MATRIX QUANTIZATION OF THE LORENZ STRANGE ATTRACTOR AND

MATRIX QUANTIZATION OF THE LORENZ STRANGE ATTRACTOR AND THE ONSET OF TURBULENCE IN QUANTUM FLUIDS M. AXENIDES E. FLORATOS (INP DEMOKRITOS) & (PHYSICS DPT Uo. A) 5 TH 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 K 41, K 62 SCALING LAWS • LANDU-LIFSHITZ BOOK, 1987 • HOLMES-LUMLEY BERKOOZ 1996

TURBULENCE IN QUANTUM FLUIDS AT VERY LOW TEMPERATURES He. IV 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 PRL 94, 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 10 fm/c~10 -23 sec 100 Me. V~1012 K <<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])

Lorenz attracting ellipsoid • E[x, y, z]=r x^2+σ y^2+(z-2 r)^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] Nx. N 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-2 r)^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 (D 0 BRANES • ARE REPLACED BY LORENZ NONLINEAR SYSTEM)

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