ

f(p)

z

p

p ⇒

t

Q

δQ

zp

p

,

Longitudinal expansion

Spatial expansion translates into redshift in pz ∼ δQs


Changes δ = αd


-1 0 1

0

1

Initial condition

d

cf~α−1

t~α−1t~α−5/2 t~α−3/2

Longitudinal expansion

Spatial expansion translates into redshift in pz ∼ δQs


Changes δ = αd


-1 0 1

0

1

Initial condition

d

cf~α−1

t~α−1t~α−5/2 t~α−3/2

Free streaming

Scattering: Elastic

Elastic scattering makes the distribution fluffier

Q

δQ

pt

Q

δQ

f(p)

zpz

p

p

t

-1 0 1

0

1

Initial condition

d

cf~α−1

t~α−1t~α−5/2 t~α−3/2

Free streamingelastic

Along the attractive solution scattering and expansion compete

When typical occupancies f ≪ 1: loss of Bose enhancement

At late times: Fixed anisotropy, dilute away

Scattering: ElasticElastic scattering makes the distribution fluffier

Q

δQ

pt

Q

δQ

f(p)

zpz

p

p

t

-1 0 1

0

1

Initial condition

d

cf~α−1

t~α−1t~α−5/2 t~α−3/2





Elastic scattering not enough for thermalization!

Scattering: ElasticElastic scattering makes the distribution fluffier

Q

δQ

pt

Q

δQ

f(p)

zpz

p

p

t

-1 0 1

0

1

Initial condition

d

cf~α−1

t~α−1t~α−5/2 t~α−3/2


Something completelydifferent




Elastic scattering not enough for thermalization!

Scattering: Inelastic

Inelastic scattering plays two significant roles (Baier, Mueller, Schiff & Son 2000)

1 soft splitting: creation of a soft thermal bath

2 hard splitting: breaking of the hard particles

Scattering: Inelastic Baier, Mueller, Schiff & Son 2000

Soft splitting: Creation of soft thermal bath

Q

δQ

pt

Q

δQ

zp

Tz

p

p

t

Soft modes quick to emitns ∼ αncol

Low p: easy to bend⇒thermalize quickly

Can dominate dynamics!

(i.e. scattering, screening, . . . )

Scattering: Inelastic Baier, Mueller, Schiff & Son 2000

Hard splitting: Qs modes break before they bend!

Q

t (Q)split

t (Q/2)split

t (Q/4)

. . .

. . .

. . .

. . .

TQ/4Q/2

split

In vacuum: on-shell particles, no splittingIn medium: Particles receive small kicks frequently

For stochastic uncorrelated kicks: Brownian motion in p-space

∆p2⊥∼ qt, tsplit(k) ∼ α

√

q/k︸︷︷︸

tform

(LPM)

Momentum diffusion coefficient q describes how the medium wiggles ahard parton

Bottom-Up Baier, Mueller, Schiff & Son 2000

Thermal bath eats the hard particles away:

δQ

pt

Q

δQ

zp

Tzp

k

p

split

t

QScales below ksplit havecascaded down to T -bath

tsplit(ksplit) ∼ t ⇒ ksplit ∼ α2qt2

”Falling” particles heat up thethermal bath

T 4 ∼ ksplit

∫

d3pf (p)

Thermalization when Qs getseaten

ksplit ∼ Qs

Needs q as an input

Old bottom up: Baier, Mueller, Schiff & Son 2000

q dominated by elastic scattering with thermal bath??

BMSS assumed: q ∼ qelastic ∼ α2T 3

Solve self-consistently:

ksplit ∼ α2qt2

T 4 ∼ ksplit(Q3s /(Qs t))

q ∼ α2T 3

for:

⇒

T ∼ α3Qs(Qs t)ksplit ∼ α13(Qst)

5Q

τ0 ∼ α−13/5Q−1s

δQ

pt

Q

δQ

zp

Tzp

k

p

split

t

Q

But: Is this all there is?

q dominated (always!) by

plasma instabilities

Plasma instabilities: Idea

Exponential growth of (chromo)-magnetic fields in anisotropic plasmas

+−+−+−+−+−+−+−+−+−+−+−+−+−+−+−+−+−+−

1/k

+−+−+−+−+−+−+−+−+−+−+−+−+−+−+−+−+−+−

How do particles deflect?


Exponential growth of (chromo)-magnetic fields in anisotropic plasmas

+−+−+−+−+−+−+−+−+−+−+−+−+−+−+−+−+−+−

Net J

Net J

Net J

Net J

+−+−+−+−+−+−+−+−+−+−+−+−+−+−+−+−+−+−

Induced current feeds the magnetic field


When chromo-magnetic field becomes strong enough to mix colors,currents no longer ”feed” the magnetic field. Growth is cut off.

No Blue Current

No Blue Current

Plasma instabilities: Slightly more quantitive

Lorentz force: F ∼ gB

Displacement: δz ∼ gBt2/p

−−+−+−+−+−+−+−+−+−+−+−

1/k

+−+−+−+−+−+−+−+−+−+−+−+−+−+−+−+−+−+−

x

z

F

t

zδ

+−+−+−+−+−+−++

Plasma instabilites: Slightly more quantitive


Dislocation: δz ∼ gBt2/p

Current:

J ∼ g︸︷︷︸

charge

∫

d3pf (p)

︸︷︷︸

# of part.s

[δz k]︸︷︷︸

disp. fraction

∼ kBt2 α

∫d3p

pf (p)

︸︷︷︸

m2

∼ kBm2t2




Current:

J ∼ g︸︷︷︸

charge

∫

d3pf (p)

︸︷︷︸

# of part.s

[δz k]︸︷︷︸

disp. fraction

∼ kBt2 α

∫d3p

pf (p)

︸︷︷︸

m2

∼ kBm2t2

Competes in Maxwell’s equation with ∇× B ∼ k B

⇒ Exponential growth if m2t2 > 1

⇒{

k instz ∼ any

k instx < m




Current:

J ∼ g︸︷︷︸

charge

∫

d3pf (p)

︸︷︷︸

# of part.s

[δz k]︸︷︷︸

disp. fraction

∼ kBt2 α

∫d3p

pf (p)

︸︷︷︸

m2

∼ kBm2t2

Competes in Maxwell’s equation with ∇× B ∼ k B

⇒ Exponential growth if m2t2 > 1

⇒{

k instz ∼ any

k instx < m

Saturation when competition in Dµ = kinst + igAµ

⇒ A ∼ kinst/g , or B ∼ kinstA ∼ k2inst/g , or f (kinst) ∼ 1/α

Don’t take my word for it.

50 100 150

m0 t

10-7

10-6

10-5

10-4

10-3

10-2

10-1

ener

gy d

ensi

ty /

g-2m

04

m0 = 1/a

m0 = 0.77/a

m0 = 0.55/a

m0 = 0.45/a

E2/2

B2/2

k*

4/4

Lasym = 4 Lmax = 16

Bodeker and Rummukainen (0705.0180) and many others. . .

Plasma instabilites: More complicated distribtutions

Strong anisotropy:

f (~p) ∼ α−cΘ(Qs − p)Θ(δQs − pz)

t

Q

m/δzp

δQ

p

m

Weak anisotropy:

f (~p) ∼ f0(|~p|)(1 + ǫF (~p))

ε1/2mzp

p

εt

Plasma instabilites: Momentum transfer

Hard parton traveling through magnetic fields receives coherent kicks frompatches of same-sign magnetic fields

−1l coh ~kinst

∆pkick ∼ g B lcoh

q t ∼ Nkick︸︷︷︸

t/lcoh

(∆pkick)2

q ∼ α lcoh B2

Weak anisotropy: qǫ ∼ ǫ3/2m3

Strong anisotropy: qδ ∼ m3/δ2

The new bottom-up

Important physics:

1 soft splitting: creation of a soft thermal bath

2 hard splitting: breaking of the hard particles

3 Plasma instabilities, screening

In particular, elastic scattering irrelevant

The new bottom-up: Early stages: Qst < α−12/5

Broadening of the hard particle distribution dominated by the plasmainstabilities (qel ≪ qinst) originating from the scale Qs

t

Q

m/δzp

δQ

p

m

-1 0 1

0

1

Initial condition

d

cf~α−1

t~α−1t~α−5/2 t~α−3/2


Instabilities

Formation of thermal cloud



t

Q

m/δzp

δQ

p

m

-1 0 1

0

1

Initial condition

d

cf~α−1

t~α−1t~α−5/2 t~α−3/2


Instabilities


Even the instabilities are not strong enough to thermalize when f ≫ 1⇒ classical theory does not thermalize under longitudinal expansion!



t

Q

m/δzp

δQ

p

m

-1 0 1

0

1

Initial condition

d

cf~α−1

t~α−1t~α−5/2 t~α−3/2


Instabilities


qδ ∼ m3/δ2, m2 ∼ α δ f Q2s , δ ∼ pz/Qs ∼

√

qδt/Qs , f ∼ 1/(αδQs t)

⇒ δ ∼ (Qst)−1/8, f (Qs) ∼ (Qs t)

−7/8, qδ ∼ Q3s (Qst)

−5/4

The new bottom-up: Late stages: α−12/5

< Qt < α−5/2:

Q

δQ

pt

Q

δQ

T

t

p

p

zpz

Splitting creates a soft T-bath


< Qt < α−5/2:

δQ

pt

Q

δQ

T

t

Q

ε

p

zpzp


The expansion and anisotropic rainmake T-bath anisotropicǫ ∼ t/tiso ∼ qt/T 2


< Qt < α−5/2:

δQ

pt

Q

δQ

m

t

Q

ε

p

zpzp

T ε1/2



Anisotropic T-bath generates itsown instabilities. Dominatesqǫ ∼ ǫ3/2m3 for (Qt) > α−12/5


< Qt < α−5/2:

t

Q

δQ

pt

Q

δQ

pt

Q

δQ

zp

Tzp

ksplit

t

Q

δQ

p

zp

ε1/2

ε

m

p

zp

T




Magnetic field induced splitting eatsthe hard particles: ksplit ∼ α2qǫt

2


< Qt < α−5/2:

t

Q

δQ

pt

Q

δQ

pt

Q

δQ

zp

Tzp

ksplit

t

Q

δQ

p

zp

ε1/2

ε

m

p

zp

T





2

q ∼ α3Q3s

T ∼ αQs(Qs t)1/4

ksplit ∼ α5(Qt)2Qs

τ0 ∼ α−5/2Q−1s


< Qt < α−5/2:

tpt

m

ε

p

zpzp

T ε1/2





2

q ∼ α3Q3s

T ∼ αQs(Qs t)1/4

ksplit ∼ α5(Qt)2Qs

τ0 ∼ α−5/2Q−1s

At (Qs t) ∼ α−5/2, ksplit ∼ Qs .Plasma instabilities continue todominate until (Qst) ∼ α−45/16.

Summary, complete catalog:

Nielsen−OlesenInstabilities

Occupancy

Broadening: non−HTL,unsaturated plasma instabilities

2

3b

Plasma instabilitiesPlay No Role

Anisotropy

α

α

c=−ln(f)/ln( )α 1d=−1/5

d=−1/7

d=−1/3

d=1/3

c=−1/3c=−1

c=1

slope

1

slope −1/3

slope −1/3

slope −1/3slope −3/8

d=ln( )/ln( )δ

d=−ln( )/ln( )ε

instabilities saturated plasma

Broadening:

5

6

9:

7:

8:

slope −9/23

3a 4a

4:

4b

We identified the relevant physics in occupancy-anisotropy plane...

Summary, complete (parametric) description of HIC:0

8/7

16/1

1

96/5

5

56/2

516

0/69

12/5 5/2

45/1

6

56/4

1

Time: a=ln(Qt)/ln(α−1)

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

Mom

entu

m: l

n(p/

Q)/

ln(α

−1)

C

B

A

D

E

F G I

H

J K O T

S

NR

M

Q

PL

a/4

3a/16 (24-9a)/64

(144-57a)/136

(16+5a)/64(184+15a)/544

(24-a)/68

(104

-41a

)/28

5-2a

(15-

6a)/1

6

(36-

15a)

/14

(52-3a)/112

1-a/4

(15a-8)/841/3

(8a-11)/24

(-56+73a)/128

pmax

kiso

kold

Tk

new-inst

ksplit

Featuring:

Obscure powers:α

184+15a554

Redshifted relics

Synchrotronabsorption

Complicated angledependence

. . . and many more

...applied to the case of longitudinal expansion.

Conclusions:

Thermalization occurs even for α ≪ 1

Outlook:

Conclusions:


Thermalization happens ”bottom up”.

Outlook:

Conclusions:



Anisotropic screening leads (always) to plasma instabilities

Outlook:

Conclusions:




In HIC: Plasma instabilities dominate dynamics! Lead to fasterthermalization

Outlook:

Conclusions:





Stronger dynamics lead to faster thermalization α−5/2 < α−13/5

Outlook:

Conclusions:






Outlook:

Here, treatment was purely parametric:Numerical treatment underway τ0 ∼ #α−5/2

Three-scale problem: Three different treatments of d.o.f.’s on lattice

Conclusions:






Outlook:



”Beyond” the leading order?

Thermalization as quenching of jets with p ∼ Qs .Hard particles perturbative, medium non-perturbative?

Conclusions:






Outlook:



”Beyond” the leading order?

Thermalization as quenching of jets with p ∼ Qs .Hard particles perturbative, medium non-perturbative?

Apply same methods to

Cosmology, reheatingNeutron stars, anomalous viscosities

Second attractor?


αδd=ln( )/ln( )

Occupancyc=−ln(f)/ln( )α

d=−1/7

d=−1/3

c=−1/3

8:

slope −9/23

Anisotropy

slope

1

c=1

InitialCondition

Attractor 2

Attractor 1

Bath Forms

Soft Particle

d=−ln( )/ln( )ε α Scattering

Plasma Instabilities

f (p) ∼ α−c (1 + α−dF (p))

Second attractor?Assume nearly isotropic distribution:

Longitudinal expansion reduces energy: ε ∝ t−4/3

pz reduces.

Plasma-instabilities keep system nearly isotropic ǫ ∼ tiso/t ∼ Q2

qǫt

Q reduces

Instability induced particle joining at hard scale

Q increases

ǫ ∼ α−d ∼ (Qst)−

8135 , f (Q) ∼ α−1(Qst)

56135 ,Q ∼ Qs(Qs t)

−

31135

Why do we think the first attractor is the relevant one?

(1/ )αc=ln(f) /ln

Condition


Scattering

PlasmaInstabilities

InstabilitiesDevelop

Plasma

Insta

bilitie

s

Plasma

Act

Attractor 2

Attractor 1

Initialt

z

p

p

Initially, unstable modes in unpopulated part of phase space

Growth from vacuum fluctuations, slowed by a log:tgrowth ∼ Q−1

s ln2(1/α)

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