1. TOWARDS QUANTIZATION OF TURBULENCE BY EMMANUEL FLORATOS PHYSICS DEPARTMENT UNIVERSITY OF ATHENS WORK DONE IN COLLABORATION WITH MINOS AXENIDES INSTITUTE OF NUCLEAR PHYSICS DEMOKRITOS,ATHENS GGI WORKSHOP 3-10/10 FIRENZE

2. PLAN OF THE TALK
1)TURBULENCE IN CLASSICAL AND QUANTUM FLUIDS-MOTIVATION (3-15)
2)THE SALTZMAN-LORENZ EQUATIONS FOR CONVECTIVE FLOW (16-17)
3)THE LORENZ STRANGE ATTRACTOR(18-19)
4)NAMBU DISSIPATIVE DYNAMICS (20-23)
5)MATRIX MODEL QUANTIZATION OF THE LORENZ ATTRACTOR (23-26)
6)CONCLUSIONS -OPEN QUESTIONS

3. TURBULENCE IN CLASSICAL AND QUANTUM FLUIDS-MOTIVATION
MOST FLUID FLOWS IN NATURE ARE
TURBULENT (ATMOSPHERE,SEA,RIVERS,
MAGNETOHYDRODYNAMIC PLASMAS IN IONIZED GASES,STARS,GALAXIES etc
THEY ARE COHERENT STRUCTURES WITH DIFFUSION OF VORTICITYFROM LARGE DOWN TO THE MICROSCOPIC SCALES OF THE ENERGY DISSIPATION MECHANISMS
KOLMOGOROV K41,K62 SCALING LAWS
LANDU-LIFSHITZ BOOK,1987
HOLMES-LUMLEY BERKOOZ 1996

4. TURBULENCE IN QUANTUM FLUIDS
AT VERY LOW TEMPERATURES HeIV
VORTICES APPEAR (GROSS-PITAEVSKI)
INTERACT BY SPLIT-JOIN CREATING
MORE VORTICES AND VORTICITY
INTERACTIONS CREATING VISCOUS EFFECTS AND TURBULENCE
KOLMOGOROV SCALING LAWS HOLD FOR
SOME SPECTRA BUT VELOCITY PDF AREN’T GAUSSIAN AND PRESSURE
SPECTRA AREN’T KOLMOGOROV
INTERESTING RECENT ACTIVITY
VERONA MEETING,BARENGHI ‘S TALK 9/2009

12. RECENT INTEREST IN QUARK-GLUON FLUID PLASMA FOUND TO BE
STRONGLY INTERACTING (RHIC EXP)
HIRANO-HEINZ et al PLB 636(2006)299,..
ADS/CFT METHODS FROM FIRST PRINCIPLE CALCULATIONS OF TRANSPORT COEFFICIENTS ,A.STARINETS(THIS CONFERENCE) OR
USING DIRECTLY QUANTUM COLOR HYDRODYNAMIC EQNS (QCHD)
REBHAN,ROMATSCHKE,STRICKLAND PRL94,102303(2005)
THERMALIZATION EFFECTS ARE IN GENERAL NOT SUFFICIENT TO DESTROY VORTICITY
AND MAY BE TURBULENCE SIGNATURES ARE PRESENT
COSMOLOGICAL IMPLICATIONS
ALREADY CONSIDERED (10^-6 SEC,COSMIC TIME)
Astro-phys ,SHILD,GIBSON,NIEUWENHUISEN

13. Dynamics of Heavy Ion Collisions
Time scale
10fm/c~10-23sec
<<10-4(early universe)
Temperature scale 100MeV~1012K
Freezeout
“Re-confinement”
Expansion, cooling
Thermalization
First contact
(two bunches of gluons)

14. History of the Universe ~ History of Matter
QGP study
Understanding
early universe

16. RAYLEIGH-BENARD CONVECTION TEMPERATURE GRADIENT ΔΤ BOUSSINESQUE APPROXIMATION

19. 3 FOURIER MODES !

20. THE SALTZMAN-LORENZ EQUATIONS FOR CONVECTIVE FLOW
x'[t]=σ (x[t]-y[t]),
y'[t]=-x[t] z[t]+r x[t]-y[t],
z'[t]=x[t] y[t]- b z[t]
3 Fourier spatial modes of thermal convection for viscous fluid
in external temperature gradient ΔΤ
σ=η/ν =Prandl number,
η=viscocity,v=thermal diffusivity
R=Rc/R ,R Reynolds number =Ratio of Inertial forces to friction forces
b=aspect ratio of the liquid container
Standard values σ=10,r=28,b=8/3 E.N.Lorenz MIT,(1963)
Saltzman(1962) ONSET OF TURBULENCE
RUELLE ECKMAN POMEAU…1971,1987..

21. THE LORENZ STRANGE ATTRACTOR

25. -20 20 40

26. Including the dissipative terms (-10 x[t],-y[t],-8/3 z[t])

27. Lorenz attracting ellipsoid
E[x,y,z]=r x^2+σ y^2+(z-2r)^2
d/dt E[x,y,z]=v.∂ E[x,y,z]=
-2 σ [r x^2+y^2+b (z-r)^2-b r^2]
<0 Outside the ellipsoid F
F: r x^2+y^2+b (z-r)^2=b r^2

29. Matrix Model Quantization of the Lorenz attractor=Interacting system of N-Lorenz attractors
X'[t]=σ (X[t]-Y[t]),
Y'[t]=-1/2(X[t]Z[t]+Z[t]X[t])
+r X[t]-Y[t],
Z'[t]=1/2(X[t] Y[t]+Y[t] X[t])- b Z[t]

30. X[t],Y[t],Z[t] NxN Hermitian Matrices
When X,Y,Z diagonal (real)we have a system of N -decoupled Lorenz Non-linear oscillators
When the off-diagonal elements are small we have weakly coupled complex oscillators
When all elements are of the same order of magnitude we have strongly coupled complex
Ones.
Special cases X,Y,Z real symmetric

31. Matrix Lorenz ellipsoid
E[X,Y,Z]=Tr[r X^2+σ Y^2+(Z-2r)^2
d/dt E[X,Y,Z]=
-2 σ Tr[r X^2+Y^2+b (Z-r)^2
-b r^2 I]
<0 Outside the ellipsoid F
F: Tr[ r X^2+Y^2+b (Z-r)^2]=N b r^2
Multidimensional attractor

32. CONCLUSIONS Construction of Matrix Lorenz attractor with U[N] symmetry
Observables … Tr[X^k Y^l Z^m]
K,l,m=0,1,2,3,…
Initial phase of development of Ideas

33. Currently
Development of the physical ideas through
Numerical work
Analytical work for weak coupling
1/N expansion
Phenomenological applications

34. OPEN QUESTIONS
EXISTENCE OF MULTIDIMENSIONAL MATRIX LORENZ ATTRACTOR
HAUSDORFF DIMENSION
QUANTUM COHERENCE OR QUANTUM DECOHERENCE
N INTERACTING LORENZ ATTRACTORS
MATRIX MODEL PICTURE (D0 BRANES
ARE REPLACED BY LORENZ NONLINEAR SYSTEM)

35. PHYSICS APPLICATIONS QUARK GLUON PLASMA COSMOLOGY QUANTUM FLUIDS
SCALING LAWS OF CORELLATION
FUNCTIONS