Ultracool Gasesfar from Equilibrium
Thomas Gasenzer
Institut für Theoretische Physik Ruprecht-Karls Universität Heidelberg
Philosophenweg 16 • 69120 Heidelberg • Germany
email: [email protected]: www.thphys.uni-heidelberg.de/~gasenzer
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Overview Lecture 1: Introduction
Ultracold gases & their dynamics, mean-field theory
Lecture 2: Non-equilibrium QFTFunctional approach, 2PI effective action.
Lecture 3: Far-from-equilibrium DynamicsThermalization. Turbulence & critical dynamics.
Lecture 2Nonequilibrium QFT
2.1 Real-time path integral
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
From classical to quantum dynamics
tfin
tini
Classical dynamics of from S.[] = 0.
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
tfin
tini
Classical dynamics of from S.[] = 0.
From classical to quantum dynamics
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
tfin
tini
From classical to quantum dynamics
Classical dynamics of from S.[] = 0.
Quantum dynamics of = from variation of an effective action, [] = 0
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
QM transition amplitude:
Z.[J.] = ∫D e i S.[]/ℏ.
D = d(x)
Path Integral Approach
tfin|tini
x=xini
xfin
Classical dynamics of from S.[] = 0.
Quantum dynamics of = from variation of an effective action, [] = 0
tini
tfin
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Effective Action
Classical dynamics of from S.[] = 0.
Quantum dynamics of = from variation of an effective action, [] = 0
tini
tfin
Generating functional:
Z.[J.] = ∫D e i S.[]/ℏ.– i ∫J.
.=.i lnZ. =.Z.-1∫D e i S.[]/ℏ
J J.=.0
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Effective Action
Classical dynamics of from S.[] = 0.
Quantum dynamics of = from variation of an effective action, []/ = –.J :
Z.[J.] = ∫D [ – ]e i .[]/ℏ.+ i ∫J.
= e i [[]/ℏ – ∫J.]
tini
tfin
Generating functional:
Z.[J.] = ∫D e i S.[]/ℏ.– i ∫J.
.=.i lnZ. =.Z.-1∫D e i S.[]/ℏ
J J.=.0
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Effective Action
Classical dynamics of from S.[] = 0.
Quantum dynamics of = from variation of an effective action, []/ = –.J :
Z.[J.] = e i [[]/ℏ + ∫J.]
[] = –.i.ℏ.ln.Z.[J.] – ∫J.
tini
tfin
Legendre transform
Generating functional:
Z.[J.] = ∫D e i S.[]/ℏ.– i ∫J.
.=.i lnZ. =.Z.-1∫D e i S.[]/ℏ
J J.=.0
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Effective Action
Classical dynamics of from S.[] = 0.
Quantum dynamics of = from variation of an effective action, []/ = –.J :
Z.[J.] = e i [[]/ℏ + ∫J.]
[] = –.i.ℏ.ln.Z.[J.] – ∫J.= S[] –.i./2 Tr.ln.G + .
tini
tfin
Gaussian integration
Generating functional:
Z.[J.] = ∫D e i S.[]/ℏ.– i ∫J.
.=.i lnZ. =.Z.-1∫D e i S.[]/ℏ
J J.=.0
2.2 Schwinger-Keldysh
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Initial value problems...
t ∣O∣t = t0.∣U†(t)O U(t)∣t0. = Tr[ρ(t0) U
†(t)O U(t) ]
G.ab(x,y) = Tr[ρ(t0) U†(x0)a(x)U(x0−y0)b(y)U(y0) ] – disc.
...require the Schwinger-Keldysh closed time path (CTP):
e.g., for x0 > y0
CTP C :
QFT
y0 x0
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Initial value problems...
t ∣O∣t = t0.∣U†(t)O U(t)∣t0.
= ∫D0D0 ρ[0,0]∫D'D' O e i (S[ ]–S[ ])/ℏ
...require the Schwinger-Keldysh closed time path, now in the Path Integral:
QFT
2.3 2PI effective action
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Dynamical Field Theory
[(t,x),†(t,y)] = (x – y) (Bose)
{(t,x),†(t,y)} = (x – y) (Fermi)
Our focus: time dependence of the lowest-order “correlation” functions:
G.ab(x,y) = T a(x)b(y) x = (t,x)
(2-point correlation function,
2-time Green function,
Single-particle density matrix, pair function)
QFT
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
2PI Effective Action (-Functional)[Luttinger, Ward (60); Baym (62); Cornwall, Jackiw, Tomboulis (74)]
Generating functional for (connected) Greens functions (2 sources)
with classical action S, e.g.,
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
1PI Effective Action
Consider for the first the 1PI effective action (1 source) for
To 1-loop order (Gaussian integral):
“mass” shift
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
2nd Legendre transform... ...yields the 2PI effective action:
where we used the K-dependence of J():
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
2nd Legendre transform... ...yields the 2PI effective action and stationarity conditions:
Plugging in the 1-loop 1PI effective action, i.e.
gives, summing beyond 1-loop terms in :
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
2PI Effective Action (in summary)[Luttinger, Ward (60); Baym (62); Cornwall, Jackiw, Tomboulis (74)]
closed for Gaussian initial conditions (only , G ≠ 0 @ t = 0)
(@ J = 0, K = 0)
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
2PI Effective Action... ...now reads:
Variation w.r.t. G gives the Schwinger-Dyson equation:
self energy
i.e.:
1PI only ⇒ 2PI only
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
2PI Effective Action
2PI 1PI⇔
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
2PI Effective Action
includes daisies, superdaisies, ladders, &tl
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Dynamic EquationsFrom the stationarity condition , one obtains, in leading order,the Gross-Pitaevskii equation:
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Dynamic EquationsFrom the stationarity conditions , one obtainsthe HFB dynamic equations:
Note:
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Dynamic Equations (⊇ Kadanoff-Baym)
From the stationarity conditions , one obtainsthe full dynamic equations:
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Numerical Demand
x0
y0
in memory
nth order Runge-Kuttapropagation in t=x0
Gij(x0,y0;x,y)
i(x0;x)
16 x 16 spatial grid 1000 x 1000 temporal grid N x N index grid, N= 2⇒ ~16 GB RAM
e.g.
Interval...
Supplementary slides
2.4 2PI truncations
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Approximations to Γ2
E.g. 4 interaction (=0) :
Loop expansion [Cornwall, Jackiw, Tomboulis PRD 10 (74) 2428]
Literature: e.g. lecture notes by J.Berges hep-ph/0409322 (AIP Conf. Proc.)
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
2PI 1/N Expansion[Berges, NPA 699 (02) 847; Aarts, Ahrensmaier, Baier, Berges, & Serreau, PRD 66 (02) 45008]
N
2PI eff. action = O(N )-singlet ⇒ these irreducible O(N )-invariants only:
with
N
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
2PI 1/N Expansion[Berges, NPA 699 (02) 847; Aarts, Ahrensmaier, Baier, Berges, & Serreau, PRD 66 (02) 45008]
N
tr(G)2/N ∝ N tr(G2)/N ∝ N 0
2PI eff. action = O(N )-singlet ⇒ these irreducible O(N )-invariants only:
with since (n) = 0, n ≥ 3
⇒ each trace ∝ N , each vertex ∝ 1/N, e.g. О(g) graphs:
N
...
b
a
a
aa
b
b
b
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
2PI 1/N Expansion[Berges, NPA 699 (02) 847; Aarts, Ahrensmaier, Baier, Berges, & Serreau, PRD 66 (02) 45008]
upto
NLO
Vertex resummation bare vertex
2.5 Equilibration of a 1D Bose gas
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
density info in F = statistical correl. function
Near-equilibrium dynamics: Damping
p p
n n
Exponential damping
n - neq ~ exp(-Γ t)
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Far-from-equilibrium dynamics
Thermal equilibrium: Loss of information about prior evolution.
Only a few conserved quantities persist.
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Conserved quantities
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Observables
We set
a(t,x) = 0 (is always one solution)
and calculate the time dependence of:
n(t,p) = ∫dr G11(t,r; t,0) eipr (no. of particles with mom. p)
QFT
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Equilibration of a 1D Bose gas
initialGaussian
final Bose-Einstein
Number of particles n(t,p) with momentum p:
[TG, J. Berges, M. Seco & M.G.Schmidt, PRA 72 (05); J. Berges & TG, PRA 76 (07)]
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Temperature appears
[J. Berges & TG, PRA 76 (07)]
flat ≙ temperature
[also: Berges & Cox (01), Berges (01), Berges, Borsanyi, & Serreau (03), Berges, Borsanyi, & Wetterich (04)]
from fit of n to
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Equilibration of a 1D Bose gasNo. of particles n(t,pi) with momentum pi:
[J. Berges & TG, PRA 76 (07)]
lowest
highest momentum
•••
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
No. of particles n(t,pi) with momentum pi:
[J. Berges & TG, PRA 76 (07)]
Equilibration of a 1D Bose gas
Far from equilibrium(no fluct.-dissip. rel.)
Near equilibrium (fluct.-dissip. rel.)
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Equilibration of a 1D Bose gasNo. of particles n(t,pi) with momentum pi:
[J. Berges & TG, PRA 76 (07)]
mean-fieldapprox.:no scattering•
••
+ ...
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Elastic scattering of classical point-like particles in one dimension
Before collision:
p − p
After collision:
− p p
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Equilibration of a 1D Bose gasNo. of particles n(t,pi) with momentum pi:
[J. Berges & TG, PRA 76 (07)]
...
Quantum vs. classicaldynamics
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Initial value problem
t ∣O∣t = t0.∣U†(t-t0) O U(t-t0)∣t0.
= Z-1∫DD O 0 e i (S[ ]–S[ ])/ℏ
Schwinger-Keldysh closed time path:
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
c
Initial value problem
2.–.2
=.(.–.)(.+.)
=:.~
Quadratic action (QM Harm. Osc.): S.[]~∫dt{(∂t)2 – 2}:
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
c
Initial value problem
2.–.2
=.(.–.)(.+.)
=:.~
Quadratic action (QM Harm. Osc.): S.[]~∫dt{(∂t)2 – 2}:
Consider QM Harm. Osc.:
S.[] – S.[] ~ –∫dt{ (∂t2 + 2) – (∂t
2 – 2) }
~ –∫dt (∂t2 + 2)
~
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
c
Initial value problem
2.–.2
=.(.–.)(.+.)
=:.~
Quadratic action (QM Harm. Osc.): S.[]~∫dt{(∂t)2 – 2}:
Path integral:
∫DD O [0,0] e i (S[]–S[])/ℏ
~ ∫DD O [0,0] exp[ –∫dt (∂t2 + 2)/ℏ]
~~ ~
[J. Berges, TG, PRA (07). ClPI goes back to Hopf (50), cf. also Phythian (75), DeDominicis et al. (76), Janssen et al. (76), Chou et al. (85), Blagoev et al. (01), Polkovnikov (03)]
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Classical Path Integral
Path integral evaluates to classical solution:
∫DD O [0,0] e i (S[]–S[])/ℏ
~ ∫DD O [0,0] exp[ –∫dt (∂t2 + 2)/ℏ]
~~ ~
[J. Berges, TG, PRA (07). ClPI goes back to Hopf (50), cf. also Phythian (75), DeDominicis et al. (76), Janssen et al. (76), Chou et al. (85), Blagoev et al. (01), Polkovnikov (03)]
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Classical Path Integral
Path integral evaluates to classical solution:
∫DD O [0,0] e i (S[]–S[])/ℏ
~ ∫DD O [0,0] exp[ –∫dt (∂t2 + 2)/ℏ]
~ ∫D O W [0,0] [(∂t2 + 2)]
~~ ~
[J. Berges, TG, PRA (07). ClPI goes back to Hopf (50), cf. also Phythian (75), DeDominicis et al. (76), Janssen et al. (76), Chou et al. (85), Blagoev et al. (01), Polkovnikov (03)]
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Classical Path Integral
Path integral evaluates to classical solution:
∫DD O [0,0] e i (S[]–S[])/ℏ
~ ∫DD O [0,0] exp[ –∫dt (∂t2 + 2)/ℏ]
~ ∫D O W [0,0] [(∂t2 + 2)]
~~ ~
Not with interactions!g (4-4) = g (3+3)~ ~
[J. Berges, TG, PRA (07). ClPI goes back to Hopf (50), cf. also Phythian (75), DeDominicis et al. (76), Janssen et al. (76), Chou et al. (85), Blagoev et al. (01), Polkovnikov (03)]
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Classical Path Integral
Path integral evaluates to classical solution:
∫DD O [0,0] e i (S[]–S[])/ℏ
~ ∫DD O [0,0] exp[ –∫dt (∂t2 + 2 + g2)/ℏ]
~ ∫D O W [0,0] [(∂t2 + 2 + g2)]
~~ ~
Not with interactions!g (4-4) = g (3+3)~ ~
Quantum vertex
[J. Berges, TG, PRA (07). ClPI goes back to Hopf (50), cf. also Phythian (75), DeDominicis et al. (76), Janssen et al. (76), Chou et al. (85), Blagoev et al. (01), Polkovnikov (03)]
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Classical vs. Quantum diagrams(2PI 1/N )
Interactions:g (4-4) = g (3+3)~ ~
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
...provides in a clear way the relation:
Classical statistical evolution...
The path integral...
...vs quantum statistical evolution:
t
x
P(x)
?
initialclassicalprobabilitydistribution
BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 2 Thomas Gasenzer
Classical vs. quantum evolution [J. Berges, TG, PRA 76, 033604 (07)]
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