Nonequilibrium Renormalization Group

J. Bergesg

Institut für KernphysikTechnische Universität DarmstadtTechnische Universität Darmstadt

Schladming lectures, 26.2. – 5.3.2011

C t tContent

• Introduction: thermal vs. nonthermal fixed points

• Scaling behavior far from equilibrium: Basics of stationary transportScaling behavior far from equilibrium: Basics of stationary transport

• Nonequilibrium functional renormalization group

• Solving truncated flow equations for self-energies

• Nonthermal fixed points: strong vs. weak wave turbulence

N ilib i i iti l l bl• Nonequilibrium initial value problems

Nonequilibrium initial value problemsThermalization process in quantum many-body systems?

Schematically:Schematically:

• Characteristic nonequilibrium time scales? Relaxation? Instabilities?

Di i ti l f f ilib i ? N th l fi d i t ?• Diverging time scales far from equilibrium? Nonthermal fixed points?

Universality far from equilibriumy q

Nonequilibrium instabilitiesØThomas Gasenzer,

this schoolNonequilibrium instabilities this school

Nonthermal fixed points

ö nonequilibrium properties independent of details of microscopic theory

Heating the Universe after inflation:t la quantum example

Schematic evolution:

(numbers ‘‘illustrative‘‘)

• Energy density of matter (~ a-3) and radiation (~ a-4) decreases

• Enormous heating after inflation to get ‘hot-big-bang‘ cosmology!

Parametric resonance instabilityM h i Kofman, Linde, Starobinsky, Phys. Rev. Lett. 73 (1994) 3195

E.g. scalar N-component λΦ4 inflaton:

Mechanism:

Fi ld t ti l φ• Field expectation value φ = ‚ΦÚ

• Fluctuation F ~ ‚{Φ,Φ}Ú

fast slow

Instability: F(t) ~ eγ t (γ > 0)

Quantum field theory:

fast slow

mbe

r‘

Berges Serreau

tion

numBerges, Serreau,

Phys. Rev. Lett. 91 (2003) 111601

Method:

Occ

upa

Berges, Nucl. Phys. A 699 (2002) 847

2PI 1/N expansion to NLO

‘O

t M

Model & Approximation• Scalar fields with quartic self interaction l and coupling to N = 2 massless• Scalar fields with quartic self-interaction l and coupling to Nf = 2 masslessDirac fermions ( symmetry group )

• 2PI effective action G:

NLO in the number Ns = 4 of

boson propagator G(x,y)

s

inflaton field components

field expectationO(g2)

fermion propagator D(x,y)

pvalue f(x)

corresponding to self-consistently dressed self-energies S:

etc.

Time evolution equationst ti ti l t ‚{Φ Φ}Ú t l f ti ‚[Φ Φ]Ústatistical propagator ~ ‚{Φ,Φ}Ú spectral function ~ ‚[Φ,Φ]Ú

Nonequilibrium:

Equilibrium/Vacuum: (fluct.-diss. relation)

slowNonperturbative: saturatedoccupation numbers ~ 1/λ

slow

pØ all processes O(1)

Ø universal

N li t b ti

fast

secondary growth rates

Nonlinear – perturbative:occupation numbers < 1/λ

secondary growth ratesc (2γ0) with c = 2,3,…

Classical/linear:primary growth rate

φ ~ (N / λ )1/2

Slow: Nonthermal fixed pointsscale invariance after parametric resonance

infrared fixed point

universal scaling‘ultraviolet‘ fixed point

(weak turbulence)

B R thk f S h idt

universal scalingexponent

(weak turbulence)

Berges, Rothkopf, Schmidt, Phys. Rev. Lett. 101 (2008) 041603

Ø - 0 + 1 + 3 = 4 for d = 3n(t,p) ~ p-κ with

Berges, Hoffmeister, Nucl. Phys. B 813 (2009) 383

Classical-statistical simulations for inflaton dynamics: Khlebnikov, Tkachev ‘96;

Comparing quantum to classicalClassical statistical simulations for inflaton dynamics: Khlebnikov, Tkachev 96; Prokopec,Roos ‘97; Tkachev, Khlebnikov, Kofman, Linde ’98; …

pBerges, Rothkopf, Schmidt, PRL 101 (2008) 041603

Practically no scalar quantum corrections at the end of preheating

Accurate nonperturbative description by (2PI) 1/N to NLO

act ca y o sca a qua tu co ect o s at t e e d o p e eat g

Test: Variation of the dimensionality of space to d = 4

with Denes SextyarXiv:1012.5944arXiv:1012.5944

IR: Well characterized by nonperturbative infrared κ = d + 1 Ø 5

‘UV‘: Perturbative Kolmogorov turbulence exponent κ = d – 3/2 Ø 5/2

with z = 1, h = 0

Nonthermal fixed points from largeclass of nonequilibrium instabilities

E.g. spinodal/tachyonic instability: classical-statistical simulation (quench)E.g. spinodal/tachyonic instability:

eff. potential

classical statistical simulation (quench)

κIR

φ

κUV

t=6000

t=500 t=1000 (Δt=1000)

with Pruschke, Rothkopf

IR fixed point rather stable, while UV (κUV = 1) already ‘thermalized‘

Fermion dynamics

LOLO:

Baacke, Heitmann, Pätzold, PRD 58 (1998) 125013; Greene, Kofman, PLB 448 (1999) 6; Giudice Peloso Riotto Tkachev JHEP 9908 (1999) 014; Garcia Bellido Mollerach Roulet

t

NLOBoson

small self coupling l leads

Giudice, Peloso, Riotto, Tkachev, JHEP 9908 (1999) 014; Garcia-Bellido, Mollerach, Roulet, JHEP 0002 (2000) 034; …

NLO:Fermion

small self‐coupling l leadsto large corrections!

+

Berges Pruschke Rothkopf PRD 80 (2009) 023522; Berges Gelfand Pruschke arXiv:1012 4632

Parametric resonancepreheating

Berges, Pruschke, Rothkopf, PRD 80 (2009) 023522; Berges, Gelfand, Pruschke, arXiv:1012.4632

my = 0

x ª g2•l

Characteristic time scales

parametric resonance nonthermal fixed pointBosons:

rate: g0 scaling behavior

tf tFermions: instability-induced

fermion productionfermion productionby inflaton decayp y y

rate: g0 rate: gy ~ (g2/l) f0

maximally amplified mode

Strongly enhanced fermion production

Fermions

IR fermions thermally occupied

BosonsBosons

Bosons still far from equilibrium!

~1/p4

nonthermal fixed point

Summarizing:

nonthermal fixed pointn(t,p) ~ egt

n(t,p) ~ p-κ

nonequilibriuminstabilities

Δtinitial

conditions

thermal equilibrium

ΔtnBEn(t=0,p)

• approached from substantial class of initial conditions (no fine tuning!)

Nonthermal fixed points:

• critical slowing down can substantially delay thermalization

• properties independent of details of the underlying microscopic theory

Fermions:

• quantum theory of preheating predicts strongly enhanced fermion production(thermally occupied in the IR while bosons are still far from equilibrium)

• strongly coupled (x~1) fermions required to speed-up thermalization of bosons

(thermally occupied in the IR while bosons are still far from equilibrium)

Nonequilibrium QCD

Relativistic heavy-ion collisions explore strong interaction matter startingfrom a transient nonequilibrium state

Short-time dynamicsA i t f th t t T i l l t f

oblate anisotropy : Txx à Tyy ~ Tzz

• Anisotropy of the stress tensor Tij in a local rest frame:

Isotropization time tiso? In the absence of nonequilibrium instabilities:

4tiso~ O(1/g4T)

• Weibel instability:

characteristic momentum of typical excitation

• Weibel instability:Weibel ’59; … Mrowczynski ’88, ’93, ’94; Arnold, Lenaghan, Moore ‘03; Romatschke, Strickland ‘03; very many since then…

(f )

x (current)

tiso ~ O(1/gT)

• Nielsen-Olesen instability:

z (force)

y (magnetic field)

Nielsen-Olesen instability:

ti ~ O(1/g1/2B1/2)

Nielsen, Olesen ’78; Chang, Weiss ’79; … Iwasaki ’08; Fujii, Itakura ’08 …B z

tiso O(1/g B )

…“homogeneous“ background field

Classical-statistical lattice gauge theory

Wilson action:

Here: β = β0 / γ = βs γ = 4, axial-temporal/Coulomb gauge

Normalized Gaussian probability functionalInitial conditions:

⟨A(t) A(t‘)Ú = ∫ DA(0) D∂tA(0) P[A(0),∂tA(0)] A(t) A(t‘)

Δ

with Δà Δz (extreme anisotropy)kT

C is adjusted to obtain a given energy density ε

Characteristic time scales

fast slow

Δ/ε1/4 = 1

primary

secondary growth rates

primary (Coulomb gauge)

Inverse primary growth rates:

Berges, Scheffler, Sexty, PRD 77 (2008) 034504 (SU(2)); + Gelfand, PLB 677 (2009) 210 (SU(3))

e.g. ¶RHIC ~ 5‐25 GeV/fm3, ¶LHC ~ 2 x ¶RHICp y g g RHIC , LHC RHIC

fast:fast:

Slow: Turbulence( )analytical (2PI)

B S h ffl S t PLB 681 (2009) 362Berges, Scheffler, Sexty, PLB 681 (2009) 362

• Scaling exponent κ close to the perturbative value κ = 4/3 See however: Arnold, Moore PRD 73 (2006) 025006; Mueller, Shoshi, Wong, NPB 760 (2007) 145

• Different infrared behavior? Nonthermal IR fixed point?

(Infrared occupation number ~ 1/g2 Ø strongly correlated)

Quark-Gluon-PlasmaInflaton( 200 MeV)( 1016 GeV) ( 200 MeV)( 1016 GeV)

Universality far from Equilibrium

k = 1.35≤ 0.02

compare also:co pa e a so

Turbulence in cold quantum gasesTurbulence in cold quantum gases

Thomas Gasenzer, this school,