Nonequilibrium Renormalization Group J. Berges Institut für Kernphysik Technische Universität Darmstadt Technische Universität Darmstadt Schladming lectures, 26.2. – 5.3.2011 C t t Content • Introduction: thermal vs. nonthermal fixed points • Scaling behavior far from equilibrium: Basics of stationary transport Scaling 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
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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
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)