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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)