Complexity and Unification in Physical Theory
Pavlos G.P Democritus University of Thrace
6th International Conference on New Frontiers in Physics Kolymbari, Crete, 28 August 2017
Democritus University of Thrace ICNFP2017 Physical Theory from Past to Future
Mechanistic reductionism, microscopical causality (from bottom to top)
Holistic multiscale distributed causality (from bottom to top and from top to bottom)
Irreversibility of time, novelty and creativity It is said that …
• Φύσις ουδέν μάτην ποιεί. (Αριστοτέλης)
• Aristotle: mathematics are abstraction from material objects • Aristotle: discrimination between potential and energetic reality • Plato: material objects are realization of mathematical forms • Einstein: space time ,finite or infinite dimensional, participates in the physical reality while fields and particles are geometrical properties the space-time manifold • Bohr Born, Heinsenberg: probabilism, as possibility or potentiality ,is prior to the observed reality (ontological probabilism according to Popper) • K.Wilson: physicist try to explain nature by using only one scale but forget that nature is a multi scale dynamical production • Prigogine: material forms as dissipative structures as well physical laws are novelty emergence through the non equilibrium and non reversible physical self-organization process • G.Nicolis: ordering and development of long range correlations in non equilibrium systems cannot be explained by local forces. • P.Davies and C. Castro: in nature acts a global ordering principle Non – Linear Physics
Dynamical Non-Equilibrium Stochastic Systems Thermodynamics Processes
Renormalization Fractional Extension Tsallis Theory Group Theory Of Non-Equilibrium (RGT) Dynamics Statistics Scale Invariance Tsallis Extension of Statistics Nonextensive Statistical Mechanics
Microscopic Level Macroscopic Level
Quantum Complexity Non – Equilibrium Quantum Phase Equilibrium Phase Transition Phase Transition Transition (E.P.T.) (N.E.P.T.) (Q.P.T.) Gaussian Statistical Mechanics
Liouville Equation – BBGKY Hierarchy Partition Function Z Langevin Equations Fokker Planck Equation (Boltzmann – Vlasov Theory) Normal Diffusion Theory Statistical Entropy
Thermodynamical Theory (Equilibrium Thermodynamics Fluctuation Theory Central Limit Theorem)
Langevin Equation
Fokker Planck Equation Distributed Dynamics q-entropy extremization (Sq=max) Basic principle of nature
• Multifractal phase space structuring ▫ Random – intermittent timeseries - signals ▫ q-CLT theory ▫ q-triplet • Power laws, multiscale self-organization • Long range spatial – temporal correlations • Fractal – multifractal spatial distribution of fields – particles ▫ (Non-equilibrium stationary states, NESS) • Fractional Dynamics – Strange dynamics on fractals ▫ Anomalous diffusion ▫ Fractional Langevin process – fractional random walk ▫ Fractional Maxwell equations ▫ FHD – FMHD ▫ Fractional accelerations – anomalous diffusions in velocity space • K-energy distributions (non-equilibrium energy spectra) • Singular functions – multifractal timeseries
What is Complexity Theory ?
Irreversibility of time, dissipative structures, Entropy production and dynamics of correlations (I.Prigogine)
Indications from statistical physics
ftˆ() BBGKY ˆ Probability density Lˆ fˆ() t Hierarchy ft() t function ˆ f { f0 , f 1 ( x 1 ),..., f ( x 1 , x 2 ,..., x ),...} fˆ V f ˆ C f ˆ Vaccum + Correlation components of pdf
en ˆ dS fˆ Uˆ ( t ) f ˆ (0) eLt f ˆ (0) 0 Entropy production dt Subdynamics Mathematical Extension
• Hilbert Space • Rigged Hilbert Space • Euclidean Space • Non-Euclidean , Fractal Space • Gaussian Statistics • Non-Gaussian Levy Statistics • Smooth Diff. Functions • Singular Fractal. Functions
Smooth ODE-PDE Fractional Dynamics Complex Laws of Nature – Time Irreversibility
Gregoire Nicolis
Classical science is a marvelous algorithm explaining natural phenomena in terms of building blocks of the universe and their interactions. Physical phenomena reducible to a few fundamental interactions
Complexity • A new attitude concerning the description of nature • Self-organization phenomena on a macroscopic scale • Spatial patterns in on temporal rythms • Scale orders of magnitude much larger than the range of fundamental interactions • Order, regularization, information • Dissipative Structures, Symmetry breaking sudden transition from simple to complex Indications from Quantum Theory
• Commutative operators • Non Locality of quantum states • Coherence • Quantum Processes • Entaglement / Superposition • Quantum Vaccum
x x dx (), x x dx p p dp () p p dp d ii i
All the physical space {} x , the momentum space {} p and the energy space {} are included in the quantum state – physical objects.
Physical object-system
Holistic description of nature
(r , r ,..., r ) ( r ) ( r )..., ( r ) Non local Quantum 1 2NNN 1 1 2 2 Entaglement Quantum Processes as Complex Processes Quantum continous creativity of Nature at every scale, creation and destruction
1/2 aˆ( ki ) n1 , n 2 ,..., n i ,... n i n 1 , n 2 ,..., n i 1,... 1/2 aˆ ( ki ) n1 , n 2 ,..., n i ,... ( n i 1) n 1 , n 2 ,..., n i 1,...
k1, k 2 ,..., kNN aˆ ( k )... a ˆ ( k 2 ) a ˆ ( k 1 ) 0 , [aˆˆ ( kj ), a ( k i )] ij (t ) ( t ) n , n ,... nn12, ,... 1 2 Quantum Field Theory : states operators and observables ˆ(xk ) (2 )3/2 aˆ ( k ) eikx d Destruction ˆ (xk ) (2 ) 3/2 aˆ ( k ) e ikx d Creation Quantum Vaccum {states of patricles-fields-matter} Creation 0 {x1 , x 2 ,..., xN , k 1 , k 2 ,..., k N ,n 1 , n 2 ,... n i ,... } Complex System Quantum Complexity (M-Theory)
Quantum Vaccum Operators Quantum Objects, Quantum 0 States aaˆ ,, ˆx ˆ • Supersymmetry • D-dimensional Space • Super-gravity • Point Particles Complex • Super-strings • Strings System • Gauge Fields • p - branes • Symmetry Breaking • Universe • Renormalization • Multiverse
• Quantum Unification
• Time Reversibility Quantum Time Evolution
• Space-Time Symmetries ˆˆ • Internal Symmetries LHunif unif • Super Symmetry
Time Evolution
ˆˆ i H ,,(,), t U t t00 t t Schroendinger-Heiseberg– t Dirac Formulation Uˆ( t , t ) 1ˆ i H ˆ U ˆ ( , t ) d 00 t0
t ' i Feynman Path Integral ',t ' , t N D exp[ L ( , ) dx ] Formulation t Reductionistic Laws of Nature
• Basic Laws (time reversible) -- Classical Theory
()action (particle and fields S L dt 0 LLLLparticles fields gravity gravity)
• Probablilistics (Probability Amplitudes Quantum Theory) Feynman Path Integral
N-point quantum particle corrleation amplitude (expectation value) i t ' qtTqtqt,ˆ [ˆ ( ), ˆ ( ),..., qt ˆ ( )] qt , NDqqtqt [ ( ), ( ),..., qt ( )]exp[ Ldx ] 1 2NN 0 0 1 2 t
N-point quantum field corrleation amplitude (expectation value) i t ' 0Tˆ [ˆ (x t ),..., ˆ ( x t )] 0 N D [ ˆ ( x t ),..., ˆ ( x t )]exp[ L ( , ) d x ] 1 1 1 1 t Produced-Phenomenological Laws
Fundamental level (no entropy production) Phenomenological Level Entropy Production
• Macroscopic Time irreversible Laws- Phases
• Probabilistic or Deterministic Linear or Statistical • Microscopic Time non Linear Dynamics Reversible Dynamics Mechanics • Diffusion-Convection Fluids or Chemical Dynamics
• Population Dynamics
• Climate Dynamics , etc. The Entropy Principle as the Fundamental Law of Nature
Irreversible Diffusion in physical space – Holistic processes
Irreversible Diffusion in velocity space – Holistic processes The Entropy Principle as the Fundamental Law of Nature
Entropy production Entropy Principle S ent dS ent relaxation equilibrium 0,Sent k f (,,)ln(,,)x v t f x v t d x d v dt
: Relaxation Time Classical Science dS en • Deterministic Fundamental Laws 0 Time Reversible dt t
en Complexity Theory dS Deterministic Time-Reversible Entropy Principle 0 Dynamical Laws dt
Fundamental Principle Derived Laws, Local
Interactions Global Holistic Entropy Dynamics Thermodynamics as Fundamental Theory of Nature d Cantor – Kelvin – Clausious – Boltzmann – Gibbs - Prigogine 0 Entropy Principle through Boltzmann Complexous dt
Skent ln Fundamental Definitnion (Boltzmann 1872)
Number of microscopic states of the thermodynamic system
Fundamental law of Nature
“ increases to the maximum value for isolated systems (microcanonical ensemble) “ Canonical Thermodynamic System Environment Entropy Principle Isolated microcanonical system dSent d ent ent ent all all all E SSS E 00 dt dt dSent d r S ent d i S ent dSr ent reversible flow of entropy Equillibrium State r ent i ent ent i ent d S d S dSE 0 dS irreversible production of entropy Thermodynamic Potentials and Extremum Principle Fundamental Laws - Entropy Law
I. dS 0 Isolated System (microcanonical) IV. dH 0 when S,., P const P pressure Uint. energy const . H U PV enthalpy
V volume const. V. dG 0 when T,., P const P pressure II. dU 0 when S,., V const S entropy G U TS PV,' Gibb s free Energy
III. dF 0 when T ,, V const T temperature
F U TS,' Helmholtz s free Energy Equillibrium-Metaequillibrium Thermodynamic States
F meta equillibrium • Local minimums of Free Energy • Local Maximums of Entropy Equillibrium-Metaequillibrium Phase Transitions
equillibrium λ Three stages of Thermodynamic Processes
First Stage : Thermodynamics of Equillibrium SS max
1 ET / (Bolzmann Probability Distribution Law) pe i i Z
probablitiy of microstate (i), energy of the system at microstate (i) pi Ei
ET/ Ze i partition function of the system (canonical statistics) i
Physical Meaning of partition function Z for canonical system
Z eETEi / ( E ) e , E E E E , 1/ T i ii
U E p E ln Z / i ii (internal energy)
P P p P 1 ln Z / (pressure) ii i i
FT/ 1 F Tln Z Z e Sln Z UT pii ln p (free energy) (Entropy) Three stages of Thermodynamic Processes First Stage : Thermodynamics of Equillibrium
S pln p ii Entropy as functional of probablity
For small Fluctuations ΔΕ of E from mean U :
U lnZ ln (UU ) Z () U e SUlnZ UT1 ln ( ) (): U number of microstates for canonical statistics
Second Stage : Near Equillibrium Thermodynamics
2 SSSS 0, S 0, = equllibrium state , = canonical system
2 SS 0 , 0 Theorem of minimum Entropy production
Three stages of Thermodynamic Processes
Second Stage : Near Equillibrium Thermodynamics
Normal Fokker - Planck Equation 2
[(,)]fxt [(,)(,)] Axtfxt 2 [(,)(,)] Dxtfxt t x x prob. density Drift Diffusion
Ornstein – Uhlenbeck Process
1 2 f(,) x t [ a f (,)] x t 2 [(,)] f x t t x 2 x 2 Gaussian Evolution Gaussian Stationary Solution 2 a 22 1/2 (x x0 ) /2 Dt 1/2ax /2 f ( x , t ) (2 Dt ) e f ()() x 2 e
Boltzmann - Gibbs entropy extremization
1 Ei / SBG p iln p i p i Z e i p E const. i ii
Three stages of Thermodynamic Processes Third Stage : Nonequillibrium Statistical Mechanics Boltzmann-Gibbs, Tsallis and other entropic forms Far from equillibrium Thermodynamics
2 2 SSSS 0, S 0 decrease of entropy
• Self-organization • Dissipative Structures • Metaequillibrium stationary states , (NESS)
Tsallis q-Entropy q q qi / p p E const. E i 1 p i ii qi i i 1 S q pe iq q-Gaussian Disribution 1 q Zq
q-Thermodynamics
qiE 1 qq Ze , SZ ln , TSU /, Uq p i E i/ p i qqi q q q qq ii
1 U q ln q Z q / , FUTS q q q ln q Z q q=1 Boltzmann-Gibbs Statistics
Three stages of Thermodynamic Processes
Third Stage : Nonequillibrium Statistical Mechanics – Tsallis and other entropic forms - Fractional Processes
Fractional Kinetics fractional order
• Fractional FPE – Anomalous Diffusion • Fractional Dynamics on Fractals (,) • Fractional Space-Time Derivatives-Integration tx
Fractal Time Contraction Fractal Space Contraction Processes Processes t x
• Supper (persistent) diffusion 1<α<2 • Levy Flights/walks , Levy-Tsallis • Sub (antipersistent) diffusion 0<α<1 • Levy non local processes • Power law waiting time distribution • Power law distributions • Non-Markovian memory property • Fractal Geometry (Physical/Phase) • Long range time correlations • Fractal Matter-Fields distributions Mean – square displacement Normal – Anomalous Diffusion Non-equilibrium Stationary States, Percolation Critical States
2 2H x t Dt Dt , 0 H 1
ds 2 , 1 2 df 2
(Non Gaussian dynamics)
0, μ 1 subdiffusion 1, 0, μ 1 superdiffusion 2H , ds=spectral fractal dimension θ=connectivity index df= Hausdorff fractal dimension of space H=Hurst exponent dw=1/H=fractal dimension of trajectories 1, H 1/2, dw 2 Normal diffusion (Gaussian dynamics) μ={competition between FTRW – Levy process} Poincare Section of Phase Space Intermittency in Phase Space
• Stochastic Sea • Fractal Set (Island, Cantori) • Trapping – Stickiness – Levy flights • Lyapunov Exponents ≥ 0 • Strange Topology – Strange Dynamics • Scale Invariance (RGT)
Levy Distribution – Anomalous Diffusion Scale Invariant Distributed Dynamics
BBGKY Hierarchy Boltzmann - Vlassov, MHD Theory, HD Theory Scale Invariance - Singularities tt 1 /3 1ht u /3 uh u bb /3
Fractional Extension Fractional Real - Space Derivative Fractional Time Derivative
xi 1 dxi α f t,r f t,r m t 1 dr xi 1 xi x x f t,r f t,r i i a m 1am t m a t 0 t r i eˆ Riemann – Liouville operator r r i xi Fractal – time random walks (FTRW) 2 2 2 i,k Fractal active time (Cantor set) r 2 r i,k xi xk 1 a r persistent (super-diffusive) process Riesz – Weyl operator 0 a 1 anti-persistent (sub-diffusive) process Levy flights – Levy walks Waiting time Power Law distribution
Power Law distribution 1 r ~ r1a 1 12 Strange Dynamics in Phase Space
FRACTIONAL KINETIC EQUATION (FRACTIONAL FOKKER–PLANCK–KOLMOGOROV EQUATION) Zaslavsky G.M., Chaos, fractional kinetics, and anomalous transport, Physics Reports 371, 461-580, 2002.
Three stages of Thermodynamic Processes Entropic Principles non q-descripable q-descripable
q 1 q 1 others
global long range correlations
local gaussian correlation C. Tsallis, Introduction to Nonextensive Statistical Mechanics, Springer, 2009
Other Entropic Forms
• Renyi , Curado • Kaniadakis κ-entropy • Arneodo – Plastino • Sharma-Mittal • Landsberg-Vedral • Beck – Cohen superstatistics • Abe entropy • Spectral Statistics Theoretical Concepts
Near Thermodynamic Equilibrium Far from Thermodynamic Equilibrium
Euclidean Geometry-Topology Fractal Geometry -singular functions Smooth Functions – Smooth differential Equations Fractional differential-integral equations Normal derivatives-integrals Fractional HD - MHD theory HD – MHD – Vlason-Blotzman theory Anomalous diffusion – motion Normal diffusion-Brownian motion Strange Kinetics, Fractional Accelerations, K- Gaussian statistics-dynamics distributions Normal Langevin-FP equations Non-Gaussian statistics-dynamics Extensive statistics – BG entropy Fractional Langevin-FP equations Infinite dimensional noise Non-Extensive statistics – Extremization of Tsallis White-colored noise entropy Normal Central Limit Theorem (CLT) q-extended CLT Intermittent turbulence
Normal Liouville theory Fractional Liouville theory Locality in space and time (multi)Fractal Topology Separation of time-spatial scales Power laws – mulitscale processes Microscopic-macroscopic locality Memory – long range correlations Equilibrium RG Nonlocality in space and time Non-Equilibrium RG Non-Equilibriun Dynamical Phase Transitions
(A) dx/ dt f (x,) Gaussian Equilibrium Non – Linear Dynamics Critical States Equilibrium Phase Transition x Power Laws
(B) Self-Organization Structure Α Dissipative Structures Β Long Range Correlations
C (C) Spatiotemporal Chaos λ(i) Critical Points Strange Dynamics (Attractors) F Levy Processes Bifurcation Points Scale Invariance Intermittent Turbulence Tsallis Entropy Fractal Topology C Anomalous Diffusion Β Fractional Acceleration Α λ K-Distributions q-Extension of Statistics
Complexity Theory Entropy Principle
Nonequilibrium self-organization 1 p q q i S kn W max Sq k pi nq pi k q q i q 1 1 F U TS n Z min Z e q q q q q q q
Fq (Free energy) = minimum => Nonequilibrium stationary states (NESS)
Tsallis Non-extensive Statistical Mechanics q-Gaussian distributions: Energy spectra, timeseries magnitudes
1/1q [1 (1 q)] eq Popt Zq Zq
q=1 => Boltzman – Gibbs Theory Kappa distributions
High Energy Spectra Tsallis Theory Non-extensive Statistical Mechanics q - Calculus
q-Product, q-Sum
q-Fourier Transform (FT) Tsallis Theory Non-extensive Statistical Mechanics q-entropy
Independents Sub-systems
Statistically Dependent Sub-systems Tsallis Theory Non-extensive Statistical Mechanics
Generalized Fokker-Planck Equations
Levy Distribution
q-Gaussian Distribution q-extension of Thermodynamics
1 q() Eiq V Ze Fq U q TS q ln qZ q qq conf
S / pq 1/ KT 1 q qi Uqqln qZ , conf TUq
qq 2 q UF q q E pi E i/ p i U q q CTTq 2 conf conf TTT
Non-equilibrium q-points correlations
1 Gn ( x , x ,..., x ) ( x ) ( x )... ( x ... Z ( J ) q1 2 n 1 2 nq q Z J()() x1 J xn q-extension of Central Limit Theorem (q-CLT) Theory of q-triplet of Tsallis Multiplicative Processes in Phase Space – Physical Space Multiplicative Processes (Holistic timeseries production) Multifractal Theory
Theiler J., Vol. 7, No. 6/June 1990/J. Opt. Soc. Am. A, 1055
generalized dimensions
most-dence points least-dence points
Information entropy
Information dimension Wikipedia
Rényi entropy Sq maximization produces multifractal phase space
•Fractional Maxwell equations, fractional distributions
•Fractional Langevin- FP equations
•Fractional- multi scale acceleration
•K distributions, singular- multi fractal time series, q-triplet Renormalization Group Theory (RGT)
Non-Equilibrium – Non-linear Complex Dynamics (Chang, 1992) i fφ,,, x t n x t t ii i 1,2,... Generalized Langevin Stochastic Equation
Ptφx , Random Field Distribution Function
Dxexp i Lφ , φ , x d x dt Stochastic Lagrangian Dynamics
Scale Invariance - RGT - Fixed Points (SOC, Chaos, Intermittency, NESS, etc.) L/ l RL
L'/' dl RL L ή '' dPm/() dl R L mn P n
L'( l ) Vk ( l ) U k V k0 exp( k l ) U k
Non-equilibrium – Non-extensive Random Field Theory (Partition Function Theory) Turbulence – Intermittent Turbulence
Uriel Frisch – Turbulence (The Legacy of A.N. Kolmogorov) ‘mother – eddy’ ‘daughters Tsallis q-entropy principle
probability density P(α)
singularity spectrum f(α)
turbulence mass exponent τ(q) β-model cascade (eddies without space filling) (q) (q 1)d(q) (β-model)
‘p-model q-triplet Tsallis One-Dimensional Systems (timeseries)
qrel
q qent=qsen stat Sq
t The q-triplet of Tsallis
q - CLT (qk-1, qk, qk+1)= qstat, qrel, qsen)
Gaussian-BG equilibrium (qstat=qsen=qrel=1) Nonequilibrium (qstat, qsen, qrel)
(qstat, qsen, qrel)
Metaequilibrium PDF Equilibrium PDF
(q-stationary)
Equilibrium BG entropy production Metaequilibrium q-entropy production
(q-sensitivity)
Equilibrium relaxation process Metaequilibrium nonextensive relaxation process
(q-relaxation) Experimental Verifications Universality of Tsallis q triplet
• Cosmic Stars • Space Plasmas • Turbulence • Energetic Particles • Climate • Seismicity • Biology • Human Body • Economic Systems • Information Systems Universality q_stat of Tsallis q triplet
q-triplet Cross Correlation q_sen q_stat, q_sen 0,590099636 q_stat, q_rel -0,101510833
q_rel Non-equilibrium Universality Renormalization Fix Points of Non-equilibrium Phase Transition
Increase Decrease Fm in Sq q_stat Cor. Dim. (D2) q_sen q_rel f(α) – Dq L1 (Lyap. Expon)
Non-equilibrium Free Energy Phase Transition
Far from Equilibrium
Far from Equilibrium
Equilibrium CHAOTIC ANALYSIS (SUNSPOT DYNAMICS) TSALLIS STATISTICS (SUNSPOT DYNAMICS)
Sunspot TMS V1 Component V2-10 Component q relaxation 2.522±0.044 5.255±0.308 2.426±0.054 q stationary 1.53±0.04 1.40±0.08 2.12±0.20 q sensibility 0.368± 0.005 0.055± 0.009 0.407±0.029
Δα=αmax-αmin 1.752± 0.003 1.133± 0.009 1.940±0.029 Interplanetary Observations (8 March 2012)
(Re -2,5)
The locations of spacecraft ACE, CLUSTER 4, THEMIS-E and C, on 8 March 2012 11:00 UT are shown. The green bow-shaped structure corresponds to the bow-shock, while(Re the -59,5) white -shaped structure to magnetopause and its axis are in the Geocentric Solar Ecliptic (GSE) system. Solar Wind – Magnetic Field
Calm - qstat= 1.25 ± 0.06 Shock - qstat=1.50 ± 0.05 ICME - qstat = 1.12 ± 0.05 Multi-spacecraft approach (Shock Period)
ACE CLUSTER 4 THEMIS E THEMIS C
(Shock Period) (Shock Period) (Shock Period) (Shock Period)
qstationary 1.69 ± 0.03 1.50 ± 0.05 1.47 ± 0.05 1.86 ± 0.04 qsensitivity -0.8002 ± 0.0223 -1.1787 ± 0.0717 -1.1145 ± 0.0843 -0.2246 ± 0.0149 qrelaxation 2.656 ±0.052 4.154 ± 0.434 4.794 ±0.345 3.155 ±0.175 Solar Energetic Particle
Quiet - qst at=1.08±0.03 SEP- qstat=1.18 ±0.05 Shock - qst at=1.77 ±0.09 ICME - qstat=1.16 ±0.04 Solar Energetic Particle (Multifractal Spectrum)
Quiet - qsen=-3.049 SEP - qsen=-0.521 Shock - qsen=0.427 ICME - qsen=-1.563 Δα=0.251 Δα=0.653 Δα=1.483 Δα=0.388
Quiet - ΔDq=0.155 SEP - ΔDq =0.556 Shock - ΔDq =1.371 ICME - ΔDq =0.287 p-model=0.546 p-model=0.612 p-model=0.753 p-model=0.566 Solar Wind – Ion Flux Extension of Thermodynamics Near Thermodynamic Equilibrium Far from Thermodynamic Equilibrium
• Boltzmann-Gibbs Entropy S BG and • Tsallis Entropy S q and other Entropies – Statistics. Statistics. • Gaussian probability density functions. • Levy – Tsallis non Gaussian distributions. • Normal Classical and Quantum diffusion. • Fractional Classical and Quantum Langevin, Fokker-Planch Equations. Diffusion, Langevin, Fokker-Planch • Classical Mechanics-Field Theory and Equations. QFT. • Fractional Mechanics-Field Theory and • Unified Quantum Fields, Electromagnetic QFT. Weak, Nuclear. • Unified E-W-N+ gravity . • Smooth Space-Time Manifolds Euclidean, • Fractal Space-Time Sets. Riemannian. • Critical Dissipative Self-Organized • Critical Self-Organized States. Structures. • Seperation of Dynamics and • Unification of Dynamics and Thermodynamics. Thermodynamics. • Matter ~ Energy • Matter ~ Energy ~ Information • Equllibrium RGT and Reduction of • Non Equllibrium RGT and Reduction of dimensionality (D-finite). dimensionality (D-infinite). • Normal Central Limit Theorem. • q-Gaussian Central Limit Theorem. From Thermodynamics to Classical and : smooth trajectory Quantum Dynamics : stochastic trajectory • State Space Manifold M • Classical Mechanics M=N qt1, • Quantum Theory = Hilbert space d • Nonequillibrium Dynamics = extended fractal space s qt, 00 No entropy production {,}HLˆc : probablility density in phase space Deterministic t d dS ˆ : density operator in quantum states time reversible 0 , 0 ˆ q dt dt : number of microstates Dynamics i[,] Hˆˆˆˆ L t S : entropy of the system
c Lˆ : Liouville operator in classical mechanics Entropy production ˆq: Liouville operator in quantum mechanics L c c Lˆ ˆ : classical Fokker-Planck operator Probabilistic FP LFP t d dS ˆq : quantum Fokker-Planck operator time irreversible 0 , 0 LFP ˆ ˆq dt dt Dynamics LFP ˆ t From Probabilism and Stochasticity to x x x x 0 1 i N Determinism s Path Integral Formulation lim ...dx d x ... d x d NN 01 N lim dDxx ... Ni i0 Fokker-Planck Process t t0 t1 ti tN dx f( x , t ) Dx exp[ Ls ( x , ) d ] f ( x , t ) s 00 Lˆ : Stochastic Lagrangian for Fokker-Planch equation dt t0
Quantum Mechanics (Feynman Formulation)
t1 2 i x,;,, t x t probability x, t ; x , t Dx exp[ L ( x , x ) d ] x,;,., t x00 t prob amplitude 00 00 t0 Probability = max Classical deterministic trajectory in phase space
Extremiztion of probablility Reversible deterministic Dynamics L s stochastic Lagrangian L deterministic Lagrangian
Correlations and Entropy Production
Stable Hamiltonial Dynamics S fln f dpdq df f df dS no entropy B {Hf , } 0,B 0 dt t p dt dt production f( p , q ) : pdf in phasespace Bogoliubov–Born–Green–Kirkwood–Yvon Hierarchy (BBGKY)
f(1) df (1) dS C { H , f (1)} C C 0, B 0 2 : two point correlations tp 22 dt dt diffusion drift C n : n-point correlations
f()() n df n dS SB f( n )ln f ( n ) {H , f ( n )} C C 0 ,B 0 tp n dt n dt dp()() n dq n drift diffusion
Fractional Extension of BBGKY Hierarchy
,{,}{,} , : n-point correlations t t q p q p non Gaussian statistics
Entropy Principles the basic tool for Unification Entropy Principle Thermodynamics : Z,F many point correlations
Quantum Field Theory n-point Green Function correlations
Thermodynamical it QFT n-point Green Functions Correlations Cn in in d-dimensional space (d+1) dimensional space ZJWJ()() E XE (,): ix0 x Wick rotation QFT Statistical Mechanics (d+1) space ~ Euclidean QFT in (d) space n GN ( x ,..., x ) 0 T [ (x ),..., ( x )] 0 , Nn WJ() 11nnG( x1 ,..., xn ) ( ) i J( x1 )... J ( xn ) W N Dexp[ i1 S ( )] Z ( J ): Generating Functi o nal EE Thermodynamics Z()() J eETi / partition functi o n QFT-Thermodynamics i 1 n ZJ ( ) t =-iτ
Cnn( x1 ,..., x ) (correlations ) {QFT , WE ( J )} {(),}Z J Thermodynamics Z J( x1 )...Jx (n ) QFT as self-organization process of Subquantum Thermodynamics
Hooft, Beck, Parisi , Stochastic – Chaotic Quantization
H (,): x t random field , (,) x tN (Random Field t Langevin equation)
, N : ucnorrelated noise Ni(,)x t N j (',') x t ij ( x x ')( t t ')
Random Field Fokker-Planck Equation 1 H ( ) P(,) t { P (,) t [ P (,)]} t t 1 Equillibrium Solution : P ( , t ) ~ exp[ H ( )] (stationary state)
Random Field FPE Quantum Field Schroendinger Equation ( Thermodynamics ) (Quantum Field Theory) QFT as self-organization process of Subquantum Thermodynamics
Random Field FPE Quantum Field Schroendinger Equation ( Thermodynamics ) (Quantum Field Theory)
Quantum Hamiltonian Operator
H ˆ pˆ 2 [()()] H ˆˆ i where [,]pˆˆ i ,[,][,] p ˆ p ˆ ˆ ˆ 0
QFT Schroendinger Equation
ˆ d ˆ ˆ ˆ Time Evolution Operator U (,) t t 0 : i U (,)(,) t t 00 H U t t dt
ˆ Schroendinger Field Equation : P((,),)xx t t U (,)((,),) t t0 P t 0 t 0 t Uˆˆ( t , t ) T exp[ i H d ] 0 t0 J. Binney et al., The Theory of Critical Phenomena: An Introduction to the Renormalization Group, Oxford Univ. Press, 1992. Why Study Phase Transitions ? First phase changes are immensely influential in every corner of the Universe-indeed it is widely argued that the very existence of the observable Universe is attributable to a phase change in the state of some pre-existing vacuum, and that the disposition of matter in and around galaxies should be understood in terms of fluctuations associated with some such transition (see for example Kolb and Turner 1989).
Not only is the creation of long-range structure by short-range inter-molecular forces intriguing, but any example of scale-freedom is worthy of close examination since this phenomenon occurs in several physical systems that are inadequately understood-the clustering of galaxies (e.g., Peebles 1980), the distribution of earthquakes (e.g., Carlson and Langer 1989, Bak et al. 1988), turbulence in fluids and plasmas (e.g., Mandlebrot 1974), polymers (de Gennes 1972), snow flakes (Ballet al. 1989, Meakin and Tolman 1989)-to name but a few. In each case there is a wide range of scales over which some phenomenon varies as a power law of the scale, presumably because there is a gross mismatch between the largest and smallest scales in the problem. J. Binney et al., The Theory of Critical Phenomena: An Introduction to the Renormalization Group, Oxford Univ. Press, 1992.
An elementary particle is represented by a structure of a certain physical size on the lattice. As the lattice is refined this structure should retain its physical size by covering more and more lattice sites. Hence, as the discrete model approaches the continuum limit of real quantum fields, the particle must be represented by correlations on the lattice of longer and longer range, and the field theory that gives rise to these correlations must be approaching what in statistical mechanics we would call a critical point. Elementary particles as Thermodynamic self- organization process
Random Field n-point Corelations in D-dimensions n-point Green’s Functions x D n-point interactions
G(x1 , x 2 ,..., xN ) ( x 1 ), ( x 2 )... ( x )
1 Fd() xD D (x12 ), ( x )... ( x )e where F ()() h J , h () field energy density, J source field
FdxD Z () J D e Random Field Partition Funnction
n-point correlation function n n-point particle interactions n 1 ZJ ( ) G ( x , x ,..., x ) n-point correlations 12 n Z J( x )... J ( x ) 1 n J 0 Elementary particles as Thermodynamic self- organization process Random Field
Random Field xn x3 ()x • Self-organization Interactive x • Phase Transitions Particle System 1 • Long-range x2 Correlations Stochastic Process as macroscopic quantity Diffusion Process 2 , velocity random variable v x/ t f(,)(,) x t D0 f x t t Statistical-Thermodynamical Uncertainty Principle
x t D0
Quanticity and Stochasticity
StochasticProcesses Quantum Processes C. Beck, Chaotic strings and standard model parameters, Physica D: Nonlinear Phenomena, 171, 72-106, 2002.
A fundamental problem of particle physics is the fact that there are about 25 free fundamental constants which are not understood on a theoretical basis. These constants are essentially the values of the three coupling constants, the quark and lepton masses, the W and Higgs boson mass, and various mass mixing angles. An explanation of the observed numerical values is ultimately expected to come from a larger theory that embeds the standard model. Prime candidates for this are superstring and M theory. However, so far the predictive power of these and other theories is not large enough to allow for precise numerical predictions.
We will report on a numerical observation that may shed more light on this problem. We have found that there is a simple class of 1+1 -dimensional strongly self-interacting discrete field theories (called ’chaotic strings’ in the following) that have a remarkable property. The expectation of the vacuum energy of these strings is minimized for string couplings that numerically coincide with running standard model couplings α(E), the energy E being given by the masses of the known quarks, leptons, and gauge bosons. C. Beck, Chaotic strings and standard model parameters, Physica D: Nonlinear Phenomena, 171, 72-106, 2002.
Chaotic strings can thus be used to provide theoretical arguments why certain standard model parameters are realized in nature, others are not. We may assume that the a priori free parameters evolve to the local minima of the effective potentials generated by the chaotic strings. Out of the many possible vacua, chaotic strings may select the physically relevant vacuum of superstring theories.
The dynamics of the chaotic strings is discrete in both space and time and exhibits strongest possible chaotic behaviour. It can be regarded as a dynamics of vacuum fluctuations that can be used to 2nd-quantize other fields, for example ordinary standard model fields, or ordinary strings, by dynamically generating the noise of the Parisi-Wu approach of stochastic quantization on a very small scale. Mathematically, chaotic strings are coupled map lattices of diffusively coupled Tchebyscheff maps TN of order N. It turns out that there are six different relevant chaotic string theories - similar to the six components that make up M-theory in the moduli space of superstring theory. T.S. Biro et al., Chaotic Quantization of Classical Gauge Fields, Foundations of Physics Letters, 14(5), 2001.
We argue that the quantized non-Abelian gauge theory can be obtained as the infrared limit of the corresponding classical gauge theory in a higher dimension. We show how the transformation from classical to quantum field theory emerges, and calculate Planck's constant from quantities defined in the underlying classical gauge theory.
Although much progress has been made in recent years, the question, how gravitation and quantum mechanics should be combined into one consistent unified theory of fundamental interactions, is still open. Superstring theory, which describes four- dimensional space-time as the low-energy limit of a ten- or eleven-dimensional theory ( “M-theory”), may provide the correct answer, but the precise form and content of the theory is not yet entirely clear. It is therefore legitimate to raise the question whether the fundamental description of nature at the Planck scale is really quantum mechanical, or whether the underlying theory could be a classical extension of general relativity. This questions was initially raised by 't Hooft, who has argued that quantum mechanics can logically arise as low-energy limit of a microscopically deterministic, dissipative theory T.S. Biro et al., Chaotic Quantization of Classical Gauge Fields, Foundations of Physics Letters, 14(5), 2001.
We here show that in some cases, specifically for non-Abelian gauge fields, the functional integral of the three-dimensional Euclidean quantum field theory arises naturally as the long-distance limit of the corresponding classical gauge theory defined in (3+1)-dimensional Minkowski space. Because of the general nature of the mechanism underlying this transformation, for which we have coined the term chaotic quantization, it is expected to work equally well in other dimensions. For example, the four-dimensional Euclidean quantum gauge theory arises as the infrared limit of the (4+1)-dimensional classical gauge theory. We emphasize that the dimensional reduction is not caused by compactification; the classical field theory does not exhibit periodicity either in real or imaginary time. Stochastic Process as macroscopic quanticity Quanticity and Stochasticity StochasticProcesses Quantum Processes Fokker-Planck Equation
2 fxt(,)(,)()(,) fxt Vxfxt ( Feynman-Kac Formuma) t t it Wick Rotation 2 ( Schroendinger Equation ) i fxt(,)(,)()(,) fxtVxfxt t Quantum Theory ~ Subquantum Stochastic Processes
Significant Analogy Entropy
S pln p x dx x ii ii i i all microstates quantum eigenstates participate Cosmic Thermodynamics – Cosmic Novelty
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The Cosmic Tree of Bifurcation .... 1 2 The Cosmif Fractal Entropic non-Equillibrium Cosmos
Symmetry Breaking-Cosmic self-organization Cosmic Renormalization
• Cosmic Fractal self-similarity • Cosmic Self-Organization • Long Range Space-Time • Multi-level Renormalization Correlations • Fixed Points • Cosmic Evolution • Bifurcation Points • Reduction of Dimensionality • Multiscale Cohenrence – Cooperations • Symmetry Breaking • Entropy Principle Cronos Principle • Multiplicative Structuring • Contraction of Fractal Space-Time • Non-Linear Evolution • Scale Relativity • Entropy and Information • Scale Unification Production • Fractal Laws • Goedel non-Computable • Dissipative Structures non-Algoritmic Dynamics • Symmetries THANK YOU for your attention