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TU Vienna
WMAP Science Team
ALICE/CERN
J. Berges
Heidelberg University
Workshop on non-equilibrium physics and holography, July 2016
Universality far from equilibrium
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Content
• Thermalization dynamics in isolated quantum systems
• Universality far from equilibrium: Nonthermal fixed points
in early-universe cosmology, relativistic heavy-ion collisions,
and ultracold quantum gases
• Infrared cascade and dynamics of Bose condensation
• Some puzzles and challenges
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Universality in thermal equilibrium
Example: Schematic phase diagram of
strong interaction matter (QCD)
Universal critical exponents: e.g.
order parameter (Tc – T)β
Ising universality class (d=3)Quark chemical potential [GeV]
typical liquid-gas system
Fine-tuning of critical parameter(s) to observe universal behavior
(relevant operators for renormalization group fixed points)
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Universality far from equilibrium
Example: Schematic thermalization for isolated quantum systems
No fine-tuning to observe far-from-equilibrium universal behavior
(irrelevant operators for nonthermal renormalization group fixed points)
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Extreme conditions
Dimensionless combination of
coupling strengthfield2 expectation value (vacuum/
thermal equilibrium/nonequilibrium)
characteristic energy/momentum2
Extreme conditions can enhance the loss of details about
microscopic properties (coupling strengths, initial conditions, …)
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Relativistic heavy-ion collision experiments
Early-universe inflaton dynamics
Table-top experiments with ultracold atoms
Preheating after inflation (~1016 GeV)
Quark-Gluon Plasma (~100 MeV ~1012 K)
Strong quenches at nanokelvins
WMAP Science Team
TU Vienna
Isolated quantum systems in extreme conditions
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Inflation
Quantum
fluctuations
WMAP Science Team
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nonequilibrium
instability
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Macroscopic fields, condensates and fluctuations
In a quantum theory the field amplitude corresponds to the expectation
value of a (here relativistic, real) Heisenberg field operator
density operator at some
`initial´ time t = 0
time-dependent
expectation value
occupation
number
distributiondispersion
Fluctuations derive from correlation functions, e.g. spatially homogeneous:
volume:`quantum-half´
condensate2
0
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Preheating: Insensitivity to initial condition details
Example: `Inflaton´ 4 theory ( 1),
Be
rge
s, B
og
usla
vski, S
ch
lich
tin
g,
Venugopala
n, JH
EP
1405 (
2014)
054
instability
1. Large initial field: 2. High occupancy:
time
resc
ale
dfi
eld
1/2
0/Q
initial over-
occupation
large initial field
1.
2.
classical-statistical simulation
(weak-coupling limit)
approach to
nonthermal fixed point
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Nonthermal fixed point: Insensitivity to coupling strength
Universal scaling behavior
for wide range of couplings!
E.g. scalar N-component 4 quantum theory (1/N to NLO 2PI):
= 0.1
= 1
= 0.01
f
p-5
p-5
p-5
Berges, Wallisch, arXiv:1607.02160
Occupation number distribution:
momentum
(similar results for strong-
field initial conditions)
N = 4
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Schematic behavior near nonthermal fixed point: dual cascade
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number conservation: energy conservation:
Particles are transported towards lower scales, energy towards higher scales
Particle versus energy transport
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Self-similarity
Scaling exponents and determine rate and direction of transport:
Time-independent scaling function:
e.g. 0, 0: particle transport towards lower momentum scales
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Self-similar dynamics: infrared scaling
Piñ
eir
oO
rioli,
Bog
usla
vski, B
erg
es,
PR
D 9
2 (
2015)
025041
,
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Mass scale separating non-relativistic infrared regime
non-relativistic infrared dynamics expected because of the
generation of a mass gap (condensate + medium)
relativistic & non-relativstic field theories have same infrared scaling
Piñeiro Orioli, Boguslavski, Berges, PRD 92 (2015) 025041
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E.g. Gross-Pitaevskii equation for dilute Bose gas:
Non-relativistic system: Dilute quantum gases
Interacting bosons with s-wave scattering length a,
interatomic distance
`diluteness‘ parameter:
`inverse coherence length‘:
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E.g. in the mean-field approximation for a spatially homogeneous system
without condensate the interaction term leads to a constant energy shift:
occupation number
Dilute bose gas in extreme conditions
For the overoccupied Bose gas with
the mean-field shift in energy is of the same order as the relevant
kinetic energy irrespective of the coupling g, since
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Universal scaling form of the distribution function
,,
,
Piñ
eir
oO
rioli,
Bog
usla
vski, B
erg
es,
PR
D 9
2 (
2015)
025041
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Estimating scaling properties
Use
Time-independent fixed point equation:
+ scaling relation:
`collision integral´
Micha, Tkachev, PRD 70
(2004) 043538
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Conservation laws
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Perturbative estimate (Gross-Pitaevskii)
Using:
gives scaling relation:
Negative perturbative exponents do not account for inverse particle cascade!
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Beyond perturbation theory: large-N expansion to NLO
based on Berges Nucl. Phys. A 699 (2002) 847; Aarts et al. Phys. Rev. D 66 (2002) 045008
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Vertex correction (NLO 1/N)
scaling behavior:
, in the infrared:
one-loop retarded self-energy:
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gives scaling relation:
Scaling solution at NLO 1/N
Positive nonperturbative exponents can describe inverse cascade!
Piñeiro Orioli, Boguslavski, Berges, PRD 92 (2015) 025041
NLO result in good agreement with full numerical simulation
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Self-similar dynamics from classical-statistical simulations
Piñ
eir
oO
rioli,
Bog
usla
vski, B
erg
es,
PR
D
92 (
2015)
02
50
41
,
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Condensation far from equilibrium
volume:
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Condensation time
Analytic estimates
agree well with
simulations!
,
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Small initial (vacuum) fluctuations:
Large initial gauge fields:
CGC: Lappi, McLerran, Dusling, Gelis, Venugopalan, Epelbaum…
plasma instabilities
Mrowczynski; Rebhan, Romatschke, Strickland; Arnold, Moore, Yaffe; Bödecker; Attems, …
Romatschke, Venugopalan; J.B., Scheffler, Schlichting, Sexty; Fukushima, Gelis …
Heavy-ion collisions in the high-energy limit
JB, Schenke, Schlichting, Venugopalan, NPA931 (2014) 348 for initial spectrum from Epelbaum,
Gelis, PRD88 (2013) 085015. Plasma instabilities from wide range of initial conditions:
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Overoccupied non-Abelian plasma
(controls “prolateness” or “oblateness”
of initial momentum distribution)
p
occupancy parameter
anisotropy parameter
• To discuss attractor: Initial overoccupation described by family of distributions at τ0
(read-out in Coulomb gauge)
~1/g2
Q
S
J.B., Boguslavski, Schlichting, Venugopalan, PRD89 (2014) 074011; 114007; PRL114 (2015) 061601
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Nonthermal fixed point
`Bottom-up´* scaling emerges as a consequence of the fixed point!
Evolution in the `anisotropy-occupancy plane´
Lattice data
*Baier et al, PLB 502 (2001) 51
J.B
., B
og
usla
vski, S
ch
lich
tin
g, V
en
ug
op
ala
n,
PR
D 8
9 (
20
14
) 0
74
011
; ib
id. 11
40
07
(25624096)
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Self-similar evolution
Scaling exponents:
and scaling distribution function fS:
stationary fixed-point distribution
τ/τ0 = 7.5 – 30
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Comparing gauge and scalar field theorieswith longitudinal expansion
1/pT
For gauge & scalar fields: Inertial range of thermal-like transverse
spectrum ~1/pT even as longitudinal distribution is being `squeezed´
p-5
1/pT
Strongly enhanced infrared regime for scalars: inverse particle cascade
leading to Bose condensation, as in isotropic superfluid turbulence
scalar 4
At latest available times for scalars a flat distribution for emerges
J.B., Boguslavski, Schlichting, Venugopalan, Phys. Rev. Lett. 114 (2015) 6, 061601
‘Coulomb gauge‘
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Universality far from equilibrium
Same universal exponents and scaling function in 1/pT inertial range
gluon distribution: scalar 4:
un
ivers
al
scalin
gfu
ncti
on
:
rescaled momentum:
Remarkably large universality class far from equilibrium!
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Some puzzles and challenges
The scaling solution seems well understood for the gauge theory (BMSS)
– but the corresponding kinetic theory arguments fail for the scalar theory
However: No dominance for small angle scattering in scalar theory!
Much more general principle underlying nonthermal fixed point?
for gauge theory
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In the weak-coupling limit, kinetic theory is expected to have an
overlapping range of validity with classical-statistical simulations
f(p) ~ 1/g2 1/g2 > f(p) > 1 f(p) < 1
kinetic theory
classical-statistical lattice gauge theory
(quantum)(classical particle)(‘overoccupied‘)
However, kinetic theory cannot reproduce important quantities such as
PL / PT characterizing isotropization of the longitudinally expanding plasma:
Scaling behavior of PL / PT
the same for scalar and
gauge field simulations!
In scalar theory behavior
of PL / PT known to arise
from infrared contributions
(Bose condensation)
Nonperturbative despite
(weak) coupling parameter
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Perturbative estimates extrapolated beyond the weak-coupling
regime suggest the absence of transient universal scaling behavior
Gauge: Kurkela, Zhu, PRL 115 (2015) 182301; Scalar: Epelbaum, Gelis, Jeon,
Moore, Wu, JHEP 09 (2015) 117
However, no such indications (yet no expansion) from nonperturbative
estimates in scalar quantum field theory
(and holographic superfluids?)
JB, Wallisch, arXiv:1607.02160
Ewerz et al., JHEP 18 (2015) 1505 (cf. also
Adams, Chesler, Liu, Science 341 (2013) 368)
By now, detailed understanding of the dynamics of Bose condensation in
scalar quantum field theory (NLO-1/N)/vertex-resummed kinetic equation
However, no such understanding in gauge theories – despite indications
for infrared contributions to gauge invariant quantities (PL / PT)
Holography / functional renormalization group / solving QFTs by
ultracold quantum gas measurements / …
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Universality far from equilibrium
extreme conditions, e.g. large fields:
f(t,p) et overoccupation self-similar evolution/
condensation:
relaxation:
f(t,p) - fT(p) e-t
f(t,p) tα fS(tβp)