Transcript matrix quantization of the lorenz strange attractor and the onset

MATRIX QUANTIZATION
OF THE LORENZ
STRANGE ATTRACTOR
AND
THE ONSET OF TURBULENCE
IN QUANTUM FLUIDS
M. AXENIDES
E.FLORATOS
(INP DEMOKRITOS)
&
(PHYSICS DPT UoA)
5TH AEGEAN HEP SUMMER SCHOOL
MILOS ISLAND 21-26/9/2009
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
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
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
• 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 09065087,SHILD,GIBSON,NIEUWENHUISEN
Dynamics of Heavy Ion Collisions
Time scale
Temperature scale
10fm/c~10-23sec
100MeV~1012K
<<10-4(early universe)
History of the Universe
~ History of Matter
QGP study
Understanding
early universe
RAYLEIGH-BENARD CONVECTION
TEMPERATURE GRADIENT ΔΤ
BOUSSINESQUE APPROXIMATION
3 FOURIER MODES !
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..
THE LORENZ STRANGE
ATTRACTOR
20
0
-20
40
20
0
-20
0
20
Including the dissipative terms
(-10 x[t],-y[t],-8/3 z[t])
20
0
-20
40
20
0
-10
0
10
20
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
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]
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
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
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
Currently
Development of the physical ideas through
• Numerical work
• Analytical work for weak coupling
• 1/N expansion
• Phenomenological applications
• 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)
PHYSICS APPLICATIONS
•
•
•
•
•
QUARK GLUON PLASMA
COSMOLOGY
QUANTUM FLUIDS
SCALING LAWS OF CORELLATION
FUNCTIONS