MATRIX QUANTIZATIONOF THE LORENZ

STRANGE ATTRACTORAND

THE ONSET OF TURBULENCE IN QUANTUM FLUIDS

M. AXENIDES (INP DEMOKRITOS)&

E.FLORATOS (PHYSICS DPT UoA)

5TH AEGEAN HEP SUMMER SCHOOLMILOS 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 scale10fm/c~10-23sec<<10-4(early universe)

Temperature scale 100MeV~1012K

History of the Universe ~ History of Matter

QGP study

Understandingearly universe

RAYLEIGH-BENARD CONVECTIONTEMPERATURE 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 TURBULENCERUELLE ECKMAN POMEAU…1971,1987..

THE LORENZ STRANGE ATTRACTOR

-20

0

20

-20

0

20

0

20

40

-20

0

20

-20

0

20

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

-10

0

10

20

-20

0

20

0

20

40

-10

0

10

20

-20

0

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