Ultracool Gases far from Equilibrium - Max Planck Societybec10/Gasenzer_Lecture3.pdf · BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 3 Thomas Gasenzer Richardson

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 3 Thomas Gasenzer

Lecture 1: IntroductionUltracold gases & their dynamics, mean-field theory

Lecture 2: Non-equilibrium QFTFunctional approach, 2PI effective action.

Lecture 3: Far-from-equilibrium DynamicsThermalization. Turbulence & critical dynamics.

Overview

Lecture 3Far-from-equilibrium dynamics

3.1 Equilibration of a1D Fermi gas


Elastic scattering of classical point-like particles in d dimensions


Elastic scattering of classical point-like particles in one dimension

Before collision:

p − p

After collision:

− p p


Elastic scattering of quantum point-like particles in one dimension

Before collision:

p − p

After collision:

− p p


Equilibration in a quantum system

Hamiltonian can be diagonalised.

Dephasing only (“Decoherence”).

Suppose: Recurrence time large.

System approaches thermal ensemble?

Special rôle of integrable systems


Heuristic StatisticsJaynes' maximum-entropy principle [PR 106, 620 (57)] :

Generalised Gibbs ensemble at t :

Maximisation of entropy under constraints for macroscopic observables.

fixes


Heuristic StatisticsJaynes' maximum-entropy principle [PR 106, 620 (57)] :


Maximisation of entropy under constraints for macroscopic observables.

Also: Rigol et al., PRL 98, 050405 (07), P. Reimann, PRL 101, 190403 (08).

fixes


Rôle of conserved quantitiesJaynes' maximum-entropy principle [PR 106, 620 (57)] :


Question: What are the operators for the conserved quantities ?

Answer: Is easy only for integrable systems.

See, e.g., Rigol et al., PRL 98 (07); Cazalilla, PRL 97 (06); Barthel and Schollwöck, PRL 100 (08)


Equilibration of the 1D Fermi gasNumber of particles n�(p,t) with momentum p:

M. Kronenwett & TG, 1006.3330 [cond-mat.quant-gas]

See also (for relativist. Fermions)Berges, Borsanyi, Serreau,NPB (03)Berges, Borsanyi, WetterichPRL (04)


Equilibration of the 1D Fermi gasNumber of particles n�(p,t) with momentum p:




Equilibration of the 1D Fermi gasInverse slope function: Thermalisation?




Equilibration of the 1D Fermi gasInverse slope function: Varying the total energy



Equilibration of the 1D Fermi gasVarying the total energy: Chemical potential



Equilibration of the 1D Fermi gasHigh-momentum tails: Power-law distributions



Fluctuation-dissipation theoremA grand-canonical stateimplies the relation



Near Equilibrium:Fluctuation-dissipation relation QFT

[A. Branschädel & TG, JPB 41, 135302 (08)]


Near Equilibrium:Linear Response QFT

[A. Branschädel & TG, JPB 41, 135302 (08)]

Kinetic (Boltzmann) case:Quasiparticles with well-defined energy

p0 = ωp


for 2-time time-ordered Green function, carrying two-fold information:

G.ab(x,y) = TC†

a(x)b(y)

= F.ab(x,y) – sgnC(x0–y0) ab(x,y)

where x = (x0, x)

Bosons: [(t,x),†(t,y)] = (x – y)

Dynamical field theoryQFT

i2

statistical ... spectral function


Equilibration of the 1D Fermi gasEnergy conservation:


3.2 Turbulence& Critical Dynamics


Nonequilibrium Stationary Dynamics


Turbulence

Kinetic energy cascade

large scales (source) → small scales

→ dissipation (sink)


Richardson Cascade

“Big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity.”

(L.F. Richardson, 1920)





Richardson Cascade

“Big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity.”

(L.F. Richardson, 1920)




Kolmogorov (1941): scale-invariant, incompressible fluid

E(k) ~ P 2/3 ρ1/3 k—5/3 (Turbulent stationarity)


Kolmogorov's theory of turbulence(1941)

Take isotropic system: Cascade = Radial momentum transport

k

kx

ky

pump

radial flux

∂t ε(k) = ―∇kp(k)

Energy density ε Energy flux p

Andrey N. Kolmogorov(1903-1987)



Cascade = Transport in momentum space:

Radial energy density E = k2ε [kg s—2]

log E(k)

log klog L-1

pump

3D:



Stationary E-distribution in cascade?

Radial energy density E = k2ε [kg s—2]

log E(k)

log k log η-1log L-1

pump

dump

3D:



Assume self-similarity:

Radial energy density E = k2ε [kg s—2] Radial energy flux P = k2|p| [kg m—1 s—3]

log E(k)

log k

E(k) ~ k—ζ

pump

dump

3D:



Assume self-similarity:

Radial energy density E [kg s—2] Radial energy flux P [kg m—1 s—3]

log E(k)

log k

E(k) ~ P 2/3 ...

pump

dump

Dimensional analysis:

3D:



Scale invariant (self-similar) stationary transport:

Radial energy density E [kg s—2] Radial energy flux P [kg m—1 s—3] Density ρ [kg m—3]

log E(k)

log k

E(k) ~ P 2/3 ρ1/3 k—5/3

pump

dump

Dimensional analysis:

3D:


Compare: Thermal Equilibrium

Constant energy ε ~ E/k2 (In 3D):

Radial energy density E [kg s—2]

log ε(k)

log k

ε(k) ~ ω n ~ T

(Rayleigh-Jeans: n ~ T/ω )

3D:


Compare: Thermal Equilibrium

Constant energy ε ~ E/k2 (In 3D):

Radial energy density E [kg s—2]

log E(k)

log k

E(k) ~ ω n k2

~ T k2

(Rayleigh-Jeans: n ~ T/ω )

3D:


SummaryScaling stationary solutions:

log E(k)

log k

E(k) ~ k—ζ

n(k) ~ k—κ

pump

dump


Origin of radial fluxBalance equation for radial flux

k

kx

ky

pump

radial flux

rad. occupation no. n rad. particle flux Q

∂t n(k) = ― ∂k Q(k)


Origin of radial fluxWith kinetic (Boltzmann) eq.

k

kx

ky

pump

radial flux

∂t n(k) = ― ∂k Q(k)

~ k2 J(k)

Boltzmannscattering integral

rad. occupation no. n rad. particle flux Q


Kinetic equation

Flow in momentum space from kinetic (Boltzmann) equation:

Scattering integral:


To derive scaling Familiarize the...

Critical dynamicsbeyond kinetic theory


Dynamical field theory

Kinetic (Quantum-Boltzmann) eq.:

Dynamic (Schwinger-Dyson) eq.:

or...



Dynamical field theory

Kinetic (Quantum-Boltzmann) eq.:

Dynamic (Schwinger-Dyson) eq.: (from 2PI effective action)


p = ( p0, p):


Strong turbulence

p = ( p0, p):

2PI to NLO in 1/N: Vertex:

[J. Berges, NPA 699 (02) 847; G. Aarts et al., PRD 66 (02) 45008]


Strong Turbulence in 2D

momentum p

κ = 4

κ = 4/3κ = 2

nonequilibriumthermalequilibrium

n ~ k -κ

C. Scheppach, J. Berges, TG PRA 81 (10) 033611 J. Berges et al., PRL 101 (08) 041603


Simulations in 2+1 D (semi-classical)

Remember:

B. Nowak, D. Sexty, TG (unpublished)











3D Simulation

relativistic Bose gas

J. Berges, A. Rothkopf, and J. Schmidt, PRL 101 (08) 041603

“ultracold” Bose gas




Remember:




Remember:




Remember:




Remember:



Turbulence as a fixed point

[Fig.: Berges 08]

The


Thanks & credits to......my work group in Heidelberg:

Cédric Bodet Matthias KronenwettBoris Nowak Denes SextyMartin TrappeJan Zill

Alexander Branschädel ( Karlsruhe)Stefan Keßler ( LMU München)Christian Scheppach ( Cambridge, UK)Philipp Struck ( Konstanz) Kristan Temme ( Vienna)

...my collaboratorsJürgen Berges (Darmstadt) • Hrvoje Buljan (Zagreb) • Keith Burnett (Oxford/ Sheffield) • David Hutchinson (Otago) • Paul S. Julienne (NIST Gaithersburg) • Thorsten Köhler (Oxford/UCL) • Otto Nachtmann (Heidelberg) • Markus K. Oberthaler (Heidelberg) • Jan M. Pawlowski (Heidelberg) • Robert Pezer (Sisak) • David Roberts (Oxford/Los Alamos) • Janne Ruostekoski (Southampton) • Michael G. Schmidt (Heidelberg) • Marcos Seco (Santiago de Compostela)

LGFG BaWue

€€€...

ExtreMe Matter Institute

Supplementary slides


Nonthermal Equilibration in a quantum system

M. Rigol, V. Dunjko, V. Yurovsky, M. Olshanii, PRL 98, 050405 (2007)

S. R. Manmana, S. Wessel, R. M. Noack, A. Muramatsu, PRL 98, 210405 (2007)

D. M. Gangardt, M. Pustilnik, PRA 77, 041604(R) (2009)

M. Cazalilla, PRL 97, 156403 (2006)

C. Kollath, A. M. Läuchli, E. Altman, PRL 98, 180601 (2007)

M. Eckstein, M. Kollar, PRL 100, 120404 (2008)

M. Moeckel, S. Kehrein, PRL 100, 175702 (2008)

A. Iucci, M. A. Cazalilla, PRA 80, 063619 (2009)

D. M. Kennes, V. Meden, arXiv:1006.0927v1 [cond-mat.str-el] (2010)


Integrable Lieb & Liniger model

Lieb-Liniger Bose gas in 1D: [PR 130, 1605 & 1616 (63)]

Solution via Bethe-Ansatz with cusp conditions @ particle contact hyperplanes

c > 0

xi = xj Dirichlet-von Neumann


Dynamics: H. Buljan, R. Pezer, & TG, PRL 100 (2008)

Integrable Lieb & Liniger model

Lieb-Liniger Bose gas in 1D: [PR 130, 1605 & 1616 (63)]

Solution via Bethe-Ansatz with cusp conditions @ particle contact hyperplanes

c > 0

xi = xj Dirichlet-von Neumann


Tomonaga-Luttinger Liquid model1D Fermi gas with contact interactions. e.g. Schultz et al. cond-mat/9807366

Bosonize by replacing as follows:

(right/left-moving particle-hole pairs)

particle-holecontinuum

Includeall energies < 0!

k k+q

}εq

TomonagaLuttinger

Lieb & Mattis


Tomonaga-Luttinger Liquid model

Bosonization ⇒ quadratic Hamiltonian

Quartic fermion interaction

TomonagaLuttinger

Lieb & Mattis

Expect nonthermal evolution after switching on interactions:

Iucci & Cazalilla, 1003.5170 [cond-mat], also Kennes & Meden 1006.0927 [cond-mat]



Evolution of mom. distr. of a 2D Bose gas following an interaction quench







Remember:




Remember:




Remember:



Scaling exponents( in d dimensions )

κ = dConstant P(k) ≡ P Constant Q(p)

κ = d −2/3

n ~ k -κ

C. Scheppach, J. Berges, T. Gasenzer PRA 81 (10) 033611

UV:

IR: κ = d +4 κ = d +2


Turbulence in a Bose Einstein Condensate

[E.A.L. Henn et al. PRL (09)]



[E.A.L. Henn et al. PRL (09)]

[Abu-Shaeer et al. (01)]



[N. Berloff & B. Svistunov, PRA (02)] [E.A.L. Henn et al. PRL (09)]

[Abu-Shaeer et al. (01)]


Wave turbulence

Turbulent thermalisation after universe inflation

[Micha & Tkachev, PRL 90 (03) 121301, PRD 70 (04) 043538]


Wave turbulence

[Micha & Tkachev, PRL 90 (03) 121301, PRD 70 (04) 043538]



V.E. Zakharov, V.S. L'vov, G. Falkovich, Kolmogorov Spectra of Turbulence I (Springer, Berlin, 1992)



V.E. Zakharov, V.S. L'vov, G. Falkovich, Kolmogorov Spectra of Turbulence I (Springer, Berlin, 1992)


(Weak) wave turbulence


∂kP(k) ~ J(k) ⇒ Stationarity if J(k) = 0



(Weak) wave turbulence


∂kP(k) ~ J(k) ~ nk3 ⇒ nk ~ P 1/3

⇒ E(k) ~ (Pρ2)1/3 ωk k—7/3



(Weak) wave turbulence Scaling solutions

Constant energy flux P:

E(k) ~ (Pρ2)1/3 ωk k—7/3


(Weak) wave turbulence Scaling solutions

Constant energy flux P:

Constant particle flux Q (P Pk = ωkQ):

E(k) ~ (Pρ2)1/3 ωk k—7/3

E(k) ~ (Qρ2)1/3 ωk4/3

k—7/3

⇒ 2 distinct scaling laws!


Scaling solutions

Want to look for scaling solutions fulfilling stationarity condition J(p) = 0

Scaling ansatz:

Implies scaling of the single-particle momentum distribution ( ):

( )


Stationarity Conditionin the full dynamical theory

p = ( p0, p):

(Bosons with 3- and 4-interactions, 2PI diagrams to O(g2)


Ultraviolet scaling exponent( large p in d dims)

κ = d

κ = d + 4(z ̶ 2 +η)/ 3

Corresponds to constant P(p)

[C. Scheppach, J. Berges, & TG, to be submitted]



κ = d −2/3

κ = d + z ̶ 4(2 ̶ η)/ 3

Corresponds to constant Q(p)




κ = d −2/3

κ = d + z ̶ 4(2 ̶ η)/ 3


Not present in general dynamical case!



Scattering integral

Strong turbulence


Infrared scaling modification

momentum p

κ = 7

κ = 3κ = 2

nonequilibrium


Stationarity Conditionin the full dynamical theory

p = ( p0, p):

2PI to NLO in 1/N: Vertex:


IR scaling exponent( small p in d dims)

κ = d + 4

κ = d + 2z

Corresponds to constant P(p)

[Berges et al., PRL 101 (2008) 041603, C. Scheppach, J. Berges, & TG, to be submitted]



κ = d + 2

κ = d + z





κ = d + 2

κ = d + z



Should not be present in general dynamical case either!


Summary

momentum p

κ = 7

κ = 3κ = 2

nonequilibrium

κ = 3/2 κ = 4

relativistic simulation

[Berges et al., PRL 101 (2008) 041603]

equilibrium


Lewis Fry Richardson, FRS (1881-1953)

Big whirls have little whirls that feed on their velocity,and little whirls have lesser whirls and so on to viscosity.

(L.F. Richardson, The supply of energy from and to Atmospheric Eddies, 1920)

Great fleas have little fleas upon their backs to bite 'em,And little fleas have lesser fleas, and so ad infinitum.And the great fleas themselves, in turn, have greater fleas to go on;While these again have greater still, and greater still, and so on.

(Augustus de Morgan, A Budget of Paradoxes, 1872, p. 370)

So, naturalists observe, a fleaHas smaller fleas that on him prey;And these have smaller still to bite 'em;And so proceed ad infinitum.

(Jonathan Swift: Poetry, a Rhapsody, 1733)

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Ultracool Gases far from Equilibrium - Max Planck Societybec10/Gasenzer_Lecture3.pdf · BEC10 Summer School · MPIPKS Dresden · 10. August 2010 · Lecture 3 Thomas Gasenzer Richardson