Critical Physics far from Equilibrium

Transcription

1 Critical Physics far from Equilibrium Institut für Theoretische Physik Philosophenweg Heidelberg, Germany www:

2 Nonequilibrium Stationary Dynamics

3 Turbulence Kinetic energy cascade large scales (source) small scales dissipation (sink)

4 Richardson Cascade Kinetic energy cascade large scales (source) small scales dissipation (sink) 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)

5 Richardson Cascade Kinetic energy cascade large scales (source) small scales dissipation (sink) 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)

6 Kolmogorov's theory of turbulence (1941) Take isotropic system: Cascade = Radial momentum transport k Andrey N. Kolmogorov ( ) radial flux t ε(k) = k p(k) pump Energy density Energy flux ky ε p kx

7 Kolmogorov's theory of turbulence (1941) Cascade = Transport in momentum space: log E(k) pump log L-1 3D: Radial energy density log k E = k2ε [kg s 2]

8 Kolmogorov's theory of turbulence (1941) Stationary E-distribution in cascade? log E(k) pump dump log L-1 3D: Radial energy density log k E = k2ε log η-1 [kg s 2]

9 Kolmogorov's theory of turbulence (1941) Assume self-similarity: E(k) ~ k ζ log E(k) pump dump log k 3D: Radial energy density Radial energy flux E = k2ε [kg s 2] P = k2 p [kg m 1 s 3]

10 Kolmogorov's theory of turbulence (1941) Assume self-similarity: Dimensional analysis: E(k) ~ P log E(k) 2/3... pump dump log k 3D: Radial energy density Radial energy flux E P [kg s 2] [kg m 1 s 3]

11 Kolmogorov's theory of turbulence (1941) Scale invariant (self-similar) stationary transport: Dimensional analysis: E(k) ~ P log E(k) 2/3 ρ1/3 k 5/3 pump dump log k 3D: Radial energy density Radial energy flux Density E P ρ [kg s 2] [kg m 1 s 3] [kg m 3]

12 Compare: Thermal Equilibrium Constant energy ε ~ E/k2 (In 3D): ε(k) ~ ω n ~T log ε(k) (Rayleigh-Jeans: n ~ T/ω ) log k 3D: Radial energy density E [kg s 2]

13 Summary Scaling stationary solutions: E(k) ~ k ζ n(k) ~ k κ log E(k) pump dump log k

14 Origin of radial flux Balance equation for radial flux t n(k) = k Q(k) k radial flux pump rad. occupation no. n rad. particle flux Q ky kx

15 Origin of radial flux Boltzmann scattering integral With kinetic (Boltzmann) eq. t n(k) = k Q(k) ~ k2 J(k) k radial flux pump rad. occupation no. n rad. particle flux Q ky kx

16 Kinetic equation Flow in momentum space from kinetic (Boltzmann) equation: Scattering integral:

17 To derive scaling Familiarize the...

18 To derive scaling Familiarize the... V.E. Zakharov, V.S. L'vov, G. Falkovich, Kolmogorov Spectra of Turbulence I (Springer, Berlin, 1992)

19 Turbulent scaling in 2+1 D κ = 4/3 n~k κ =2 thermal equilibrium nonequilibrium momentum p C. Scheppach, J. Berges, TG PRA 81 (10) κ

20 Simulations in 2+1 D (semi-classical) Evolution of mom. distr. of a 2D Bose gas following an interaction quench B. Nowak, D. Sexty, TG (unpublished)

21 Simulations in 2+1 D (semi-classical) B. Nowak, D. Sexty, TG (unpublished)

22 Critical dynamics beyond kinetic theory

23 Dynamical field theory Kinetic (Quantum-Boltzmann) eq.: Scattering integral: Dynamic (Schwinger-Dyson) eq.: or...

24 Dynamical field theory Kinetic (Quantum-Boltzmann) eq.: Scattering integral: Dynamic (Schwinger-Dyson) eq.: (from 2PI effective action) p = ( p0, p):

25 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]

26 Strong Turbulence in 2D n~k κ =4 κ = 4/3 nonequilibrium κ =2 thermal equilibrium momentum p J. Berges et al., PRL 101 (08) C. Scheppach, J. Berges, TG PRA 81 (10) κ

27 Simulations in 2+1 D (semi-classical) Remember: B. Nowak, D. Sexty, TG (unpublished)

28 Simulations in 3+1 D (semi-classical) B. Nowak, D. Sexty, TG (unpublished)

29 Simulations in 3+1 D (semi-classical) B. Nowak, D. Sexty, TG (unpublished)

30 Simulations in 3+1 D (semi-classical) B. Nowak, D. Sexty, TG (unpublished)

31 3D Simulation ultracold Bose gas relativistic Bose gas B. Nowak, D. Sexty, TG (unpublished) J. Berges, A. Rothkopf, and J. Schmidt, PRL 101 (08)

32 1D Simulation = B. Nowak, D. Sexty, TG (unpublished)

33 1D Equilibration of Fermions M. Kronenwett, TG (unpublished)

34 1D Fermiongas: Scaling M. Kronenwett, TG (unpublished)

35 1D Fermiongas: Scaling M. Kronenwett, TG (unpublished)

36 Turbulence as a fixed point [Fig.: Berges 08]

37 Thanks & credits to......my work group in Heidelberg:... Cédric Bodet Matthias Kronenwett Boris Nowak Denes Sexty Martin Trappe Alexander Branschädel Stefan Keßler Christian Scheppach Philipp Struck Kristan Temme ( ( ( ( ( Karlsruhe) LMU München) Cambridge, UK) Konstanz) Vienna) LGFG BaWue ExtreMe Matter Institute