J Mech Behav Mater 2017; 26(5–6): 139–180
George P. Pavlos* Complexity theory, time series analysis and Tsallis q-entropy principle part one: theoretical aspects https://doi.org/10.1515/jmbm-2017-0023 level for a unified description of macroscopic and micro- scopic levels, the relation between the observer and the Abstract: In this study, we present the highlights of com- examined object, etc. In general, as far as the complex- plexity theory (Part I) and significant experimental verifi- ity theory and every new level of the physical reality are cations (Part II) and we try to give a synoptic description concerned, new concepts and new classifications are of complexity theory both at the microscopic and at the required. Initially, the first principles of the physical macroscopic level of the physical reality. Also, we pro- theory modeled the entire cosmos, using a reductionist pose that the self-organization observed macroscopically point of view, as a deterministic, integrable, conserva- is a phenomenon that reveals the strong unifying char- tive, mechanistic and objectifying whole. However, acter of the complex dynamics which includes thermo- Boltzmann’s probabilistic interpretation of entropy dynamical and dynamical characteristics in all levels of as well as the non-integrability of the large Poincare the physical reality. From this point of view, macroscopi- systems rose for the first time serious doubts concern- cal deterministic and stochastic processes are closely ing the universality and plausibility of the primitive related to the microscopical chaos and self-organization. classical physical theory. Moreover, two great revolu- The scientific work of scientists such as Wilson, Nicolis, tions reconfigured the absolute objectifying character Prigogine, Hooft, Nottale, El Naschie, Castro, Tsallis, of the classical physical theory. The first one, concern- Chang and others is used for the development of a uni- ing the macrocosm, was the relativity theory of Einstein fied physical comprehension of complex dynamics from and the second one, concerning the microcosm, was the microscopic to the macroscopic level. Finally, we pro- the Quantum Theory of Heisenberg, Schrodinger and vide a comprehensive description of the novel concepts Dirac. Finally, the reductionist character of the physical included in the complexity theory from microscopic to theory has been disputed after the scientific job of Fei- macroscopic level. Some of the modern concepts that can genbaum, Poincare, Lorenz, Prigogine, Nicolis, Ruelle, be used for a unified description of complex systems and Takens and other scientists who founded the chaos and for the understanding of modern complexity theory, as complexity theory. Recently, Ord, Nottale, El Naschie, it is manifested at the macroscopic and the microscopic Tsallis and other scientists introduced new theoretical level, are the fractal geometry and fractal space-time, and fertile concepts for the complex nature of the physi- scale invariance and scale relativity, phase transition and cal reality and the unification of the physical theory at self-organization, path integral amplitudes, renormaliza- the microscopic and macroscopic levels. In particular, tion group theory, stochastic and chaotic quantization complexity theory includes: chaotic dynamics in finite and E-infinite theory, etc. or infinite dimensional state space, far from equilibrium Keywords: fractal acceleration; fractal dissipation; fractal phase transitions, long range correlations and pattern dynamics; intermittent turbulence; non-equilibrium formation, self-organization and multi-scale coop- dynamics; Tsallis non-extensive statistics. eration from the microscopic to the macroscopic level, fractal processes in space and time and other significant phenomena. Complexity theory is considered as the third scientific revolution of the last century (after Rela- 1 Introduction tivity and Quantum Theory). However, there is no sys- tematic or axiomatic foundation of complexity theory. The great challenge of complexity theory emerges from In this direction, a significant contribution concerning old and essential problems such as: the time arrow, the the question “what is Complexity”, can be found in the existence or not of a simple and fundamental physical book “exploring complexity”, written by G. Nicolis and Ilya Prigogine, where some complementary definitions
*Corresponding author: George P. Pavlos, Department of Electrical of complexity can be found. and Computer Engineering, Democritus University of Thrace, 67100 Generally, we can summarize the basic concept of Xanthi, Greece, e-mail: [email protected] complexity theory as follows: 140 G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one a) Complexity theory is the generalization of statistical at the macroscopical level of the microscopic reality. physics for critical states of thermodynamical equilib- However, in Complexity theory happens the opposite to rium and for far from equilibrium physical processes. this. Namely, entropy is the fundamental physical cause b) Complexity is the extension of dynamics to the non- of becoming, while Newtonian force is the macroscopic or linearity and strange dynamics. the microscopic result and the macroscopic-microscopic c) Also, according to Ilya Prigogine complexity theory phenomenology of the holistically working entropy prin- is related to the dynamics of correlations instead of ciple. We note here that the concept of force is absent even dynamics of trajectories or wave functions. in the general relativity theory as well as in the quantum theory where the physical description of reality is seceded Complexity theory was entranced in the physical theory in abstract mathematical forms, as Riemannian geom- simultaneously with the entropy principle after Carnot, etries or operators in Hilbert spaces. Also according to the Thomson, Clausius and finaly by the statistical theory of complexity theory, physical beings and physical struc- Boltzmann who unified the macroscopic and the micro- tures, are holistically sustained dissipative structures, scopic theory of entropy by introducing the concept of produced by the general process of nature aiming to the microscopical “complexions”. According to the Complex- maximization of entropy. This happens even at the level ity theory, different physical phenomena occurring at dis- of “elementary” or fundamental particles, as in any kind tributed physical systems such as space plasmas, fluids of physical process, at the macro or the microscopic level, or materials, chemistry, biology (evolution, population nature must satisfy the entropy principle. After this, from dynamics), ecosystems, DNA or cancer dynamics, social- the complexity point of view, there is no significant dif- economical or informational systems, networks, and in ferentiation between the group of galaxies, the stars, general defect development in distributed systems, can be the animals, the flowers, or the elementary particles, as described and understood in a similar way. This descrip- everywhere we have open, dynamical and self-organized tion is based at the principle of entropy maximization. systems and everywhere nature works to maximize the Therefore, complexity theory can also be considered as entropy. Therefore, the generalization by Tsallis of the the entropy theory in all its generalizations and implica- Boltzmann-Gibbs (BG) entropy to the q-entropy through tions, as it is described in the following sections. his non-extensive statistical theory constitutes the unifi- Entropy principle includes all the new and non- cation of all the distinct characters of natural complexity, Newtonian characteristics of physics, which constitutes near or far from the thermodynamic equilibrium. the complexity theory in its deepest manifestation. In The exponential character of BG statistics permits the Newtonian physics when we ask the question, how the distinction between scales at micro and macro level while nature works? We can answer: by natural forces, gravity the power law character of Tsallis non-extensive statistics or any other kind of force producing mechanical work unifies all the physical scales through the development of and energy. This is the basic physical concept, which strong long range correlations and multilevel interactions. includes also dynamical fields as the local extension of Tsallis q-extension of statistics includes the BG statistics the force at distance concept. From the mathematical as the limit of statically theory when q tends to the value point of view even relativity or quantum theory, except one (q = 1). Here we have similar behavior with quantum the quantum measurement reduction phenomenons, both mechanical theory where the classic mechanics corre- of them belong to the Newtonian type of theory as they sponds to the limit h = 0 and relativistic mechanical theory conserve the more general Newtonian characteristics of where classic mechanics corresponds to the limit c = infi- determinism and temporal reversibility, which constitute nite. That is the BG entropy principle describes the world the nucleus of Newtonian physics. In the non-Newtonian near thermodynamic equilibrium while as Tsallis [1], [2] and non-reductionistic theory of complexity, the answer has shown this principle must be extended for the far from to the question, how nature works is this: Nature tries equilibrium processes as the Tsallis q-entropy principle. to maximize the entropy. In Newtonian physical theory This generalization of entropy principle by Tsallis can the force is the fundamental reality and it is the cause of produce the multilevel, multi scale or long range correla- becoming while entropy is a subjective phenomenologi- tions observed at the complex systems. cal characteristic out of the basic theory. That is At Newto- In this way the extended by Tsallis entropy principle nian physical theory entropy is the secondary result at the applied everywhere, near or far from thermodynamical macroscopical level of forces and dynamics acting at the equilibrium manifests itself as a universal physical prin- microscopic level. Although however there is no final or ciple, which can explain the holistic-complex character physically accepted explanation of such a manifestation of physical processes. That is the whole is richer than its G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one 141 parts as it includes some kind of reality more than the this way the physical principle of the entropy maximiza- reality of its parts. The part studied as itself and indepen- tion describes how the whole creates its parts as it was dently from the whole, is characterized by some kind of supported before. In this way the entropy maximization mechanistic behavior, which is manifested as force, flow creates and structures the physical states, in the classical of energy and momentum, as noise and flow of informa- or the quantum phase space, or the structures in the phys- tion. This corresponds to the random Walk modeling of ical space. Parallels the entropy principle leads the physi- every dynamical process, or the Langevin-Fokker Planck cal processes in the dynamics phase space or the physical process type description of the dynamics applied to the space. Namely, there is a internal self-consistency between part embedded in the whole. The random Walk process entropy maximization principle, the topological charac- near thermodynamic equilibrium is known as normal ter of the created physical states in the phase space and diffusion process, while far from equilibrium is known the physical processes in the state space and the physi- as anomalous diffusion and strange dynamic process. cal space. That is mathematical structuring of the set of However, the entropy maximization principle reveals that physical states in phase or physical space corresponds to the blind forces and noises acting at the random Walk fractal sets and fractal topology. Near the thermodynamic process are not “blind” but they work in harmony with equilibrium the physical states structuring corresponds to the entropy maximization. May be the entropy principle the Euclidean geometry and topology, while far from equi- causes the random Walk in a far unified physical theory. librium it corresponds to the strange or anomalous non It is significant to note here that the strong nonlinear- Euclidean topology. All these are described quantitatively ity of the deterministic dynamics which includes strong by the topological parameter known as the connectiv- dynamical instabilities, as sensitivity to initial conditions, ity index θ which takes values zero (θ = 0) and greater or acts equivalently with the random Walk entropic process lower than 0 (θ > 0, θ < 0) if the topology is Euclidean or near or far from the thermodynamical equilibrium state. non Euclidean correspondingly. The physical processes in The near equilibrium random Walk process is memoryless the phase or physical space are described correspondingly Markov process but far from equilibrium obtains memory by fractional differential or integral equations for topolo- as non-Gaussian and non-Markov process. The wholis- gies with (θ ≠ 0) different than zero and normal differen- tic character of entropy maximization principle creates tial or integral equations for θ = 0. Analogously with the probabilistically the whole of the possible physical states above description, the physical magnitudes functions in at which the whole and its parts can be found. More over the phase space or physical space are related with time or the entropy principle can unify quantum and complex between them can be normal smooth that is infinitely dif- systems through the general theory of many point correla- ferentiablefunctions for θ = 0 or singular fractional func- tions. It is already known that the probability amplitudes tions for θ ≠ 0. of Quantum Field Theory (QFT) can be estimated through In general, the fractal character of the state sets in the statistical correlations at thermodynamical equilib- phase or physical space and the correspondent fractional rium of the quantum system embedded in a higher dimen- physical processes, they can satisfy spatial and temporal sion, according to the stochastic or chaotic quantization scale invariance principles, as well as power law relations. [3, 4]. This is the deeper meaning of the development of long From this point of view, the long range correlations range and multi scale correlations as the entropy as the of the complexity theory are nothing else than the macro- extended entropy of the system must be maximized. That scopic manifestation of the quantum entanglement where is the non-Gaussian character of non-equilibrium statisti- the parts cannot be divided from the whole as they are cal theory, known as the non-extensive Tsallis theory can strongly correlated in it, while the quantum entanglement create many point correlations (long range correlations) could be considered as the the manifestation also of sub- estimated by the functional derivative of the q-extended dynamical complexity and self organization [5, 6]. partition function. Therefore, while the BG entropy prin- Also, it is significant to note that the equivalence of ciple is related with two point Gaussian correlations, the quantum theory and the stochastic processes according Tsallis entropy principle is related with many points cor- to chaotic and stochastic quantization indicates the uni- relations which means long range correlations. Also the versality of Tsallis entropy principle both at the macro- statistics depends upon the topological character of the scopical as well as at the microscopical level. That is the state space as the normal Central Limit Theorem (CLT) cor- entropy extramization at the microscopic level causes responds to connectivity θ = 0 while for the strange topol- the elementary particle structuring as a microscopic self- ogy of state space the extended statistics causes the well organization process. From a general point of view in known as q-extended central limit theorem (q-CLT). Also 142 G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one
The Tsallis q-extension of CLT produce a series of charac- fractal topology structures, of percolation states and non- teristic indices corresponding to different physical pro- extensive statistics in phase space. cesses, the most significant of which constitute the Tsallis The intermittent turbulent states and the complex spa- q-triplet. tial-temporal behavior of nonlinear distributed dynamical The scale invariance character of fractal structures systems, can be considered as the spatial-temporal mir- and the fractional character of processes on fractals near roring of the strange or fractal topology of the distributed or far from thermodynamical equilibrium is related to the dynamics phase space and the included strange fractional Renormalization Group Theory (RGT) which leads to the dynamics in the physical space. As the control parameters reduction of dimensionality of the degrees of freedom. of the system change, the system dynamics can generate This process of the reduction of the degrees of freedom topological and dynamical phase transition processes is harmonized to or caused also by the maximization of developing new equilibrium or meta-equilibrium states entropy near equilibrium critical or far from equilibrium corresponding to local extremization of q-entropy and stationary critical states. In this way, the entropy maximi- new fixed points, according to non-equilibrium renormal- zation creates low dimensional stationary structures and ization theory [10]. low dimensional dynamical processes. The theoretical description of complex systems or The q-extension of the CLT as well as the far from equi- complex dynamics according to the extended entropy librium extension of the RGT are related with the general principle can be studied by using experimental signals in theory of strange dynamics-strange kinetics [7] which cor- the form of Time-space series, which are the one dimen- responds to strange topology of the phase space. sional projection of complex processes in the phase or the The fractal topology of the phase space is produced physical space. That is significant characteristics of the by the fractional dynamics of the complex system related physical processes are mirrored in the time- space series to anomalous diffusion processes, multiscale coopera- which can manifest the extensive-non extensive charac- tion and long-range correlations as well as development ter of entropy, as well as the normal or fractional-strange of memory processes and percolation clusters [8, 9]. The character as well as the dimensionality of the mirrored phase space of nonlinear dynamics can reveal strong in dynamics according to the embedding theory of Takens homogeneity including a complex, multi scale and hier- [11]. In this way by using the time series observations we archical system of islands and stochastic sea islands can estimate the Tsallis q-triplet which give useful infor- correspond to stable nonrandom orbitals immersed in mation for the q-extended statistics of the dynamical the stochastic sea of chaotic phase space. Cantorus or process. Such information concerns the multifractality cantori are the boundaries between the island or system of the state space, the relaxation of the dynamics toward of islands and the stochastic sea, which can create trap- the equilibrium state and the entropy production rate of ping of the dynamics and creates flights of the dynamics the observationally mirrored physical system at the time perpendicular to the boundaries of the islands. Cantorus series numbers. can be imagined as a fractal curve or line including an Parallel to the physical manifestation of complexity infinite number of gaps immersed in the stochastic sea there exist also the mathematical manifestation of com- of phase space corresponding to the nonlinear dynam- plexity. Cantor was the founder of mathematical theory ics. All points of a cantorus belong to the same orbit if the by introducing the set theory including smooth Euclidean initial condition has been chosen from the cantorus mani- sets as well as fractal sets as was the first fractal set known fold. Every islands can be enclosed with an infinite set of as the Cantor set or the Candor dust. Smooth differentia- cantori the system of which can produce trapping or stick- ble functions and smooth differentiable spaces are related ing (“stickiness”) and flights of dynamics as the comple- with Gaussian statistics which permits the discrimina- mentary features of anomalous diffusion of the nonlinear tion of macroscopic smooth Euclidean scales excluding dynamics in the complex phase space. This is the meaning the non-Euclidean or singular microscopic scales. On the of complex or strange dynamics of complex and nonlinear other side the Tsallis non-Gaussian and non-extensive systems, which is related with the intermittency character statistics unifies all the scales so that the microscopic of anomalous diffusion in the phase space or the physical non-Euclidean geometry and singularity of physical mag- space in the case of nonlinear and distributed dynamics nitudes can be emerged to the macroscopic scales. as the case of space plasma dynamics. Moreover, strange After this general description of the basic concepts of dynamics creates multiscale and multifractal structures the complexity theory and the extended Tsallis entropy in the phase space or the physical space. Strange dynam- principle, we present in the following analytically the ics is the fractional dynamical manifestation also of the plurality of different kind of theoretical or experimental G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one 143 manifestations of the complexity theory at distributed level. If we accept this extreme scientific concept then we complex systems in generally. must accept also the new point of view, that the classical In the following, in Sections (2–6) we present the ther- kinetics is insufficient to describe the emerging complex oretical framework and methodology of data analysis for character as the the physical system lives far from equi- understanding the experimental observations by distrib- librium. Moreover, recent evolution of the physical theory, uted dynamical systems. Finally, in Section (7) we summa- centered on nonlinearity, non-extensivity and fractality, rize and discuss the highlights of complexity theory and shows that Prigogine’s point of view was not as extreme as applications. it was considered at the beginning. After all, Tsallis exten- sion of statistics [2] and the fractal extension for dynamics of complex systems as it was developed by Zaslavsky [24], Castro [12], Tarasov [25, 26], El Naschie [27], Milovanov 2 Theoretical concepts and Rasmussen [28], El-Nabulsi [29], Nottale [13], Goldfain [30] and many others scientists, are the double face of a 2.1 Complexity theory and the cosmic unified novel theoretical framework. In this way they con- ordering principle stitute the appropriate base for the modern study of non- equilibrium dynamics, since the q-statistics is related, at According to previous introductory remarks, the com- its foundation, to the underlying fractal dynamics of the plexity theory concerns every kind of non-equilibrium non-equilibrium states. distributed dynamics or equilibrium critical distributed For complex systems near equilibrium, the underly- dynamics. ing dynamics and the statistics are Gaussian as it is caused That is conceptual novelty of complexity theory by a normal Langevin type stochastic process with a white embraces all the physical reality from equilibrium to non- noise Gaussian component. The normal Langevin stochas- equilibrium states. This is stated by Castro [12] as follows: tic equation corresponds to the probabilistic description of “…it is reasonable to suggest that there must be a deeper dynamics related to the well-known normal Fokker-Planck organizing principle, from small to large scales, operating equation. For Gaussian processes only the moments- in nature which might be based in the theories of complex- cumulants of first and second order are non-zero, while ity, non-linear dynamics and information theory in which the central limit theorem inhibits the development of long dimensions, energy and information are intricately con- range correlations and macroscopic self-organization as nected.”. Tsallis non-extensive statistical theory [2] can be any kind of fluctuation quenches out exponentially to the used for a comprehensive description of complex physi- normal distribution. Also at equilibrium, the dynamical cal systems, as recently we became aware of the drastic attractive phase space ofdistributed system is practically change of fundamental physical theory concerning physi- infinite dimensional as the system state evolves in all cal systems far from equilibrium. After the theory of scale dimensions according to the famous ergodic theorem of BG relativity introduced by Nottale [13], the new concept of statistics. However, according to Tsallis statistics, even for holistic and multiscale dynamics of distributed systems the case of thermodynamic equilibrium where the Gauss- was supported by many scientists as Castro [12], Iovane ian statistics lives the non-extensive character permits the [14], Marek-Crnjac [15], Ahmed and Mousa [16] and Agop development of long-range correlations produced by equi- et al. [17]. librium phase transition multi-scale processes according The dynamics of complex systems is one of the most to the Wilson RGT [19]. From this point of view, the classi- interesting and persisting modern physical problem, cal mechanics of fluids, materials or as the theory of parti- including the hierarchy of complex and self-organized cles and fields, including also general relativity theory, as phenomena such as: anomalous diffusion-dissipation well as the quantum mechanics – quantum field theories, and strange kinetics, fractal structures, long range cor- is nothing more than a near thermodynamical equilibrium relations, far from equilibrium phase transitions, reduc- approximation of a wider theory of physical reality, char- tion of dimensionality, intermittent turbulence, etc. [7, acterized as complexity theory. This theory can be related 18–23]. More than other scientists, Prigogine, as he was to a globally acting ordering process, which includes the deeply inspired by the arrow of time and the chemical statistical and the fractal extension of dynamics classical complexity, supported the marginal point of view that the or quantum. That is at every level of nature physical reality dynamical determinism of physical reality is produced is continuisly creating. by an underlying ordering process of entirely holistic Generally, the experimental observation of a complex and probabilistic character which acts at every physical system presupposes non-equilibrium process of the 144 G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one physical system which is subjected to observation, even field physical theory. The unified character of macro- if the system lives thermodynamically near to equilibrium scopic and microscopic complexity is further verified by states. Also, experimental observation includes discovery the fact that the point-Green functions produced by the and ascertainment of correlations in space and time, as generating functional of QFT, after the Wick rotation, can the spatiotemporal correlations are related to or caused by be transformed to point correlation functions produced by the statistical mean values fluctuations. The theoretical the partition function of statistical theory. This indicates interpretation and prediction of observations as spatial the presence at every level-scale of reality the existence of and temporal correlations – fluctuations is based on sta- self-organization process underlying also at the creation tistical theory which relates the microscopic underlying and interaction of elementary particles. This is related to dynamics to the macroscopic observations, indentified to the development of correlations in complex systems and statistical moments and cumulants. Moreover, it is known classical random fields [37]. For this reason lattice theory that statistical moments and cumulants are related to the describes simultaneously microscopic and macroscopic underlying dynamics by the derivatives of the partition complexity [3, 19, 36]. function to the external source variables [31–33]. From this In this way, instead of explaining the macroscopic point of view, the main problem of complexity theory is complexity by a fundamental physical theory such as how to extend the knowledge from thermodynamical equi- QFT, Superstring theory, M-theory or any other kind of librium states to the far from equilibrium physical states. fundamental theory, we become witnesses of the oppo- We believe that the non-extensive statistics introduced site fact, according to what Prigogine imagined. That is, by Tsallis [2] as the extension of BG equilibrium statisti- macroscopic self-organization process and macroscopic cal theory is the appropriate base for the non-equilibrium complexity install their kingdom in the heart of reduction- extension of complexity theory. Tsallis non-extensive sta- ism and fundamentalism of physical theory. That is at tistics as the appropriate far from equilibrium statistical the microscopic level the Renormalization field theories theory can produce the q-partition function and the cor- combined with Feynman diagrams that were used for the responding q-moments and q-cumulants, in correspond- description of high energy interactions or the statistical ence with BG statistical interpretation of thermodynamics. theory of critical phenomena and the nonlinear instabili- The observed miraculous consistency of physical ties for example of plasmas [38], lose their efficiency when processes at all levels of physical reality, from the mac- the complexity of the process scales up [32, 39]. roscopic to the microscopic level, as well as the ineffi- The universality of Tsallis non-extensive statistics as ciency of existing theories to produce or to predict this it is presented in the follllowing of this study is the mani- harmony and hierarchy of structures inside structures festation of the more general theory known as fractal from the macroscopic or the microscopic level of cosmos dynamics which was developed rapidly the last years [24– reveals the nessecity of new theoretical approaches. This 26]. Fractal dynamics are the modern fractal extension of completely supports or justifies new concepts as such physical theory in every level. On the other hand, the frac- indicated by Castro [12]: “of a global ordering principle” tional generalization of modern physical theory is based or by Prigogine [34], about “the becoming before being at on fractional calculus: fractional derivatives of integrals every level of physical reality.” The problem however with or fractional calculus of scalar or vector fields and frac- such beautiful concepts is how to transform them into an tional functional calculus [25, 26, 29, 40]. The efficiency experimentally testified scientific theory. In this direc- of fractional calculus to describe complex and far from tion, in his book “Randmonicity” Tsonis [35] presents a equilibrium systems which display scale-invariant prop- significant sinthesis of holistic and reductinisitc (ana- erties, turbulent dissipation and long range correlations lytic) shientific approach. The word radmnocity includes with memory preservation is very impressive, since these both meanings: chance (radomness) and mnimi-memory characteristics cannot be illustrated using traditional ana- (determnism). lytic and differentiable functions, as well as ordinary dif- The Feynman path integral formulation of quantum ferential operators. Fractional calculus permits the fractal theory after the introduction of imaginary time transfor- generalization of Lagrange-Hamilton theory for fields or mation by the Wick rotation indicates the inner relation particles, the Fokker-Planck equation Liouville theory and of quantum dynamics and statistical mechanics. In this BBGKI hierarchy, or the fractal generalization of QFT and direction, the stochastic and chaotic quantization theory path integration theory [25, 26, 29, 40, 41]. was developed [3, 4, 23, 36], which opened the road for the According to the fractional generalization of dynam- introduction of macroscopic complexity and self-organi- ics and statistics, we maintain the continuity of functions zation concepts in the region of fundamental quantum but abolish their differentiable character based on the G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one 145 fractional calculus which is the non-differentiable gener- and globally active ordering principle. This concept can be alization of differentiable calculus. At the same time, the related to the fractional generalization of dynamics that deeper physical meaning of fractional calculus is the uni- can be indentified with the dynamics of correlations sup- fication of microscopic and macroscopic dynamical theory ported by Balescu [18], Prigogine [34] and Nicolis [49], as based at the space-time fractality [13, 24, 42–44]. Also, the the non-equilibrium generalization of Newtonian theory. space-time itself is related to the fractality-multifractality This conjecture concerning the fractional unification of of the dynamical phase-space, which can be manifested macroscopic and microscopic dynamics can be strongly as non-equilibrium complexity and self-organization. supported by the Tsallis nonextensive q-statistics theory Moreover, fractal dynamics leads to a global generali- which is verified almost everywhere from the microscopic zation of physical theory as it can be related to the infinite to the macroscopic level. From this point of view, it is rea- dimensional Cantor space, as the microscopic essence of sonable to support that the q-statistics and the fractional physical space-time, the non-commutative geometry and generalization of space-time dynamics is the appropriate non-commutative Clifford manifolds and Clifford algebra, framework for the description of their non-equilibrium or the p-adic physics [45, 46]. According to these new con- complexity. cepts, introduced the last two decades, at every level of physical reality, from the microscopic to the macroscopic level, we observed and describe complex structures which 2.2 Chaotic dynamics and statistics cannot be reduced to underlying simple fundamental entities or underlying simple fundamental laws. Also, the The macroscopic description of complex distributed non-commutative character of physical theory and geom- systems can be approximated by non-linear partial differ- etry [46, 47] indicates that the scientific observation is ential equations of the general type: nothing more than the observation of undivided complex ∂ux(, t) geometrical physical structures in every level. Cantor was = Fu(, λ) (1) ∂ the founder of the fractal concept, creating fractal Cantor t sets by contraction of the continues real number set. On where ux() belongs to an infinite dimensional state the other hand the set of continues systems can be under- (phase) space which is a Hilbert functional space. stood as the result of the observational coarse graining of The control parameters (λ) measure the distance from the fractal Cantor reality [48]. From a philosophical point the thermodynamical equilibrium as well as the critical of view the mathematical forms are nothing else than or bifurcation points of the system for given and fixed self-organized complex structures in the mind-brain, self- values, depending upon the global mathematical struc- consistent to all the physical reality. On the other hand, ture of the dynamics. As the system passes its bifurcation the generalization of Relativity theory to scale relativ- points a rich variety of spatio-temporal patterns with dis- ity by Castro [12] or Nottale [13] indicates the unification tinct topological and dynamical profiles can be emerged of microscopic and macroscopic dynamics through the such as: limit cycles or torus, chaotic or strange attractors, fractal generalization of dynamics. turbulence, vortices, percolation states and other kinds of After all, we conjecture that the macroscopic self- complex spatiotemporal structures [18, 23, 49–52]. Gen- organization connected with the novel theory of complex erally, chaotic solutions of the mathematical system (1) dynamics, as it can be observed at far from equilibrium transform the deterministic profile of dynamics to non- dynamical physical states, is the macroscopic emergence linear stochastic system: of the microscopic complexity which can be enlarged as ∂u′ the system arrives at bifurcation or far from equilibrium =+Φλ(,ux′ )(δ , t) (2) ∂t critical points. That is, far from equilibrium the observed physical self-organization process the manifestation at where u′ is the reduced slow part of the non-linear dynam- the globally active ordering principle which acts in prior- ics and δ(,xt ) corresponds to the random force fields ity or self-consistenly from local interactions processes. produced by strong chaoticity-fast part of the non-linear In this framework of understanding, we could conjecture dynamics [53]. that the concept of local interactions themselves are the The non-linear mathematical models for distributed local manifestation of the universal and holistically active dynamical systems (1, 2) include plethora of mathemati- ordering principle. Namely, what until now is known as cal solutions and phase transition processes at the bifur- fundamental physical laws is nothing else than the equi- cation points as the control parameters increases. This librium manifestation or approximation of a universall scenario of non-linear dynamics can represent plethora of 146 G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one non-equilibrium physical states included in mechanical, in physical sciences in many regions of the physical sci- electromagnetic, chemical and other physical complex ences by the work of Prigogine, Nicolis, Yankov and others systems. The random component δ(,xt ) in equation (2), [22, 34, 49]. Moreover, non-linearity and chaos is the top related to the BBGKY hierarchy as it is presented in the of a hidden mountain including new physical and math- next section. As we presented at Figure 1, the low values ematical concepts such as fractal calculus, p-adic physi- of control parameters λ which means weak coupling for cal theory, non-commutative geometry, fuzzy anomalous the environment the system lives near to thermodynamic topologies, fractal space-time etc. [7, 13, 42, 43, 45, 56–58]. equilibrium, as the system departs from the thermody- These novel physical-mathematical concepts obtain their namic equilibrium by increasing the control parameters physical power when the physical system lives far from λ we have emergence of self-organising structures as limit equilibrium. cycles, limit torus, strange attractors or more complex and Furthermore and following the traditional point of multifractal structures. Far from equilibrium, the systems view of physical science, we arrive at the central con- can lives at metaequilibrium states (local minimum of free ceptual problem of complexity science. That is, how is it energy) reaviling complex structures, fractional dynam- possible the local interactions in a spatially distributed ics, non-Gaussian sattistics and multiscale and long-range physical system can cause long-range correlations or can correlations [18, 25, 26, 54]. create complex spatiotemporal coherent patterns non as These forms of the non-linear mathematical systems dissipative structures, as the previous non-linear math- (1, 2) correspond to the original version of the new science ematical systems reveal, when they are solved arithmeti- known today as complexity science. This new science has cally, or as the analysis of in situ observations of physical a universal character, including an unsolved scientific and systems shows. For non-equilibrium, physical systems conceptual controversy, which is continuously spreading the above questions lead us to seek how the develop- in all directions of the mathematical descriptions of the ment of complex structures and long range spatio-tem- physical reality concerning the integrability or comput- poral correlations can be explained and described by ability of the dynamics [55]. The concept of universality local interactions of particles and fields. At a first glance, was supported by many scientists, after the Poincare’s the problem looks simple supposing that it can be discovery of chaos and its non-integrability, as is it shown explained by the self-consistent particle-fields classical interactions. This is the reductioning dogma of science. However, the existed rich phenomenology of complex non-equilibrium phenomena reveals the non-classical and strange character of the universal non-equilibrium critical dynamics [2, 34, 35, 49, 51]. In the following and for the better understanding of the new concepts we follow the road of non-equilibrium statistical theory [18, 21, 23, 31, 55, 59]. Today the study of complex systems press as for the extension of physical theory to a new theoretical dogma known as the dynamics of correla- tions which at its limit includes the old dogma of local foreces and local interactions. The stochastic equation (2) belongs to the general type of Langevin equations. According to previous studies, the stochastic Langevin equations can take the general form:
Figure 1: This figure decribes the basic scenario toward the devel- ∂u δH i =−ΓΓ()xx + ()nx(, t) opment of complexity and emergence of complex structures as the i (3) ∂tuδ i (,xt ) control parameters λ increases. For weak coupling (low values of λ) the system lives near thermodynamical equilibrium (region A), where free energy obtains the absolute minimum. As the coupling of where H is the Hamiltonian of the system, δH/δui its func- the system with its envriromnet becomes stronger by increasing the tional derivative, Γ is a transport coefficient and ni are the control parameters λ the system passes through bifurcation points components of a Gaussian white noise: and develops complex structures with strong space-time correla- tions. Far from equilibrium, the system lives at local minimus of free <>nx(, t)0= energy where it appears complex-dissipative structures, fractional i (4) <>nx(, tn)(xt′′, )2=−Γδ()xxδδ()xt′′()−t dynamics and Tsallis distributions. ij ij G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one 147
The above stochastic Langevin Hamiltonian equation as with scale relativity principles [12, 13, 40] and fractal can be related to a probabilistic Fokker-Planck equation [21]: extension of dynamics [7, 24–26, 29, 30, 40, 61] or the older Prigogine’s correlations dynamics theory [34]. Here, we 1 ∂PHδδδ =⋅ Px+ [(Γ )]P (5) can argue in addition to previous description that far from Γδ()xt∂ uuδδu equilibrium quantum mechanics is transformed to a frac- where P = Pu({ (,xt )}, t) is the probability distribution tional mechanics theory. The fractional generalization of i function of the dynamical configuration {(uxi , t)} of the QM-QFT drifts along also the tools of quantum theory into system at time t. the correspondent non-equilibrium generalization of RG The solution of the FP equation can be obtained as a theory or the path integration and Feynman diagrams. functional path integral in the state space {(uxi )} : This generalization implies also the generalization of sta- tistical theory as the new road for the unification of mac- P({ux()}, tQ)e ∆ xp()−SP({ux()}, t ) (6) ii∫ 00 roscopic and microscopic complexity through the theory of many points theoretical functions. where P ({ux()}, t ) is the initial probability distribution 00i If P[(ux, t)] is the probability of the entire field function in the extended configuration state space and path in the field state space of the distributed system, S = i∫Ldt is the stochastic action of the system obtained by then we can extend the theory of generating function of the time integration of it is stochastic Lagrangian (L) [21]. moments and cumulants for the probabilistic description The stationary solution of the Fokker-Planck equation of the paths [31]. The n-point field correlation functions corresponds to the statistical minimum of the action and (n-points moments) can be estimated by using the field corresponds to a Gaussian state: path probability distribution and field path (functional)
P({uuii}) ∼−exp[ (1/)ΓΗ({ })] (7) integration: The path integration in the configuration field state 〈〉ux(,11 tu)(xt22, )(…uxnn, t ) space corresponds to the integration of the path prob- = ∆uP[(ux, tu)] (,xt )(…ux, t ) (10) ability for all the possible paths which start at the con- ∫ 11 nn figuration state ux(, t ) of the system and arrive at the 0 For Gaussian andom processes which described the final configuration state ux(, t). Langevin and F-P equa- equilibrium the nth point moments with n > 2 are zero. tions of classical statistics include a hidden relation with This corresponds to the Markov processes while far from Feynman path integral formulation of QM [3–5, 23, 37, 60]. equilibrium it is possible to be developed non-Gaussian As the F-P equation can be transformed to a Schrödinger (with infinite nonzero moments) processes. According to type equation: Haken [31] the characteristic function (or generating func- d iUˆˆ(,tt )(=⋅HUˆ tt, ) (8) tion) of the probabilistic description of paths: dt 00 [(ux, tu)] ≡ ((xt, ), ux(, tu), …, (,xt )) (11) by an appropriate operator Hamiltonian extension 11 22 nn H((ux, tH)) ⇒ ˆ ((uxˆ , t)) of the classical function (H) is given by the relation: where now the field (u) is an operator distribution. From this point of view, the classical stochasticity of the macro- N Φ ((jt), jt(),,… jt())e=〈 xpijux(, t )〉 (12) scopic Langevin process can be considered as caused by a path 11 22 nn ∑ iii path i=1 macroscopic quandicity revealed by the complex system as the F-K probability distribution P satisfies the quantum while the path cumulants K ()tt… are given by the sa1 as relation: relations: ˆ P(,ut |,ut 00)|=〈uU(,tt )|u0 〉 (9) Φpath ((jt11), jt22(),,… jtnn())
∞ s This generalization of classical stochastic process as i n =⋅exp(Kt……tj) j (13) ∑∑aa,1… = sa aa a a quantum process could explain the spontaneous devel- s! 1 s 11ss s=1 opment of long-range correlations at the macroscopic and the n-point path moments are given by the functional level as an enlargement of the quantum entanglement derivatives: character at critical states of complex systems. This inter- pretation is in faithful agreement with the introduction 〈〉ux(,11 tu), (,xt22 ),…, ux(,nn t ) of complexity in sub-quantum processes and the chaotic ==((δΦn {}jj)/δδ… jt){j }0 (14) – stochastic quantization of field theory [3, 30, 37], as well in1 i 148 G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one
For Gaussian stochastic field processes the cumulants From this point of view phase transition processes except the first two vanish (k3 = k4 = …0). For non-Gaussian can be developed between different critical states, when processes it is possible to be developed long range correla- the order parameters of the system are changing. The far tions as the cumulants of higher than two order are non- from equilibrium development of chaotic (weak or strong) zero [31]. This is the deeper meaning of non-equilibrium or other critical states include long-range correlations self-organization and ordering of complex systems. The and multiscale internal self-organization. Now, these far characteristic function of the dynamical stochastic field from equilibrium self organized states, cause the equilib- system is related to the partition functions of its statistical rium BG statistics and BG entropy, to be transformed and description, while the cumulant development and multi- replaced by the Tsallis q-statistics and Tsallis q-entropy. point moments generation can be related with the BBGKY The extension of renormalization group theory and criti- statistical hierarchy of the statistics as well as with the cal dynamics, under the q-extension of partition function, Feynman diagrams approximation of the stochastic field free energy and path integral approach has been also indi- system [32]. For dynamical systems near equilibrium only cated [2, 62–64]. The multifractal structure of the chaotic the second order cumulants are non-vanishing, while far attractors can be described by the generalized Rényi from equilibrium field fluctuations with higher-order non- fractal dimensions: vanishing cumulants can be developed. N λ Finally, using previous decriptions we can now log pq 1 ∑ i (15) understand how the non-linear dynamics correspond to D = lim,i=1 q q − 1lλ→0 og λ self-organized states as the high-order (infinite) non-van- ishing cumulants can produce the non-integrability of the where p ~ λα(i) is the local probability at the location (i) of dynamics. From this point of view the linear or non-linear i the phase space, λ is the local size of phase space and a(i) instabilities of classical kinetic theory are inefficient to is the local fractal dimension of the phase space dynamics. produce the non-Gaussian, holistic (non-local) and self- The Rényi q̅ numbers (different from the q-index of organized complex character of non-equilibrium dynam- Tsallis statistics) take values in the entire region (−∞, +∞) ics. That is, far from equilibrium complex states can be of real numbers. The spectrum of distinct local pointwise developed including long-range correlations of field and dimensions α(i) is given by the estimation of the function particles with non-Gaussian distributions of their dynamic f(α) defined by the scaling of the density n(a, λ) ~ λ−f(a), variables. As we show in the next section such states reveal where n(a, λ)da is the number of local regions that have the necessity of new theoretical tools for their understand- a scaling index between a and a + da. This reveal f(a) as ing which are much different from those used in classical the fractal dimension of points with scaling index a. The linear or non-linear approximation of kinetic theory. fractal dimension f(a) which varies with a shows the mul- tifractal character of the phase space dynamics which includes interwoven sets of singularity of strength a, by 2.3 Strange attractors and self-organization their own fractal measure f(a) of dimension [65, 66]. The multifractal spectrum D of the Renyi dimensions can be When the dynamics is strongly nonlinear then far from q̅ related to the spectrum f(a) of local singularities by using equilibrium it is possible to occur strong self-organization the following relations: and intensive reduction of dimensionality of the state space, by an attracting low dimensional smooth set with pdqf= αα′′pd()λα− ()α′ ′ ∑ i ∫ (16) parallel development of long-range correlations in space and time. The attractor can be smoothly periodic (limit τ()qq≡−(1)mDq =−ina ((qa fa)) (17) cycle, limit m-torus), simply chaotic (mono-fractal) or strongly chaotic with multiscale and multifractal profile. dq[(τ )] aq()= (18) Also, attractors with weak chaotic profile known as SOC dq states are also well knowh. This spectrum of distinct dynamical profiles can be obtained as distinct critical f ()αα=−qqτ() (19) points (critical states) of the nonlinear dynamics, after successive bifurcations as the control parameters change. According to Arneodo et al. [67] the physical meaning The fixed points can be estimated by using a far from of these quantities included in relations (16–19) can be equilibrium renormalization process as it was indicated obtained if we identify the multifractal attractor as a by Chang et al. [21]. thermodynamical object, where its temperature (T), free G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one 149 energy (F), entropy (S) and internal energy (U) are related of an ordering principle in physical reality, which cannot to the properties of the multifractal attractor as follows: be exhausted with the local interactions in the physical systems, as we mentioned in previous sections. 1 qq⇒=,(τ )(qD−⇒1) F T q (20) αα⇒⇒Uf,() S 2.4.1 The non-extensive entropy (Sq) This correspondence presents the relations (18–20) Usually any extension of physical theory is related to some as a thermodynamical Legendre transform [68]. When q̅ special kind of mathematics. Tsallis non-extensive statis- increases to infinite (+∞), which means, that we freeze tical theory is connected with the q-extension of exponen- the system (T →0), then the trajectories (fluid lines) (q=+∞) tial-logarithmic functions as well as the q-extension of are closing on the attractor set, causing large probability Fourier Transform (FT). The q-extension of mathematics values at regions of low fractal dimension, where α = α min underlying the q-extension of statistics, are included in and D = D . Oppositely, when q̅ decreases to infinite q̅ −∞ the solution of the non-linear equation: (−∞), that is we warm up the system (T(q=−∞)→0) then the trajectories are spread out at regions of high fractal dy q ==yy, ((0) 1, qR∈ ) (21) α ⇒ α ′ > < dx dimension ( max). Also for q̅ q̅ we have Dq′̅ Dq ̅ and D ⇒ D (D ) for α ⇒ α (α ) correspondingly. It is also q̅ +∞ −∞ min max According to (21) its solution is the q-exponential function known the Renyi’s generalization of entropy according x eq : Rq1 to the relation: SP= log. However, the above xq1/(1− ) qi∑ eq≡+[1 (1− )]x (22) q − 1 i q description presents only a weak or limited analogy The q-extension of logarithmic function is the reverse x between multifractal and thermodynamical objects. The of eq according to: real thermodynamical character of the multifractal objects x1−q − 1 and multiscale dynamics was discovered after the defini- ln x ≡ (23) q 1 − q tion by Tsallis [2] of the q-entropy related with the q-statis- tics as it is summarized in the next section. As Tsallis has The q-logarithm satisfies the property: shown Renyi’s entropy as well as other generalizations of ln ()xx =+ln xxln +−(1 qx)(ln)(ln)x (24) entropy cannot be used as the base of the non-extensive qABqAqBqAqB generalization of thermodynamics. In relation of the pseudo-additive property of the q-logarithm, a generalization of the product and sum to q-product and q-sum was introduced (22): 2.4 The highlights of Tsallis theory
lnqqxy+ln xy⊗≡qqe (25) As we show in the second part next of this study, every- xy⊗≡xy++(1− qx) y where in physical systems we can ascertain the presence q (26) of Tsallis statistics. This discovery is the continuation of a Moreover in the context of the q-generalization of the more general ascertainment of Tsallis q-extensive statis- central limit theorem the q-extension of Fourier transform tics from the macroscopic to the microscopic level [2]. was introduced (1): In our understanding the Tsallis theory, more than +∞ a generalization of thermodynamics for chaotic and q−1 F []pd()ξ ≡≥xeixξ[(px)] px(), (1q ) (27) complex systems, or a non-equilibrium generalization of qq∫ −∞ B-G statistics, can be considered as a strong theoretical foundation for the unification of macroscopic and micro- It was for the first time that Tsallis [2], inspired by scopic physical complexity. From this point of view, Tsallis multifractal analysis, conceived that the BG entropy statistical theory is the other side of the modern fractional Sk=− ppln =
Tsallis non-extensive statistics, based on the extended statistical theory can faithfully be described the maximum concept of q-entropy: (extreme) of the Tsallis q-entropy function. The extremiza- tion of Tsallis q-entropy corresponds to the q-generalized N =− q −=<> form of the normal distribution function: Skqi1(∑ pq1) kplnqi(1/) (29) i=1 2 −<β()xx>q pxqq()= Aeβ q (33) for continuous state space, we have where Sk=−1[px()](qdx q − 1) q ∫ (30) Aqq =−(1)/πΓ(1/(qq−−1))/Γ((3)/[2/(1q − )]) For a system of particles and fields with short range for q > 1, correlations inside their immediate neighborhood, the and Tsallis q-entropy Sq asymptotically leads to BG entropy (S ) corresponding to the value of q = 1. For probabil- BG Aqq =−(1 )/πΓ((53−−qq)/[2(1 )])/Γ((2)−−qq/(1)) istically dependent or correlated systems A, B it can be for q < 1, Γ(z) being the Riemann function. proven that: The q-extension of statistics includes also the q-exten-
SAqq()+=BS()AS++qq(/BA)(1)− qS()ASq (/BA) sion of central limit theorem which can describe also (31) faithfully the non-equilibrium long range correlations in =+SBqq() SA(/Bq)(+−1)SBqq()SA(/B) a complex system. The normal central limit theorem con-
A B cerns Gaussian random variables (xi) for which their sum where SAqq()≡ Sp({ i }), SBqq()≡ Sp({ i }), Sq(B/A) and N = Sq(A/B) are the conditional entropies of systems A, B. Z ∑ xi gradually tends to become a Gaussian process as When the systems are probabilistically independent, then i=1 N→ ∞, while its fluctuations tend to zero in contrast to the relation (31) is transformed to: possibility of non-equilibrium long range correlations. By
SAqq()+=BS()AS++qq()Bq(1− )(SA)(SBq ) (32) using q-extension to Fourier transform, it can proved that q-independence means independence for q = 1 (normal The first part of S (A + B) is additive S (A) + S (B) q q q central limit theorem), but for q ≠ 1 it means strong cor- while the second part is multiplicative including long- relation (q-extended CLT). In this case (q ≠ 1) the number range correlations supporting the macroscopic ordering of WAA++…+A of allowed states, in a composed system by phenomena. Zelenyi and Milovanov [8] showed that the 12 N the (A1, A2,…, AN) sub-systems, is expected to be smaller Tsallis definition of entropy coincides with the so-called N than WWAA++…+=Ai= Π 1 A where WWAA, ,,… WA are the “kappa” distribution, which appears in space plasmas 12 Ni 12 N possible states of the subsystems. and other physical realizations [69–74]. Also, they indi- In this way, Tsallis q-extension of statistical physics cate that the application of Tsallis entropy formalism cor- opened the road for the q-extension of thermodynam- responds to physical systems whose the statistical weights ics and general critical dynamical theory as a non-linear are relatively small, while for large statistical weights the system lives far from thermodynamical equilibrium. For standard statistical mechanism of BG is better. This result the generalization of BG nonequilibrium statistics to means that when the dynamics of the system is attracted Tsallis nonequilibrium q-statistics we follow Binney et al. in a confined subset of the phase space, then long – range [32]. In the next we present q-extended relations, which correlations can be developed. Also according to Tsallis can describe the non-equilibrium fluctuations and n- [2] if the correlations are either strictly or asymptotically point correlation function (G) can be obtained by using inexistent the BG entropy is extensive whereas S for q ≠ 1 q the Tsallis partition function Z of the system as follows: is non-extensive. q ∂n n 1 Zq Giqn(,12 ii,,…… ),≡〈ssii ,, si 〉=q (34) 12 n zj∂⋅∂∂jj… ii12 in 2.4.2 The q-extension of statistics and thermodynamics
where {si} are the dynamical variables and {ji} their sources According to the Tsallis q-extension of the entropy princi- included in the effective – Lagrangian of the system. Cor- ple, any stationary random variable can be described as relation (Green) equations (35) describe discrete variables, the stationary solution of generalized fraction diffusion the n-point correlations for continuous distribution of var- of equation (1). At metastable stationary solutions of a iables (random fields) are given by the functional deriva- stochastic process, the maximum entropy principle of BG tives of the functional partition as follows: G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one 151
n ∂ Gx(, xx,,…… )(≡〈ϕϕxx)( )(ϕ x 〉 ∂ 1 Sq qn12 12 nq Uq==ln Z , (45) qq∂∂β TU 1 δδ q (35) = … ZJq () ZJδδ()xJ()x 2 1 n ∂∂δ UF∂ CT≡=qq=−T q (46) where ϕ()x are random fields of the system variables and q ∂∂TT ∂ 2 T j()x their field sources. The connected n-point correla- n tion functions Gi are given by: The q-exponential probability distributions described previously, can describe the nonequilibrium plasma states n δδ Gx(, xx,,…… )l≡ og ZJ() (36) including the random profile of fields or particles. The qn12 δδJx() Jx() q 1 n nonequilibrium plasma states correspond to the extremi- The connected n-point correlations correspond to cor- zation of Tsallis q-entropy under appropriate conditions relations that are due to internal interactions defined as [2]. Especially, for energetic (nonthermal) particle popula- [32]: tions the q-exponential probability distributions take the form of kappa distributions of two main types: Gxn(, xx,,…… )(≡〈ϕϕxx)()(〉−〈〉ϕϕxx)(… ) (37) qn12 11nq nq −κ 1 εκ−UT(, ) P(1)()ε ~1+⋅κ ∗∗ , The probability of the microscopic dynamical configu- κ κ * kTB ∗ rations is given by the general relation: −−κ 1 (2) 1 εκ −UT() − βSconf = P ()εκ ~1+⋅ (47) P(conf) e (38) κ kTB where β = 1/kT and S is the action of the system, while conf where U is the main kinetic energy U = 〈ε 〉 [73]. the partition function Z of the system is given by the κ According to Livadiotis and McComas [70], the con- relation: nection between kappa distributions and the entropic
− βSconf Z = ∑e (39) index q of Tsallis non-extensive statistical mechanics is conf given by the transformation k = 1/(q − 1).
The q-extension of the above statistical theory can be obtained by the q-partition function Zq. The q-partition 2.4.3 Fractal generalization of dynamics function is related with the meta-equilibrium distribution of the canonical ensemble which is given by the relation: Fractional integrals and fractional derivatives are deriva-
−−βqE()VZ/ tives or integrals on fractals which are related to the pe= iq q iq (40) fractal contraction transformation of phase space as well with as contraction transformation of space-time in analogy −−βqE()V = iq with the fractal contraction transformation of the Cantor Zqq∑e (41) conf set [47, 75]. Also, the fractional extension of dynamics includes the non-Gaussian scale invariance, related to the and multiscale coupling and non-equilibrium extension of the β = β q renormalization group theory [24]. Moreover, Tarasov [25, qi∑ p (42) conf 26], Goldfain [30], Cresson and Greff [40], El-Nabulsi [29] and other scientists generalized the classical or quantum where β = 1/KT is the Lagrange parameter associated with dynamics in a continuation of the original break through the energy constraint: by Ord [76], El-Naschie [27], Nottale [13], Castro [12] and 〈〉≡=qq others concerning the fractal generalization of physical Epqi∑∑EpiiUq (43) conf conf theory. According to Tarasov [25] the fundamental theorem The q-extension of thermodynamics is related to the of Riemann-Liouville fractional calculus is the generaliza- estimation of q-Free energy (F ) the q-expectation value of q tion of the known integer integral – derivative theorem as internal energy (U ) the q-specific heat (C ) by using the q q follows: q-partition function: If 1 = a F ≡−UTSq=− ln Z (44) F()xIα x fx() (48) qq qqβ Then 152 G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one
a axDF()xf= ()x (49) for the surface fractal integration and a where axI is the fractional Riemann-Liouville according (58) dlγ = Cx11(,γ )dl to: 1−γ 2(Γ 1/2) γ− x ′′ 1 a 1(fx)dx Cx1(,γ )|= x | (59) axIf()x ≡ − (50) Γγ(/2) Γ()a ∫a ()xx− ′ 1 a
a and axD is the Caputo fractional derivative according to: For the line fractal integration. By using the fractal generalization of integration and the corresponding gen- DFan()xI==−anDF()x ax ax x eralized Gauss’s and Stoke’s theorems we can transform 1(x dx′ dnF x) (51) = fractal integral laws to fractal and non-local differential Γ()na− ∫ ()xx− ′ 1+−an dxn a laws. The fractional generalization of classical dynamics (Hamilton Lagrange and Liouville theory) can be obtained for f(x) a real valued function defined on a closed interval by the fractional generalization of the quantative descrip- [a, b]. tion of the phase space. For this, we use the fractional In the next, we summarize the basic concepts of the power of coordinates: fractal generalization of dynamics as well as the fractal aa generalization of Liouville theory following Tarasov [25]. Xx= sgn( )|x | (60) According to previous descriptions, the far from equi- librium dynamics includes fractal or multi-fractal distri- where sgn(x) is equal to +1 for x ≥ 0 and equal to −1 for bution of fields and particles, as well as spatial fractal x < 0. temporal distributions. This state can be described by the The fractional measure Ma(B) of a n-dimension phase fractal generalization of classical theory: Lagrange and space region (B) is given by the equation: Hamilton equations of dynamics, Liouville theory, Fokker- MB()= ga()dqµ (,ˆˆ p) Planck equations and Bogoliubov hierarchy equations. In aa∫ (61) B general, the fractal distribution of a physical quantity (M) obeys a power law relation: where dqµa (,ˆˆ p) is a phase space volume element: D R dqˆˆaaΛdp MMD ∼ 0 (52) dµΠa = 2 (62) R0 [(aaΓ )] where (M ) is the fractal mass of the physical quantity (M) D where g(a) is a numerical multiplier and dqaaΛdp means in a ball of radius (R) and (D) is the distribution fractal KK the wedge product. dimension. For a fractal distribution with local density The fractional Hamilton’s approach can be obtained ρ()x the fractal generalization of Euclidean space inte- by the fractal generalization of the Hamilton action gration reads as follows: principle: MW()= ρ()xdV DD∫ (53) Sp=−[(ˆˆqHtp, ˆˆ, qd)] t (63) W ∫ where The fractional Hamilton equations:
a dV = CD(, xd) V (54) dqˆ − D 33 =−Γ(2 ap) ˆ aa1DH (64) at pˆ and
3−D 2(Γ 3/2) D−3 aaˆ =− CD(, xx)|= | (55) DtqpDˆ H (65) 3 Γ(/D 2)
while the fractional generalization of the Lagrange’s Similarly the fractal generalization of surface and line action principle: Euclidean integration is obtained by using the relations: SL= (,tq , ud) t (66) (56) ∫ dSd = Cd22(, xd) S 2−d Corresponds to the fractional Lagrange equations: 2 − Cd(, xx)|= |d 2 (57) 2 Γ(/d 2) DaaLa−−Γ(2 )[DDa L]0= qtˆ UU=q (67) G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one 153
Similar fractional generalization can be obtained for dis- with a < 1 for sub-diffusion and a > 1 for super-diffusion. sipative or non-Hamiltonian systems [41]. The fractal gen- In the next, we follow Zaslavsky [24] in order to present eralization of Liouville equation is given also as: the internal relation of Tsallis theory and fractal extension of dynamics. The equilibrium normal-diffusion process is ∂p N = Lpˆ (68) described by a chain equation of the Markov-type: ∂ NN t Wx(, tx;, td)(= xW xt, ;,xt )(Wx, tx;, t ) 3311 ∫ 233222211 (71) ˆ where pN and LN are the fractional generalization of the N-particle probability distribution function and the where W(x, t; x′, t′) is the probability density for the Liouville operator correspondingly. The fractal gener- motion from the dynamical state (x′, t′) to the state (x, t) alization of Bogoliubov hierarchy can be obtained by of the phase space. The Markov process can be related using the fractal Liouville equation as well as the fractal to a random differential Langevin equation with addi- Fokker Planck hydrodynamical-magnetohydrodynamical tive white noise and a corresponding Fokker-Planck approximations. The fractional generalization of classi- (FP) probabilistic equation [24] by using the initial cal dynamical theory for dissipative systems includes the condition: non-Gaussian statistics as the fractal generalization of W (,xy ;)∆δtx=−()y BG statistics. Finally the far from equilibrium statistical ∆t→0 (72) mechanics can be obtained by using the fractal extension of the path integral method. This relation means no memory in the Markov process and The fractional Green function of the dynamics is given help to obtain the expansion: by the fractal generalization of the path integral: Wx(, yt;)∆δ=−()xy+−ay(;∆δtx)(′ y) x f i 1 Kx(, tx;, tD)[ xS()τγ]exp() +−by(;∆δtx)(′′ y) (73) af fiia∫ a 2 x h i (69) i exp(S γ) ∑ h a where a(y; Δt) and b(y; Δt) are the first and second moment {}γ of the transfer probability function W(x, y; Δt): where K is the probability amplitude (fractal quantum ∆∆=− ≡〈〈〉∆ 〉 a ay(; td)(∫ xx yW)(xy, ;)ty (74) mechanics) or the two point correlation function (statis- tical mechanics), D[x (τ)] means path integration on the ∆∆=−22≡〈〈〉∆ 〉 a by(; td)(∫ xx yW)(xy, ;)ty() (75) sum {γ} of fractal paths and Sa(γ) is the fractal generaliza- tion of the action integral: By using the normalization condition:
x f dyWx(, yt;)∆ = 1 1 aa−1 ∫ (76) SL[]γτ=−((Dq ), ττ)(td) τ a ∫ γ (70) Γ()a x i we can obtain the relation:
1 ∂by(;∆t) ay(;∆t) =− (77) 2 ∂y
2.4.4 The Tsallis extension of statistics via the fractal extension of dynamics The Fokker-Planck equation which corresponds to the Markov process can be obtained by using the relation: At the equilibrium thermodynamical state the underlying +∞ statistical dynamics is Gaussian (q = 1). As the system goes ∂px(, t)1 =−lim(dyWx, yt;)∆ py(, tp)(xt, ) (78) far from equilibrium the underlying statistical dynamics ∆t→0 ∫ ∂tt∆ −∞ becomes non-Gaussian (q ≠ 1). At the first case the phase space includes ergodic motion corresponding to normal where p(x, t) ≡ W(x, x0; t) is the probability distribution diffusion process with mean-squared jump distances pro- function of the state (x, t) corresponding to large time portional to the time 〈x2〉 ~ t whereas far from equilibrium asymptotic, as follows: the phase space motion of the dynamics becomes cha- ∂Px(, t)12 otically self-organized corresponding to anomalous diffu- =−∇+xx((AP xt, )) ∇ ((BP xt, )) (79) ∂t 2 sion process with mean-squared jump distances 〈x2〉 ~ ta, 154 G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one where A(x) is the flow coefficient: bt(;ξ∆) B()ξ = lim (90) ∆t→0 ()∆t β 1 Ax(, tx)l≡〈im 〈〉∆ 〉 (80) ∆t→0 ∆t Using the above equations we can obtain the fFPK and B(x, t) is the diffusion coefficient: equation. Far from equilibrium the non-linear dynam- ics can produce phase space topologies corresponding to 1 2 B(,xt )l≡〈im 〈〉∆x 〉 (81) various complex attractors of the dynamics. In this case the ∆t→0 ∆t extended complexity of the dynamics corresponds to the generalized strange kinetic Langevin equation with corre- The Markov process is a Gaussian process as the moments lated and multiplicative noise components and extended lim〈〈∆xm 〉〉 for m > 2 are zero. The stationary solutions of ∆t→0 fractal fFPK equation [7, 24]. The q-extension of statistics by FP equation satisfy the extremal condition of BG entropy: Tsallis can be related to the strange kinetics and the frac- tional extension of dynamics through the Levy process: SK=− px()ln px()dx BG B ∫ (82) P(,xt ;)xt = dx ……dx Px(, tx;, t ) nn00 ∫ 11NN−−NN11N− corresponding to the known Gaussian distribution: Px(,11 tx;,00 t ) (91) 22 px()∼−exp( x /2σ ) (83) The Levy process can be described by the fractal F-P equation: According to Zaslavsky [24] the fractal extension of FP equation can be produced by the scale invariance prin- ∂∂βPx(, t) aa∂ +1 =+[(Ax)(Px, tB)] [(xP)(xt, )] ciple applied for the phase space of the non-equilibrium ∂∂txβ ()−∂aa()−x +1 dynamics. As it was shown by Zaslavsky [24], for strong (92) chaos the phase space includes self similar structures of islands inside islands dived in the stochastic sea. The where ∂β/∂tβ, ∂a/∂(− x)a and ∂a+1/∂(− x)a+1 are the fractal time fractal extension of the Fokker-Planck-Kolmogorov equa- and space derivatives correspondingly [24]. The stationary tion (fFPK) can be derived after the application of a Renor- solution of the FP equation for large x is the Levy distri- malization Group of anomalous Kinetics (RGK): bution P(x) ~ x−(1+γ). The Levy distribution coincides with the Tsallis q-extended optimum distribution q-exponen- Rˆ :,sS′′==λλ t t (84) Ks t tial function for q = (3 + γ)/(1 + γ). The fractal extension of dynamics takes into account non-local effects caused by where s is a spatial variable and t is the time. Correspond- the topological heterogeneity and fractality of the self- ingly, to the Markov process equations: organized phase-space. ∂ βPt(,ξ )1 Also, the fractal geometry and the complex topol- ≡+lim[Wt(,ξξ ;)∆ξtW− (, ξ ;)t ] (85) ∂ttββ∆t→0 ()∆ 00ogy of the phase-space introduce memory in the complex
dynamics which can be manifested as creation of long Wn(,ξ∆ ;)tn=−δξ()+−an(;∆δtn)(()α ξ ) range correlations, while, oppositely, in Markov process 1 we have complete absence of memory. +−bn(;∆δtn)((2a) ξ ) +… 2 (86) In general, the fractal extension of dynamics as it was done until now from Zaslavsky, Tarasov and other scien- as the space-time variations of probability W are consid- tists indicate the internal consistency of Tsallis q-statistics ered on fractal space-time variables (t, ξ) with dimensions as the non-equilibrium extension of BG statistics with the (β, a). For fractional dynamics a(n; Δt), b(n; Δt) satisfy the fractal extension of classical and quantum dynamics. equations: Finally, we must mention the fact that the fractal α a an(;∆ξtn)|=−∫ |(Wnξ∆, ;)tdξ∆≡〈〈〉||ξ 〉 (87) extension of dynamics identifies the fractal distribution of a physical magnitude in space and time according to the 22α a a bn(;∆ξtn)|=−∫ |(Wnξ∆, ;)tdξ∆≡〈〈〉||ξ 〉 (88) scaling relation M(R) ~ R with the fractional integration as an integration in a fractal space. From this point of view and the limit equations: it could be possible to conclude the novel concept that the non-equilibrium q-extension of statistics and the fractal at(;ξ∆) A()ξ = lim (89) extension of dynamics are related to the fractal space and ∆t→0 ()∆t β time themselves [7, 13, 77]. G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one 155
D()hp=+min[ hJ3(− p)] (97) 3 Intermittent turbulence p The multifractal character of the turbulent state can Intermittent turbulence structuring of materials, fluids be apparent at the spectrum of the structure function and other far from equilibrium distributed dynamical scaling exponents J(p) by the relation: systems can be also dedcribed through the extremization of Tsallis q-entropy. That is the extremization of q-entropy dJ()p dh ()p =+hp()[(pD−=′ hp( ))] ∗ hp() (98) structures the phase space as the multifractal set which dp ∗∗dp ∗ can produce multifractal structures and long-range cor- as the minimum value of the relation (96) corresponds the relations in space and time. In this case, we can assume maximum of 3–J(p) function for which: the mirroring relationship between the phase space mul- tifractal attractor of the distributed dynamics and the cor- dJ(3− ()pd) =−()Dph = 0 (99) responding multifractal turbulence dissipation process of dp dp the dynamical system in the physical space. Multifractal- ity and multiscaling interaction, chaoticity and mixing and or diffusion (normal or anomalous), all of them can be D′((hp)) = p (100) manifested in both the state (phase) space and the physi- ∗ cal (natural) space as the mirroring of the same complex According to Frisch [33] the energy (ε ) dissipation is dynamics. We could say that turbulence is for complex- i said to be multifractal if there is a function F(a) which ity theory, what the blackbody radiation was for quantum maps real scaling exponents a to scaling dimensions theory, as all previous characteristics of complexity can be F(a) ≤ 3 such that: observed in turbulent states. The multifractal character of turbulence can be char- a−1 (101) εl ()rl~ as l → 0 acterized: a) in terms of local velocity of other variables p 3 increments δurl () and the structure function Slpl()≡〈δu 〉 for a subset of points r of R with dimension F(a). first introduced by Kolmogorov [78], b) in terms of local In correspondence with the structure function theory dissipation and local scale invariance of fluids equations presented above in the case of multifractal energy dissipa- a q 1− tion the moments ε follow the power laws: upon scale transformations rr′′==λλ, uua/3 , tt′ = λ 3 . l qq The turbulent flow is assumed to process a range of 〈〉εl ~ l (102) scaling exponent, hmin ≤ h ≤ hmax while for each h there is a subset of point of R3 of fractal dimension D(h) such that: where the scaling exponents σ(q̅) are given by the relation: h (93) σ()qq=−min[ (1aF)3+−()a ] (103) δurl ()∼→ll as 0 a The multifractal assumption can be used to derive the In the case of one dimensional dissipation process structure function of order p by the relation: the multifractal character is described by the function f(a) instead of F(a) where f(a) = F(a) − 2. In the language ph 3−Dh( ) Sl()~(dhµ )(ll) (94) p ∫ of Renyi’s generalized dimensions and multifractal theory the dissipation multifractal turbulence process corre- where dμ(h) gives the probability weight of the different sponds to the Renyi’s dimensions D according to the scaling exponents, while the factor l3−D(h) is the probability q̅ σ ()q of being with a distance l in the fractal subset of R3 with relation D =+0 3. Also, according to Frisch [33] the q q − 1 dimension D(h). By using the method of steepest descent relation between the energy dissipation multifractal for- [33] we can derive the power-law: malism and the multifractal turbulent velocity increments
PJ()P formalism is given by the following relations: Slpl()≡→δul~, l 0 (95) app hD==, ()hF()af=+()aJ2, ()p =+σ (104) where: 333
J()pp=+min[ hD3(− h)] (96) h In the next of this section we follow Arimitsu and The above relation is the Legendre transforma- Arimitsu [79] connecting the Tsallis non-extensive statis- tion between J(P) and D(h) as D(h) can be derived by the tics and intermittent turbulence process. Under the scale relation: transformation: 156 G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one
α−1 εεn ∼ 0n(l \)l0 (105) d-dimensional natural space [79]. The gradient singulari- ties cause the anomalous diffusion in physical or in phase the original eddies of size l can be transformed to consti- 0 space of the dynamics. The total dissipation occurring in tuting eddies of different size l = l δ , n = 0, 1, 2, 3, … after n 0 n a d-dimensional space of size l scales also with a global n steps of the cascade. If we assume that at each step of n dimension D for powers of different order q̅ as follows: the cascade eddies break into δ pieces with 1/δ diameter q̅ (1qD− ) then the size l = l δ−n. If δu = δu(l ) represents the velocity εqd∼=q τ()q n 0 n n ∑ nnllnnl (109) n difference across a distance r ~ ln and εn represents the rate of energy transfer from eddies of size l to eddies of size l n n+1 Supposing that the local fractal dimension of the then we have: set dn(a) which corresponds to the density of the scaling a a−1 exponents in the region (α, α + dα) is a function f (a) ll3 d ==nn(106) δδuunn and εε according to the relation: ll00 − f ()α dn()α ∼ ln d da (110) where a is the scaling exponent under the scale transfor- where d indicates the dimension of the embedding mation (33). space, then we can conclude the Legendre transforma- The scaling exponent a describes the degree singular- tion between the mass exponent τ(q̅) and the multifractal ∂ux() δun ity in the velocity gradient = lim as the first spectrum fd(a): l →0 ∂xln n f ()aa=−qq(1−−)(Dd++1) d −1 equation in (106) reveals. The singularities a in velocity dq d (111) gradient fill the physical space of dimension d, (a < d) with =−−+ aq[( 1)(1Ddq )] dq a fractal dimension F(a). Similar to the velocity singularity other frozen fields For linear intersections of the dissipation field, that is can reveal singularities in the d-dimensional natural d = 1 the Legendre transformation is given as follows: space. After this, the multifractal (intermittency) charac- ter of fluids is based upon scaling transformations to the dd fa()=−aq ττ()qa, =−[(qD1) q ](= q), space-time variables (,Xt ) and velocity ()U : dq dq df ()a q = (112) α/3 1/−a 3 XX′ ==λλ, UU′ , tt′ = λ , (107) da and corresponding similar scaling relations for other The relations (105–107) describe the multifractal and physical variables [20, 80, 81]. Under these scale transfor- multiscale turbulent process in the physical state. The mations the dissipation rate of turbulent kinetic or dynam- relations (108–112) describe the multifractal and multi- ical field energy En (averaged over a scale ln = l0δn = Roδn) scale process on the attracting set of the phase space. rescales as εn: From this physical point of view, we suppose the physi-
α−1 cal identification of the magnitudes D , a, f(a) and τ(q̅) εε∼ (\ll) (108) q̅ nn00 estimates in the physical and the corresponding phase space of the dynamics. By using experimental timeseries Kolmogorov assumes no intermittency as the locally we can construct the function D of the generalized Rényi averaged dissipation rate [78], in reality a random vari- q̅ d-dimensional space dimensions, while the relation (107) able, is independent of the averaging domain. This means allow the calculation of the fractal exponent (a) and the in the new terminology of Tsallis theory that Tsallis q-indi- corresponding multifractal spectrum f (a). For homo- ces satisfy the relation q = 1 for the turbulent dynamics d geneous fractals of the turbulent dynamics the general- in the three dimensional space. That is the multifractal ized dimension spectrum D is constant and equal to the (intermittency) character of the HD or the MHD dynam- q̅ fractal dimension of the support [33, 65, 68]. Kolmogorov ics consists in supposing that the scaling exponent a [78] supposed that D does not depend on q̅ as the dimen- included in relations (107, 108) takes on different values at q̅ sion of the fractal support is D = 3. In this case, the mul- different interwoven fractal subsets of the d-dimensional q tifractal spectrum consists of the single point [a = 1 and physical space in which the dissipation field is embed- f(1) = 3]. The singularities of degree (a) of the dissipated ded. The exponent α and for values a < d is related to the fields, fill the physical space of dimension d with a fractal ∂Ax() degree of singularity in the field’s gradient in the dimension F(a), while the probability P(a)da to find a ∂x G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one 157 point of singularity (a) is specified by the probability relation i.e. W = eS/k where entropy is not a statistical but density P(a)da ~ lnd−F(a). The filling space fractal dimen- a dynamical magnitude. Already, Boltzmann himself was sion F(a) is related to the multifractal spectrum function t Ω using the relation W ==lim RR where t is the total R T→∞ R fd(a) = F(a) −(d − 1), while according to the distribution T Ω function Πdis(εn) of the energy transfer rate associated with amount of time that the system spends during the time T the singularity a, it corresponds to the singularity prob- in its phase space trajectory in the region R while ΩR is the ability as Πdis(εn)dεn = P(a)da [79]. phase space volume of region R and Ω is the total volume q of phase space. The above concepts of Boltzmann and Moreover, the partition function ∑ Pi of the Rényi i Einstein were innovative as concerns modern q-exten- fractal dimensions estimated by the experimental time sion of statistics which is internally related to the fractal series includes information for the local and global dis- extension of dynamics. However according to Quantum sipation process of the turbulent dynamics as well as for mechanics the physical system every time materializes its the local and global dynamics of the attractor set, as it microstates probabilistically. That is the physical system is transformed to the partition function PZq = of the ∑ iq every instant is informed for all the acceptable micro- Tsallis q-statistic theory. i states and select one of all. From this point of view the Boltzmann entropy principle is related to the potential- ity of the system to create states of maximum entropy or minimum free energy, corresponding to long range cor- 4 Non-equilibrium phase transition related metastable critical states. The Wilsonian point of process view, more than the description of macroscopic phase transition critical states, includes also the possibility to According to Wilson [19] the essence of phase transition explain the experimentally observed elementary particles process in distributedphysical systems is the multiscale as correlations on the lattices which at the continuum character of dynamics. That is, there is no fundamental limit of quantum fields represent elementary particles in scale which can be used for constructing the dynamics of similarity with the “dissipative structures” at the macro- higher scales. Also multiscale dynamics is the essence of scopic level. self organization processes in complex systems [31, 34, 49]. After Wilson [19] the multiscale character of dynam- The multiscale character of dynamics at phase transition ics obtained clear mathematical description through process is related to the efficiency of nature to create long- the RGT. According to RGT the long range correlated range correlations which cannot be understood by the critical states are scale invariant and correspond to the local character of interactions of inter-molecular dynam- fixed points of the group of scale transformations. The ics. The statistical explanation of long range correlated non-equilibrium extension of phase transition multi- physical states encloses in itself the Boltzmann’s revolu- scale dynamics based at the non-equilibrium RGT was tionary concept of the probabilistic explanation of dynam- presented by Chang [21, 38]. ics, as the macroscopic (statistical) state of the system is Moreover, the multiscale character of dynamics of created by infinitive acceptable microstates according to phase transition process concerning the efficiency of the famous Boltzmann equation of entropy: nature to develop long range correlated states is in faithful agreement with novel recent extensions of physical theory Sk=− ppln ∑ ii (113) such as: i I. The dynamics of correlations based at the BBGKY or (Bogoliubov – Born – Green – Kirkwood – Yvon) hierarchy of generalized Liouville theory. In this Sk=− lnW (114) direction the Brussels school guided by I. Prigogine where W is the number of microstates and pi is the prob- and G. Nikolis tried to unify complex dynamics and ability for the realization of microstates. statistics as well as the microscopic reversibility The deterministic character of classical dynamics in of dynamics and the macroscopic irreversibility of the phase space of microscopic states is clearly in con- thermodynamics [34, 49, 83]. tradiction with the well established principle of entropy II. Tsallis extension of BG extensive statistical theory to which cannot be defined by one microstate at every the non-extensive statistical physics by the generali- time instant. It is known how Einstein preferred to put zation of BG entropy to the q-entropy of Tsallis and dynamics in priority of statistics [82], using the inverse the q-extension of thermodynamics [2]. 158 G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one
III. Scale relativity theory obtained by Nottale [13] and magnetic flux change in plasmas [85]. However, the fractal extended scale relativity theory obtained by Castro extension of dynamics and Tsallis extension of statistics and Granik [58]. indicate the possibility for a mechanism of fractal dis- IV. El Naschies E-infinity Cantorian space-time unifica- sipation and fractal acceleration process in fluids and tion of physical theory [57]. plasmas. V. Fractional extension of dynamics by Zaslavsky, Tara- According to Tsallis statistics and fractal dynamics sov and other scientists [24–26, 29, 30, 41, 61, 84]. the super-diffusion process:
2 〈〉Rt∼ γ After all, the transition phase process near or far from (115) equilibrium indicate the multiscale character of complex with γ > 1 (γ = 1 for normal diffusion) can be developed at dynamics which can create holistic complex states as the systems far from equilibrium. Such process is known as quantum dynamics creates holistic quantum states. The intermittent turbulence or as anomalous diffusion, which macroscopic phase transition process can be understood can be caused by Levy flight process included in fractal as the generalization of quantum state transition pro- dynamics and fFPK. The solution of fFPK equation (92) cesses in accordance with scale relativity theory, accord- corresponds to double (temporal, spatial) fractal charac- ing to which macroscopic dynamics is nothing else than teristic function: the scale transformation of quantum dynamics [13]. β From this point of view, intermittent turbulent P(,kt )e=−xp(constxt ||ka) (116) states, self-organized critical (SOC) states, chaos states or defect structures in distributed systems or other forms where P(k, t) is the Fourier transform of asymptotic distri- of self-organized states-structures of distributed complex bution function: dynamics are metastable stationary states caused by the βα1+ P(,ξξ t)c∼→onstxt /, ()ξ ∞ (117) principle of q-entropy of Tsallis statistical dynamics. In similarity with the microscopic quantum vacuum and This distribution is scale invariant with mean its quantum excitations we can understand the meta- displacement: stable and multiscale correlated complex states as the αβ “bounded” macroscopic states of the generalized complex 〈〉||ξ constxt, ()t →∞ (118) dynamics, while the equilibrium thermodynamical state corresponds to the state of vacuum of correlations accord- According to this description, the flights of multi- ing to sub-dynamic theory of dynamics of correlations scale and multi-fractal profile can explain the intermittent [18]. Thus as elementary particles and quantum struc- turbulence of fluids, the bursty character of dynamical ture are the excitations of the quantum vacuum state in energy dissipation and the bursty character of induced the same way the non-equilibrium metastable stationary electric fields and charged particle acceleration in space macroscopic states are the “excitation” of the state of the plasmas as well as the non-Gaussian dynamics of brain- thermodynamic vacuum of correlations. heart dynamics, or the defect-diseases spreading. The fractal motion of charged particles across the fractal and intermittent topologies of magnetic-electric fields 4.1 Fractal acceleration and fractal energy is the essence of strange kinetics [7, 24]. Strange kinet- dissipation ics permits the development of anomalous diffusion or defects or local sources with spatial fractal-intermittent The problem of kinetic or dynamical energy dissipation in condensation of induced electric-magnetic fields in brain, materials, fluid and plasmas as well as the bursty accel- heart and plasmas parallel with fractal-intermittent dis- eration processes of particles at flares, magnetospheric sipation of magnetic field energy in plasmas and fractal plasma sheet and other regions of space plasmas is an acceleration of charged particles. Such kinds of strange old and yet resisting problem of fluids or space plasma accelerators in plasmas or defects structuring in materi- science. als can be understood by using the Zaslavsky studies Normal Gaussian diffusion process described by the for Hamiltonian chaos in anomalous multi-fractal and Fokker-Planck equation is unable to explain either the multi-scale topologies of phase space [24]. Generally, the intermittent turbulence in fluids or the bursty character anomalous topology of phase space and fractional Ham- of energetic particle acceleration following the bursty iltonian dynamics correspond to dissipative non-Hamilto- development of inductive electric fields after turbulent nian dynamics in the usual phase space [25, 26]. The most G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one 159 important character of fractal kinetics is the wandering of the dynamical state through the gaps of cantori creates effective barriers for diffusion and long-range Levy flights in trapping regions of the phase space [7, 24]. Similar Levy flights processes can be developed by the fractal dynam- ics and intermittent turbulence of the complex systems. In this theoretical framework it is expected the exist- ence of Tsallis non extensive entropy and q-statistics in non-equilibrium distributed complex systems such as, materials, fluids, plasmas, seismogenesis or brain and heart dynamics, DNA structuring systems which are studied in the part II of this study. The fractional dynamics corresponding to the non-extensive Tsallis Figure 2: As the dynamics evolves through the multifractal phase space it produce multifractal and singular time series. The phase q-statistical character of the probability distributions in space is structured as a multifractal set of dynamical microscopical the distributed complex systems indicate the develop- states as the dynamics aims at the Tsallis q-entropy maximaization. ment of self-organized and globally correlated parts of In this figure, we have used the Wikipedia multifractal picture in active regions in the distributed dynamics. This charac- order to show the different regions of phase space corresponding to ter can be related also to deterministic low dimensional definite fractal dimension. The black broken line correspond to the chaotic profile of the active regions according to Pavlos the anomalous diffusion and Random Walk of the dynamics in the multifral phase space of the distributed dynamics. et al. [86].
process and the stationary fluctuations according to the 5 Theoretical expectations through q-CLT. These physical processes are described by the q-tri- Tsallis statistical theory and plet of Tsallis as presented in Figure 3. fractal dynamics 5.1 The q-triplet of Tsallis Tsallis q-statistics as well as the non-equilibrium fractal dynamics indicate the multi-scale, multi-fractal chaotic The non-extensive statistical theory is based mathemati- and holistic character of distributed dynamics. Observa- cally on the nonlinear equation: tions in space and time of complex variables correspond- ing to complex complex distributed dynamics produce ramdom signals in the form of space-time series. These observational signals mirror in themselves the non-exten- sive and multifractal character of the complex dynamics in the phase space of the complex system according to Figure 2. Tsallis q-triplet estimation by using observation sigmals is the basic tool for the experimental verification of the nonextensive and multifractal character of the dis- tributed dynamics. Moreover, Tsallis q-entropy principle can be used for the theoretical estimation of the multi- fractal singularity spectrum developed in the phase space and mirrored at the observed signals. That is the multi- fractal structure of the phase space and the production of multifractal observational sigmals corresponds to the Figure 3: As the dynamics of the system evolves aiming to the extremixzation of Tsallis q-entropy. The theoretically pre- q-entropy maximization it produce q-entropy. We can distinguish dicted by using Tsallis q-entropy principle, multifractal three different time periods. The first period corresponds to the entropy production through the q parameter of the Tsallis singularity spectrum can be compared to the experimen- sen q-triplet. The second period corresponds to some kind of relaxation tally singularity spectrum. Also the observed multifractal process through the qrel parameter of Tsallis q-triplet. Finally, as the sigmals included information for three sigtnifical physi- system lives at the stationary state with maximaized q-entropy it cal processes as the q-entropy production, the relaxation reveals fluctuations through the qstat parameter of Tsallis q-triplet. 160 G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one
dy q ==yy, ((0) 1, q ∈ℜ) (119) 5.1.2 The qsen index and the entropy production process dx The entropy production process is related to the general with solution the q-exponential function defined previ- profile of the attractor set of the dynamics. The profile of ously. The solution of this equation can be realized in the attractor can be described by its multifractality as well three distinct ways included in the q-triplet of Tsallis: (q , sen as by its sensitivity to initial conditions. The sensitivity to q , q ). These quantities characterize three physical pro- stat rel initial conditions can be described as follows: cesses which are summarized here, while the q-triplet values characterize the attractor set of the dynamics in the dξ q =+λξ ()λλ− ξ (124) τ 11q phase space of the dynamics and they can change when d the dynamics of the system is attracted to another attrac- where ξ describes the deviation of trajectories in the phase tor set of the phase space. The equation (119) for q = 1 cor- space by the relation: ξ ≡ lim {Δx(t)\Δx(0)} and Δx(t) is responds to the case of equilibrium Gaussian BG world [2]. Δ(x)→0 the distance of neighbouring trajectories. The solution of In this case of equilibrium BG world the q-triplet of Tsallis equation (121) is given by: is simplified to (qsen = 1, qstat = 1, qrel = 1). 1 1−q λλqq(1−qt) λ ξ =−1 sens+ en e sen1 (125) λλ 11 5.1.1 The qstat index and the non-extensive physical states The qsen exponent can be also related to the multifractal profile of the attractor set by the relation: According to [2] the long range correlated metaequilib- rium non-extensive physical process can be described by 111 =− (126) the nonlinear differential equation: qaa senmin max
dP()Z q where a (a ) corresponds to the zero points of the mul- i stat =−βqP()Z stat (120) min max dE stat i stat tifractal exponent spectrum f(a). That is f(a ) = f(a ) = 0. i min max The deviations of neighbouring trajectories as well as The solution of this equation corresponds to the probabil- the multifractal character of the dynamical attractor set ity distribution: in the system phase space are related to the chaotic phe- nomenon of entropy production according to Kolmogorov- − βstatEi Piq= eZ/ q (121) stat stat Sinai entropy production theory and the Pesin theorem.
1 − βqE The q-entropy production is summarized in the equation: where β = = stat j Then the probabil- q , Zstat ∑eq . stat KT stat stat j <>St() ity distribution function is given by the relations: K ≡ limlim lim q (127) q tW→∞ →∞ N→∞ t 1/1−q P ∝−[1 (1− qE)]β stat (122) iqstat i The entropy production (dS/dt) is identified with Kq, for discrete energy states {Ei} by the relation: as W are the number of non-overlapping little windows in
2 1/1−q phase space and N the state points in the windows accord- stat W P()xq∝−[1 (1− )]βq x (123) stat NN= . ing to the relation ∑ i=1 i The Sq entropy is estimated by the probabilities P (t) ≡ N (t)/N. According to Tsallis the for continuous X states {X}, where the values of the i i entropy production K is finite only for q = q . magnitude X correspond to the state points of the phase q sen space.
The above distributions functions (122, 123) corre- 5.1.3 The qrel index and the relaxation process spond to the attracting stationary solution of the extended (anomalous) diffusion equation related to the nonlinear The thermodynamical fluctuation – dissipation theory is dynamics of system. The stationary solutions P(x) describe based on the Einstein original diffusion theory (Brownian the probabilistic character of the dynamics on the attrac- motion theory). Diffusion process is the physical mecha- tor set of the phase space. The non-equilibrium dynamics nism for extremization of entropy. If ΔS denote the devia- can be evolved on distinct attractor sets depending upon tion of entropy from its equilibrium value S0, then the the control parameters values, while the qstat exponent can probability of the proposed fluctuation that may occur is change as the attractor set of the dynamics changes. given by: G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one 161
P ∼ exp(∆sk/) (128) where the partition function Zq is given by the relation: The Einstein-Smoluchowski theory of Brownian motion Zq =−2/Xq[(1)ln2]Bq(1/2, 2/1)− (133) was extended to the general FP diffusion theory of non- equilibrium processes. The potential of FP equation may and B(a, b) is the Beta function. The partition function Zq include many metaequilibrium stationary states near or as well as the quantities X and q can be estimated by using far away from the basic thermodynamical equilibrium the following equations: state. Macroscopically, the relaxation to the equilib- =+22−−− rium stationary state of some dynamical observable O(t) 2(Xa0 1)qq(1 ) b −− (134) related to the evolution of the system in phase space can bq=−(1 2)(1 q) /[(1− )ln] 2 be described by the form of general equation as follows: We can conclude for the exponent’s spectrum f(a) by using dΩ 1 q =− Ω rel (129) the relation P(a)≈lnd−F(a) as follows: dt T qrel 2 ()aa− f ()aD=+log1−−(1 qq)/o (1− )−1 (135) where Ω(t) ≡[O(t) − O( ∞)]/[O(0) − O( ∞)] is the relaxing 022/X ln2 relevant quantity of O(t) and describes the relaxation of the macroscopic observable O(t) relaxing towards its sta- where a corresponds to the q-expectation (mean) value of tionary state value. The non-extensive generalization of 0 a through the relation: fluctuation – dissipation theory is related to the general correlated anomalous diffusion processes [2]. Now, the <−()aa2 >= daPa()qq()aa− daPa()q (136) 00q ()∫∫ equilibrium relaxation process is transformed to the metaequilibrium non-extensive relaxation process: while the q-expectation value a0 corresponds to the
maximum of the function f(a) as df(a)/da | a0 = 0. For the dΩ 1 q =− Ω rel (130) Gaussian dynamics (q→1) we have mono-fractal spectrum dt T qrel f(a0) = D0. The mass exponent τ(q̅) can be also estimated by using the inverse Legendre transformation: τ(q̅) = aq̅ − f(a) the solution of this equation is given by: as follows: −tT/ Ω rel ()te q (131) 2 rel 21Xq τ =−−−−+ ()qqaC0211log1 q (137) + 1 − q () The autocorrelation function C(t) or the mutual informa- 1 Cq tion I(t) can be used as candidate observables Ω(t) for 2 the estimation of qrel. However, in contrast to the linear where Cq̅ = 1 + 2q̅ (1 − q)X ln 2. profile of the correlation function, the mutual information The relation between a and q can be found by solving includes the non-linearity of the underlying dynamics and the Legendre transformation equation q̅ = df(a)/da. Also it is proposed as a more faithful index of the relaxation if we use the equations (135–136) we can obtain the process and the estimation of the Tsallis exponent qrel. relation:
aa−=1[−−Cq(1 q)ln2] (138) qq0 () 5.2 Measures of multifractal intermittence turbulence The q-index is related to the scaling transformations of the multifractal nature of turbulence according to the relation In the following, we follow Arimitsu and Arimitsu [79] for q = 1 − a. Arimitsu and Arimitsu [79] estimated the q-index the theoretical estimation of significant quantitative rela- by analyzing the fully developed turbulence state in terms tions which can also be estimated experimentally. The of Tsallis statistics as follows: probability singularity distribution P(a) can be estimated 111 as extremizing the Tsallis entropy functional S . Accord- =− (139) q 1 − qa a ing to Arimitsu and Arimitsu the extremizing probability −+ density function P(a) is given as a q-exponential function: where a± satisfy the equation f(a±) = 0 of the multifractal 1 exponents spectrum f(a). This relation can be used for the ()aa− 2 1−q P()aZ=−−1 1(1)− q 0 (132) estimation of q -index included in the Tsallis q-triplet q 2/X ln2 sen (see next section). 162 G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one
−−βξ(1q ) 2 F [(Gxβξ; )]()= e x (143) 6 Theoretical interpretations of qq−−11 q 3 − s = −1 ≤ < β = data analysis results where q − 1 z (q), 1 q 3 and x ()s 22−−ss(1) . 8β Cs The function z(s) permits the following q-Fourier In this section we present more analytically some useful transform mappings: theoretical concents that need for understanding the experimental evidence of complex systems that are pre- F :,ℑ→ℑ= qz()qq, 13≤< qq q1 1 sented in part two of this study. These concepts concern −1 (144) Fqq−−11:,ℑ→ℑ=q qz−1 ()qq, 13≤< the q-CLT and its relation with the q-triplet of Tsallis as well as the fractional dynamics and the anomalous as well as the inverse mappings by using the inverse diffusion-random Walk process. q-Fourier transforms:
−1 Fqq:,ℑ→ℑ=q qz1 ()q 1 (145) 6.1 The q-extension of central limit theorem −−11 Fqq−−11:,ℑ→ℑ=q qz−1 ()qq, 13≤< and the q-triplet of Tsallis We can introduce the sequence q = z (q) = z(z (q)), According to the classical central limit theorem, the prob- n n n−1 n = 1, 2, … starting from the initial q = z (q), q < 3. The q ability density functions of a sum of independent random 0 n sequence can be extended for negative integers n = − 1, variables are Gaussian when the single variables satisfied −2, … by q = z (q) = z−1(z (q)). The sequence q can be typical presuppositions. However, the non-Gaussian with −n −n 1−n n given through the following mappings: heavy tails probability distribution functions, which were zzzzzz observed which are observed in distributed dynamical →qq−−21→→=qq01→qq→2 →… systems, the statistics of which is related to the q-exten- −−11−−11−−11 ←zz←qq←zz=qq←←qqzz←… sion of central limit theorem. −−21012 Tsallis non-extensive statistical mechanics includes while the duality relations hold: the q-generalization of the classic CLT as a q-generali- zation of the Levy-Gnedenko central limit theorem [87] 1 qz− +=, n ∈Z (146) applied for globally correlated random variables. The n 1 q n+1 q-generalization of CLT based at the q-Fourier transform of a q-Gaussian can produce an infinite sequence (qn) of The q-generalization of the central limit theorem con- 1 + s sistent with non-extensive statistical mechanics is as q-parameters by using the function Z()s = , s ∈( − ∞, 3 − s follows: −1 3) and its inverse z (t), t ∈( − 1, ∞). It can be shown that For a sequence qk, k ∈ ℤ with qk ∈[1,2] and a sequence −1 z(1/z(s)) = 1/s and z(1/s) = 1/z (s). Then with q1 = z(q) x1, x2, xN, … of qk-independent and identically distributed 1111random variable then the z = x + x + x + … is also a q - = −1 == N 1 2 N k−1 and q−1 z (q) it follows that: zz, qqqq− 1 normal distribution as N→ ∞, with corresponding statisti- 1 1 and q−1 +=2. The set of all q-Gaussians Gq(β, x) be cal attractor Gxqk()β . The q-independence corresponds q kj−1 1 to the relations: denoted by:
Fqq()xy+=()ξξFx[]()⊗qqFy[]()ξ (147a) ϒβqq=>{(bG , xb): 0, β > 0} (140)
For q-Gaussians the q-Fourier transform hold as follows: Fqq−−11()xy+=()ξξFx[]()⊗qqFy[]()ξ (147b)
2 − βξx ()q Fqq[(Gxβξ, )]()= eq (141) 1 where q = z(q−1). The q-independence means independ- ence for q = 1 and strong correlation for q ≠ 1 [2, 88]. where q = z(q), 1 ≤ q < 3. 1 The q-CLT states that an appropriately scaled limit of The q-Fourier transform is defined by Umarov et al. sums of qk correlated random variables is a qk-Gaussian, [87] by the formula: * * which is the qk -Fourier image of a qk -Gaussian. The qk, F []fe()ξ =⊗i×ξ fx()dx q* are sequences: qq∫ q (142) k
2(qk+−1)q ∗ For the inverse q-Fourier transform we have the following qq== a nd qk for =±0, 1, ± 2, … (148) kk2(+−kq1) k−1 formula: G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one 163
−−≤−h including the triplet (Patt, Pcor, Pscl), where Patt, Pcor and Pscl |(fx)(Px xC∗∗)| ||xx (153) are parameters of attractor, correlation and scaling rate Where P (x) is a polynomial vorter n. If h(x ) ∈(n, n + 1) respectively and corresponds to the q-triplet (q , q , q ) n * sens rel stat then f(x) is n-times but not n + 1 times differentiable at the according to the relations [87]: point x*. The smaller the exponent h(x*) is the more irreg- ≡≡ (,PPattc or , Pqscl1)(kk−+, qq, k 1s)(qqensr, el , qstat ) (149) ular (singular) is the function f at the point x*. On a self similar fractal set the singularity strength of the measure The parameter P ≡ q ≡ q describes the non-ergodic att sens k−1 μ at the point x is described by the scaling relation: q-entropy production of the multiscale correlated process as the system shifts to the state of the q – Gaussian, µ((Bdεµ)) ()y ∼ εax() att x ∫ (154) Βε() where the q-entropy is extremized in accordance with the x generalization of the Pesin’s theorem [88]: Where Bx(ε) is a ball of size (ε) centered at x and μ(Bx) is SP((tk))/ the fractal mass in the B region. Homogeneous measures K ≡=limlim lim qi λ (150) x qqsenstW→∞ →∞ M→∞ t en are characterized by a singularity spectrum supported by
a single point (a0, f(a0)) while multifractal sets involve sin-
The parameter Pcor ≡ qrel ≡ qk describes the q-correlated gularities of different strengths (a) described by the singu- random variables participating to the dynamical process larity spectrum f(a). of the q-entropy production and the relaxation process The singularity spectrum is defined as the Hausdorff toward the stationary state. dimension of the set of all points x such that a(x) = a. The
The parameter Pslc ≡ qstat ≡ qk+1 describes the scale singularity strength a(x) of the fractal mass measure is invariance profile of the stationary state corresponding related to the local fractal dimension D of the fractal set. to the scale invariant q-Gaussian attractor as well as to More generally, the singularity spectrum associated with an anomalous diffusion process mirrored at the variance singularities h (Hölder exponents) of a distributed random scaling according to general, asymptotically scaling, form: variable is given follows:
D x D()hd=={:xh()xh} (155) N Px()∼ G (151) H x N D That is D(h) is the Hausdorff dimension of the set of all
Where Px(x) is the probability function of the self-similar points x such as h(x) = h. For the case of space plasmas or statistical attractor G and D is the scaling exponent char- fluids the singularity spectrum D(h) corresponds to the acterizing the anomalous diffusion process [89]: scaling invariance of the MHD description according to
22D the scaling transformation of the MHD equations: 〈〉xt∼ (152) −−1 hh− rr→→′λλ, uurr′′, bbrr→ λ (156) The non-Gaussian multi-scale correlation can create the intermittent multi-fractal structure of the phase space where h is a free parameter and ur, br are the velocity and mirrored also in the physical space multi-fractal distri- magnetic fields of the solar wind plasma system, cor- bution of the turbulent dissipation field. The multi-scale respondingly [90]. The a(x) singularity exponent of the interaction at non-equilibrium critical NESS creates the fractal mass measure and the f(a) singularity spectrum heavy tail and power law probability distribution func- describes the scaling of the energy dissipation in the solar tion obeying the q-entropy principle. The singularity wind turbulent system. spectrum of a critical NESS corresponds to extremized The fractal-multifractal structure of the general Tsallis q-entropy. dynamical systems (particles and fields) indicates the gen- In this framework of theoretical modeling of distributed eralization of the classical field-particle dynamics to the random fields, the fractal-multifractal character of the dis- fractional dynamics, since the functions of the distributed tributed dynamical systems at every NESS (week or strong physical system’s variables are irregular and they are pro- turbulence) as it was shown in this study, indicate the exist- duced by fractional dynamics on fractal structures. The ence of physical local singularities for the spatial-temporal differentiable nature of smooth distribution of the mac- distribution of distributed physical variables. The singular- roscopic picture of physical processes is a natural conse- ity behavior of a given random field function f(x) at a given quence of the Gaussian microscopic randomness which, point x* is defined as the greatest exponent h so that f is Lip- through the classical CLT, is transformed to the macro- schitz h at x* [67]. The Hölder exponent h(x*) measures how scopic, smooth and differentiable processes. The classical irregular f is at the point X* according to the relation: CLT is related to the condition of time-scale separation, 164 G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one
∞ aa1 −1 where at the long-time limit the memory of the microscopic where ()If()zx≡−()xf′′()xdx′ is the Riemann zx Γ()a ∫ non-differentiable character is lost. On the other hand, z Liouville fractional integral [91, 92]. The last relation con- the q-extension of CLT induces the nonexistence of time- nects the integral on fractals with fractional integrals scale separation between microscopic and macroscopic and permits the application of different tools of the frac- scales as the result of multiscale global correlations which tional calculus for the fractal medium. Respectively to the produce fractional dynamics and singular functions of Riemann Liouville fractional integral on a fractal set F we spatio-temporal dynamical physical variables. can define the Riemann Liouville fractional derivative by:
1(∂n x fx′) Dfa ()xd=−xn′, 1 < ∞ the Fokker-Planck equation are transformed to fractional f ()xX= β ()x (157) equations [25, 26, 41, 91]. The solution of the fractional ∑ iEi i=1 equations correspond to fractional non-differentiable singular self-similar functions as we can observe at the where X is the characteristic function of E. The continu- E experimental data. ous function f(x) is defined as follows: Milovanov and Zelenyi [93] introduced the fractional lim(fx)(= fy) whenever dx(, y)0= for the metric wave equation for the description of space plasma or other xy→ xy→ distributed systems where fractons excitations, namely d(x, y) defined in Rn and for points (x, y) ∈ f. The Hausdorff fractional waves on fractals can be excistense. The frac- measure μ of a subset E ∈ F is defined by: H tional extension of wave equations for spherically sym- ∞ metric vibrations of fractals includes temporal and spatial µ (,ED )(=WD)lim inf|dE()|D (158) HidE()→0{E } ∑ ii fractional derivatives as follows: i=1 ∂ψ ∞ − γ 1 ∂ D−1 ∂t where E ⊂UEii=1 , D is the Hausdorff dimension of E ⊂ U, ψ =−( ) << (163) Dt (,tx ),D−1 xx na1 n x ∂x d(Ei) are the diameters of {Ei} and W(D) for balls {Ei} cover- ing F is given by: where DD/2 − Π 2 γ m t WD()= γ ∂∂ψψ−1 (,τ x) D (159) D ψΓ(,tx )(≡=md− γτ) (164) Γ + 1 t ∂∂ttγγmm∫ ()t − τ 1+− 2 0 The Lebesque – Stieltjes integral over a D-dimensional is the fractional time derivative, D is the fractal dimen- fractal set of a function f(x) is defined by: sion of space and θ is the connectivity index of the fractal ∞ space. The nonlocal character of the fractional wave equa- fdµβ≡ µ ()E ∫ ∑ iH i (160) tion on fractals is related to the correlated and coherent = F i 1 character of local dynamical events on a self-similar struc- and it can be proved to be given by the relation: ture. The solution of the fractional wave equations con- tains exponentially decaying factors which are absent in 2(ΠΓD/2 D) fdµ = ()IfD , Euclidian solutions. These factors are responsible for the ∫ HoD (161) F Γ spatial localization of fractons, while the strength of local- 2 ization is described by the connectivity index θ, through G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one 165 the localization exponent 2 + θ/2, which coincides with the where P ≡ P(ξ, t) is the probability density of the state (ξ) Hausdorff dimension of the “geodesic lines” on the fractal at the time (t). space. The localization of waves on fractals (fractons) can The critical components (a, β) correspond to the be used for the explanation of local variation of solar wind fractal dimension of spatiotemporal nongaussian dis- characteristics, especially during shock events. tribution of the temporal-spatial functions-processes or probability distributions. The quantities A, B are given by: 〈〈||∆ξαα〉〉 〈〈||∆ξ2 〉〉 Al==im ,lΒ im (167) 6.3 Anomalous diffusion and strange ∆∆tt→→00()∆∆ttββ() dynamics where << … > > denotes a generalized convolution opera- Nonlinear dynamics can create fractal structuring of the tor [94]. phase space and global correlations in the nonlinear system. The FFPK equation is an archetype fractional equa- For nonextensive systems the entire phase space is dynami- tion of fractional stochastic dynamics in a (multi)-fractal cally not entirely occupied (the system is not ergodic), but phase space with fractal temporal evolution caused by only a scale-free –like part of it is visited yielding a long- the self similar and multiscale structure of islands around standing (multi)-fractal-like occupation. According to Milo- islands, responsible for the flights and trappins of the vanov and Zelenyi [69], Tsallis entropy can be rigorously dynamics. The “spatial” random variable can be any phys- obtained as the solution of a nonlinear functional equation ical variable, such as position in physical space, velocity in referred to the spatial entropies of the subsystems involved the velocity space or a dynamic field (magnetic or electric) including two principal parts. The first part is linear (addi- at a certain position in physical space, stresses in materi- tive) and leads to the extensive Boltzman-Gibbs entropy. als etc, underlying to the nonlinear chaotic dynamics. The The second part is multiplicative corresponding to the non- fractional dynamics of plasma includes fractal distribu- extensive Tsallis entropy referred to the long range corre- tion of field and currents, as well as fractal distribution of lations. The fractal –multifractal structuring of the phase energy dissipation field. space makes the effective number W of possible states, eff The fractional temporal derivative ∂β/∂tβ in kinetic namely those whose probability is nonzero, to be smaller equations allows one to take fractal-time random walks (W < W) than the total number of states. eff into account, as the temporal component of the strange According to Zaslavsky [24] the topological struc- dynamics in fractal-turbulent media. The waiting times ture of phase space of nonlinear dynamics can be highly follow the power law distribution P(τ) ~ τ−(1+β) since the complicated including trapping and flights of the dynam- “Levy flights” of the dynamics also follow the power law ics through a self-similar structure of islands. The island of distribution. boundary is sticky making the dynamics to be locally The asymptotics (root mean square of the displace- trapped and “stickiness”. The set of islands is enclosed ment) of the transport process is given by 〈〉||ξ 2/= 2,Dt βα within the infinite fractal set of cantori causing the com- while the generalized transport coefficient D depends on plementary features of trapping and flight being the the value of the anomalous scaling exponent b/a. essence of strange kinetics and anomalous diffusion. The solution of the fractional kinetic equation corre- The dynamics in the topologically anomalous phase sponds to Levy distributions and asymptotically to Tsallis space corresponds to a random walk process which is q-Gaussians. According to Alemany and Zanette [95], the scale invariant in spatial and temporal self-similarity set of points visited by the random walker can reveal a transform: self-similar fractal structure produced by the extremiza- ˆ tion of Tsallis q-entropy. The q-Gaussian distribution of R:,tt′′→=λξt λξξ (165) the fractal structure created by the strange dynamics and The spatial-temporal scale invariance causes strong the extremized q-entropy asymptotically corresponds spatial and temporal correlations mirrored in singular to the Levy distribution P(ξ) ~ ξ−1−γ where the q-exponent self-similar temporal and spatial distribution functions 3 + γ is related to the Levy exponent γ by q = . The Levy which satisfy the fractional generalization of classical 1 + γ Fokker-Planck-Kolmogorov equation (fFPK equation) [24]: exponent γ corresponds to the fractal structure of the points visited by the random walker. ∂∂βαP 1 ∂2 α =+()AP ()BP (166) The well-known Boltzmann’s formula S = k log W ∂∂t βα()−∂ξξ2 ()− 2 α where S is the entropy of the system and W is the number 166 G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one of the microscopic states corresponding to a macroscopic matter (particles, fluids, fields) can be replaced by a frac- state indicates the priority of statistics over dynamics. tional continuous model. In this generalization, the frac- However, Einstein preferred to put dynamics in prior- tional integrals can be considered as approximations of ity of statistics, using the inverse relation i.e. W = eS/k integrals on fractals. Also, the fractional derivatives are where entropy is not a statistical but a dynamical magni- related with the development of long range correlations tude. Already, Boltzmann himself was using the relation and localized fractal structures. t Ω W ==lim RR where t is the total amount of time that R T→∞ T Ω R the system spends during the time T in its phase space tra- 6.4 Fractal topology, critical percolation and Ω jectory in the region R while R is the phase space volume stochastic dynamics of region R and Ω is the total volume of phase space. The above concepts of Boltzmann and Einstein were innova- The nonlinearity in the dynamics of the solar wind plasma tive as concerns modern q-extension of statistics, which system including fields (B, E) and particles can create is internally related to the fractal extension of dynamics. random distributions in space of dynamical fields and According to Zaslavsky [96] and Tsallis [97] the fractal material fields (bulk, velocity, pressure, temperature, extension of dynamics includes simultaneously the fluxes, currents etc). In this section we follow Milovanov q-extension of statistics as well as the fractal extension and Zimbardo [99] and Milovanov [9] and present some RNG theory in the fFPK: basic concepts concerning topological aspects of percolat- ing random fields. ∂∂βαPx(, t) = [(Ax)(Px, t)] For any random field distribution ψ(x) in the ∂∂βtx()− a n-dimensional space (En) there exists a critical percola- a+1 ∂ n + [(Bx)(Px, t)] tion threshold which divides the space E into two topo- a+1 (168) ∂−()x ψ < logical distinct parts: Regions where (x) hc marked as “empty”, and regions where ψ(x) > hc, marked as “filled”. where the variables x corresponds to physical distributed When ψ(x) ≠ hc, one of these parts will include an infinite variables of the system, while P(x, t) describes the prob- connected set which is said to percolate. As the threshold ability distribution of the particle-fields variables. h changes we can find the critical threshold hc where the The variables A(x), B(x) corresponds to the first and topological phase transition occurs, namely the nonper- second moments of probability transfer and describe the colating part starts to percolate. wandering process in the fractal space (phase space) and The geometry of the percolating set at the critical state β β time. The fractional space and time derivatives ∂ /∂t , (h → hc) is a typical fractal set for length scales between ∂α/∂xα are caused by the multifractal (strange) topology of microscopic distances and percolation correlation length phase space which can be mirrored in the spatial multi- which diverges. The statistically self-similar geometry fractal distribution of the dynamical variables. includes power-law behavior of the “mass” density of the The q-statistics of Tsallis corresponds to the meta- fractal set such as “fractal mass density” ~xD–n, where x equilibrium solutions of the fFPK equation [26, 97]. Also, is the length scale, D is the Hausdorff fractal dimension the metaequilibrium states of fFPK equation correspond which must be smaller than the dimensionality (n) of the to the fixed points of Chang non-equilibrium RGT theory embedding Euclidean space. In addition to the parameter for space plasmas [24, 98]. The anomalous topology of D of the fractal dimension, there is the index of connec- phase space dynamics includes inherently the statistics as tivity θ which describes the “shape” of the fractal set and a consequence of its multiscale and multifractal charac- may be different for fractals even with equal values of the ter. From this point of view the non-extensive character of fractal dimension D. The index of connectivity θ is defined thermodynamics constitute a kind of unification between as characterizing the shortest (geodesic) line connect- statistics and dynamics. From a wider point of view the ing two different points on the fractal set by the relation fFPK equation is a partial manifestation of a general dθ =(2 + θ)/2, where dθ is the minimal Hausdorff dimension fractal extension of dynamics. According to Tarasov [25], of the minimal (geodesic) line. The geodesic line on a self- the Zaslavsky’s equation can be derived from a fractional similar fractal set (F) is a self affine fractal curve whose generalization of the Liouville and BBGKI equations. own Hausdorff fractal dimension is equal to (2 + θ)/2. According also to Tarasov [25, 26, 41], the fractal exten- The index of connectivity plays an essential role in many sion of dynamics including the dynamics of particles or dynamical phenomena on fractals, while it is a topologi- fields is based on the fact that the fractal structure of cal invariant of the fractal set F. G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one 167 From the fractal dimension D and the connectivity The fractional extension of Maxwell equations is index θ we can define a hybrid parameter ds = 2D/2 + θ caused by the non-local self-similar hierarchical structur- which is known as the spectral or the fracton dimension ing of the plasma system, while the degree of non-locality which represents the density of states for vibrational exci- is quantified by the connectivity index included in the tations in fractal network termed as fractions [9]. The root power exponent γ = 2/(2 + β) in the singular kernel (s − w)−γ. mean square displacement of the random walker on the The interaction of plasma charged particles and dynami- fractal set is given by: cal fields described by fractional Maxwell equations and −1 fractional transport equations causes self-consistency in 〈〉||ξ 22∼=tt/2+θ dθ (169) the fractal distribution of dynamical (magnetic-electric) fields, of bulk plasma flow fields and energy dissipation Where d is the fractal dimension of the self-affine trajec- θ fields. In this way, the fractal dimension and connectivity tory on the fractal set. Also, the spectral dimension which index (θ) of dynamical field’s distribution is self-consist- measures the probability of the random walker to return ently related to the fractal dimension (D+) and the index of to the origin, is given by: connectivity (θ+) of material fields. −d /2 P()tt∼ s (170) while the Hausdorff fractal dimension D is a structural 6.5 Renormalization Group (RNG) theory and characteristic of the fractal structure F, the spectral phase space transition dimension ds mirrors the dynamical properties such as wave excitation, diffusion etc. The fractal dimension D The multifractal and multiscale intermittent turbulent of the fractal structure F of a percolating random field character of the distributed dynamics can be verified after distributed in the En Euclidian space is given by D = n– the estimation of the spectrum f(a) of the point wise dimen- β/ν, where β, ν are the universal critical exponents of the sions or singularities (a). This justifies the application of critical percolation state [9]. According to the Alexander- RG theory for the description of the scale invariance and Orbach (AO) conjecture [100], the spectral dimension d s the development of long-range correlation of the distrib- has been established to be equal to the value d = 4/3, for s uted systems and intermittent turbulence state. Generally all embedding dimensions n ≥ 2. Especially, for embed- the non-equilibrium distributed random dynamics can be ding dimensions 2 ≤ n ≤ 5, Milovanov [100] has improved described by generalized Langevin stochastic equations the AO conjecture to the value d = c1327, where c is s of the general type: the percolation constant. This constant determines the minimal fractional number of the degrees of freedom ∂ϕ i =+ft(,ϕ xx, )(nt, ), i = 1, 2, … (172) ∂ ii that the random walker must have to reach the infi- t nitely remote point in the Euclidian embedding space n E . According to Milovanov [9, 100, 101] and Milovanov where fi corresponds to the deterministic process as con- and Zelenyi [69] the fractal topology and critical perco- cerns the plasma dynamical variables φ(x, t) and ni to the lation theory transform the description of plasma diffu- stochastic components (fluctuations). Generally, fi are sion, bulk flow and electrodynamical-MHD phenomena nonrandom forces corresponding to the functional deriva- from classical smooth equations to fractional equations tive of the free energy functional of the system. According description. to Chang [38, 102] the behavior of a nonlinear stochastic The diffusion coefficient is now expressed in terms system far from equilibrium can be described by the prob- of the connectivity index θ =(μ–β)/ν and the Hausdorff ability density functional P, defined by: dimension of the infinite percolation cluster d = n–β/ν f Pt((ϕ x, )) = where β, μ, ν are the universal critical percolation para- (173) Di()xxexp{−⋅ Ld(,ϕ ϕ, )}x dt meters. The Maxwell equations for the plasma current ∫ ∫ system are transformed to fractional equations including fractional derivatives of the magnetic field given by: where L(,ϕ ϕ, x) is the stochastic Lagrangian of the s system, which describes the full dynamics of the stochas- ∂∂γ ()BB1(w) = dw (171) tic system. ∂sγ Γγ(1−∂)(ss∫ − w) 0 Moreover, the far from equilibrium renormalization group theory applied to the stochastic Lagrangian L gen- where s, w are spatial variables. erates the singular points (fixed points) in the affine space 168 G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one of the stochastic distributed system. At fixed points the harmonized with the nonlinear dynamics of the distrib- system reveals the character of criticality, as near critical- uted system as well as the non-equilibrium extension of ity the correlations among the fluctuations of the random RGT. This is the base for the development of anomalous dynamic field are extremely long-ranged and there exist diffusion plasma processes, long-range correlations and many correlation scales. Also, close to dynamic criticality scale invariance which can be amplified as the system certain linear combinations of the parameters, character- approaches far from equilibrium dynamical critical points. izing the stochastic Lagrangian of the system, correlate Also, it is known that nonlinear dissipative dynamics with with each other in the form of power laws and the sto- finite or infinite degrees of freedom includes the possibil- chastic system can be described by a small number of ity of self-organizing reduction of the effective degrees of relevant parameters characterizing the truncated system freedom and bifurcation to periodic or strange (chaotic) of equations with low or high dimensionality. This is the attractors with spontaneous development of macroscopic nonequilibrium RG theory explanations of self-organ- ordered spatiotemporal patterns. ization process and dynamical reduction of degrees of The bifurcation points of the nonlinear dynamics, freedom. According to these theoretical results, the sto- corresponds to the critical points of far from equilibrium chastic system can exhibit low dimensional chaotic or non-classical statistical mechanics and its generaliza- high dimensional SOC like behavior, including fractal or tions, as well as to the fixed points of the RGT. The RGT multifractal structures with power law profiles. The power is based in the general principle of scale invariance of the laws are connected to the near criticality phase transition physical processes as we pass from the microscopic statis- process which creates spatial and temporal correlations tical continuum limit to the macroscopic thermodynamic as well as strong or weak reduction (self-organization) of limit. For introducing the mechanism of the RGT at dis- the infinite dimensionality corresponding to a spatially tributed dynamics we start with the generalized Langevin distributed system. Critical phase transition processes can equation: be related to discrete fixed points in the affine dynamical δF (Lagrangian) space of the stochastic dynamics. The SOC or ΨΓ=− + N (174) iiδΨ chaos like behavior of distributed dynamics corresponds i to the second phase transition process as the system lives Where F is the free energy of the system and Γ is the at different fixed points of RG. The probabilistic solution relaxation rate and N is the noise component with the (173) of the generalized Langevin equations may include correlation: Gaussian or non-Gaussian processes as well as normal or 〈〉′′=−Γδδδ′′− Ψ (175) anomalous diffusion processes depending upon the criti- Nxii(, tN)(xt, )2()xxji()xt()tx(, t) cal state of the system. corresponding to the distributed physical magnitude From this point of view, a SOC or low dimensional of the solar wind plasma system. The structure of these chaos interpretation or distinct q-statistical states with equations (174, 175) can be produced by using the different values of the Tsallis q-triplet depends upon the BBGKY (Bogoliubov-Born-Green-Kirkood-Yvon) hierar- type of the critical fixed (singular) point in the functional chy included in the Liouville equation applied at the sto- solution space of the system. When the stochastic system chastic system particles and fields. Also, the structure is externally driven or perturbed, it can be moved from a of Langevin equation ensures that the metaequilibrium particular state of criticality to another characterized by distribution is always attained as t→ ∞. In the following a different fixed point and different dimensionality or and according to Tsallis q thermodynamic theory we con- scaling laws. Thus, the old SOC theory could be a special jecture in relation with Langevin equations the generali- kind of critical dynamics of an externally driven stochas- zation of the Free energy functional to the q-Free energy tic system. After all SOC and low dimensional chaos can functional Fq so that the Langevin equation can be related coexist in the same dynamical system as a process mani- to a q-generalized Fokker-Planck equation of. By using fested by different kinds of fixed (critical) points in its the q-generalized Fokker – Planck equation we can esti- solution space. Due to this fact, the distrtibuted dynamics mate the N-point correlation function (GN) of the plasma may include high dimensional SOC process or low dimen- system, as well as, the partition function Zq as the func- sional chaos or other more general dynamical process tional integrations: corresponding to various q-statistical states. The non- q GxN (, xx, ……, )(=〈ΨΨxx), (), ,(Ψ x )〉= extensive character of the distributed dynamics related 12 N 12 Ν − D (176a) with q-statistical metaequilibrium thermodynamics as 1 Fdq x =⋅Dx[]ΨΦ()⋅⋅ΦΦ()xx… ()⋅e ∫ ∫ 12 Ν well as the existence of long-range correlations must be 2 G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one 169 − FJ()dDx ∫ q Z ((Jx)) =⋅De[]Ψ physical critical point corresponds to the flow of the para- q ∫ (176b) meter vector K at the neighborhood of the fixed point. The flow of the parameter vector K at the neighborhood of the where JZ is a source field and = lim(ZJ()x ) (176c) fixed point K * is a nonlinear flow in a finite dimensional q J→0 space which survives the most significant physical charac- The N-point q-correlation function is related to the teristics of the original dynamics of the plasma system with functional derivative of partition function as follows: infinite degrees of freedom. The representation of the infi- δΖΝ ((Jx)) nite dimensional dynamics to finite dimensional and is pos- q 1 q (177) GxNN(,12 xx, …, ) = sible at every instant the infinite dimensional dynamical ZJδδ()xJ⋅⋅()xJ… δ ()x 12 Ν state (state of infinite degrees of freedom) is transformed by the scale invariance vehicle to a finite dimensional dynam- For the estimation of the system’s partition function it ics in the parameter space. According to this theoretical makes no sense to consider fields, which varies rapidly description the distributed dynamical system can exist on scales shorter than a characteristic microscopic at district fixed points in the parameter space of RGT cor- dimension and such field should be excluded from the responding to the shock chaos or intermittent turbulence estimation of the partition function, as well as, from the states. Also the development of the strong turbulence cor- application of scale transformation according to the RGT. responds to global topological phase transition process This can be succeeded by a suitable cutoff parameter of from the calm period to the shock period fixed point in the the Fourier transform of the distributed plasma proper- solar plasma dynamical parameter space. From the above ties. According to Wilson [19] and Chang [38] the dynamics theoretical point of view the quantative change of the non- of distributed system as it approaches the critical points extensive Tsallis statistics as it can be observed by the esti- includes a cooperation of all the scales from the micro- mation of Tsallis q-triplet can be related to the change of the scopic to macroscopic level. The multiscale and holistic RGT fixed point in the dynamical parameter space. distributed dynamics can be produced by scale invariance principle included in the RG transformation correspond- ing to the flow: 7 Summary and discussion nn−10n (178) K ==RKll()……=⋅RK(), n = 0, 1, 2 Figure 4 presents the most significant concepts and theo- 0 0 where K is the original parameter vector K and Rl is retical characteristics of the dynamics and statistics as the the renormalization-group operator of the partition func- distributed dynamical system departs from the thermo- tion and the density of free energy. The parameter vector dynamic equilibrium and lives at metaequiilibrium states K has as components the parameters K12, KK, …, m the with exramized q-entropy. At the thermodynamic equi- coupling constants upon which the free energy depends. librium state the system reveals weak complex character The RG flow in the dynamical parameter space of vectors while far from equilibrium the full complex character is K is caused by a spatial change of scale as at every step manifastated. Near thermodynamic equilibrium statis- of flow in the parameter space the spatial scale is rescaled tics and dynamics are two separated but fundamental according to the relation: rl'1=⋅− r. As the free energy and elements of the physical theory. Also, at thermodynami- the partition function are rescaled by the RG flow in the cal equilibrium, nature reveals itself as a Gaussian and parameter space the correlation length (ξ) of the plasma macroscopically uncorrelated process simultaneously fields is rescaled also according to the relation: with unavoidable or inevitable and objective determinis- tic character. However, modern evolution of the scientific nn−−11 −n 0 (179) ξ()Κξ=⋅ll()Κξ==……⋅=()Κ , n 0, 1, 2 knowledge reveals the equilibrium characteristics of the physical theory as an approximation or the limit of a more At the fixed points K * of the RG flow, the relation: synthetic physical theory, which is characterized as com- ξ()Κξ∗−=⋅l 1 ()Κ∗ (180) plexity. The new physical characteristics of complexity theory can be manifested as a physical system is driven in implies that the correlation length at the fixed point must far from equilibrium states. be either zero or infinite. Also, as the zero value is without Moreover, far from equilibrium statistics and dynam- physical interest we conclude the infinite correlation of the ics can be unified through the Tsallis nonextensive sta- system at the fixed point K * or long-range correlation in tistics included in Tsallis q-entropy theory [2] and the near the fixed point. The dynamics of the system near the fractal generalization of dynamics included in theories 170 G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one Figure 4: In this table we summarize the q-extension of dynamics and statistics as the system departs from the state of thermodynamic equilibrium and lives at meta equilibrium stationary states. developed by Ord [76], Nottale [13], Castro [12], Zaslavsky following probabilistic interpretation of Heisenberg opera- [24], Shlesinger et al. [7], Kroger [60], Tarasov [25, 26, 41], tionalistic concepts and Schrodinger wave functions done El-Nabulsi [29, 61], Cresson and Greff [40], Goldfain [30, by Max Born [54], probability obtained clearly a new real- 84] and others. Also, the new non-extensive character- istic and ontological character concerning the foundation istics of non-equilibrium physical theory for distributed of physics [108]. Moreover, the development of complexity physical systems were verified experimentally until now theory caused the extension of the probabilistic character by Pavlos [64, 86, 103, 104], Iliopoulos et al. [105] and of dynamics from microscopic (quantum theory) to mac- Karakatsanis and Pavlos [106, 107] in many cases. roscopic (classical theory) level of reality. This showed Generally, the traditional scientific point of view is the deeper meaning of the Boltzmann entropy principle the priority of dynamics over statistics. That is dynam- through the fractal extension of dynamics, as well as the ics creates statistics. However, for a complex system the q-extension of statistics according to Tsallis non-extensive holistic behaviour does not permit easily such a sim- entropy theory. plification or division of dynamics and statistics. From Prigogine [34] and Nicolis [49] were the principal this point of view Tsallis statistics and fractal or strange leaders of an outstanding transition to the new epistemo- kinetics are two faces of the same complex and holistic logical ideas from microscopic to the macroscopic level, (non-reductionist) reality. As Tsallis statistics is an exten- as they discovered the admirable self Self-Organization sion of Boltzmann-Gibbs statistics, we can support that operation of the physical-chemical systems. That is, the the thermal and the dynamical character of a complex possibility for the development of long-range spatiotem- system are the manifestation of the same physical process poral correlations, as the system lives far from equilib- which creates extremized thermal states (extremization of rium. Thus, Prigogine and Nicolis opened a new road Tsallis entropy) or extremized mechanical actions, as well towards the physical understanding of random fields and as dynamically ordered states. From this point of view statistics, related to the non-Gaussian character of the the Feynman path integral formulation of the statistical physical processes. This behaviour of nature is known as physical theory and of the quantum probabilistic theory non-equilibrium development of dissipative structrures indicates the indivisible thermal and dynamical character or self-organization process. Prigogine’s, Haken’s and of physical reality [31]. After Heisenberg’s revolutionary Nicoli’s self-organization concepts inspired us to intro- substitution of physical magnitudes by operators and the duce the self organization theory as a basic tool for the G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one 171 description of many complex systems. In this direction the calculus (fractal integrals-fractal derivatives) [25, 26, 30, 40, dynamics of the space plasmas [86, 104, 109], seismogen- 41, 48]. Also, the fractal structure of space-time includes esis [105], as well as the dynamics of brain activity [110] intrinsically a stochastic character as the presupposition has been decribed as a non-equilibrium and self-organi- for determinism is differentiability [13, 40]. In this way, zation process. Parallely to the scientific work of Brussels’ statistics is unified with dynamics automatically, while school, Lorenz [111] discovered the Lorenz’s attractor as the notion of probability obtains a physical substance, the weather’s self organization process, while other sci- characterized as dynamical probabilism. The ontological entists had already observed the self organization process character of probabilism can be the base for the scientific in the dynamics of fluids (e.g. dripping faucet model) or interpretation of self-organization and ordering principles elsewhere, verifying the Feigenbaum [112] mathemati- just as Prigogine [34] had imagined, following the Heisen- cal scenarios to complexity included in nonlinear maps berg’s concept. From this point of view, we could say that or Ordinary Differential Equations-Partial Differential contemporary physical theory returns to the Aristotle’s Equations [113]. However, although there was such a wide potentiality point of view, as Aristotelianism’s potentiality development of complexity science, scientists till now theory includes the Newton’s and Democritus’ mechanis- prefers to follow the classical theory, namely that macro- tical determinism only as one component among others cosmos is just the result of fundamental laws which can in the organism-like behaviour of Nature [115]. Modern be traced only at the microscopical level. Therefore, while evolution of physical theory, as it was described previ- the classical reductionistic theory considers the chaotic ously, is highlighted in Tsallis q-generalization of the Boltz- and the self-organization macroscopic processes as the mann-Gibbs (B-G) statistics which includes the classical result of the fundamental Lagrangian or the fundamental (Gaussian) statistics, as the q = 1 limit of thermodynami- Hamiltonian of nature, there is an ongoing different per- cal equilibrium. Far from equilibrium, the statistics of the ception. Namely, that macroscopic chaos and complexity dynamics follows the q-Gaussian generalization of the B-G cannot be explained by the hypothetical reductionistic statistics or other more generalized statistics. At the same point of view based at the microscopic simplicity, but they time, Tsallis q-extension of statistics is related to the fractal are present also in the microscopic level of physical reality. generalization of dynamics. However, for complex systems, Therefore, scientists like Nelson [114], Hooft [5], Parisi their holistic behaviour does not easily permit such a simpli- [37], Beck [3] and others, used the self-organization concept fication and division of dynamics and statistics. The Tsallis process of the complexity science for the explanation of extension of statistics and the fractal extension of dynam- the microscopic “simplicity”, introducing theories like sto- ics as strange kinetics are two faces of the same complex chastic quantum field theory or chaotic field theory. This and holistic (non-reductionist) reality. Moreover, the Tsallis new perception started to appear already through the Wil- statistical theory including the Tsallis extension of entropy, son’s theories of renormalization, which showed the mul- known as q-entropy principle, [2] the fractional generaliza- tiscale cooperation in the physical reality [19]. At the same tion of dynamics [7, 24] as well as the scale relativity exten- time, the multi-scale dynamical cooperation produces sion of the Ainstein Relativity theory, parallely to the scale the self-similar and the fractal character of nature allow- extension of relativity theory [12, 13] are the cornerstones of ing the application of renormalization methods at non- modern physical theory related to non-linearity and non- equilibrium critical states of physical systems. This leads integrability as well as to the non-equilibrium ordering and to the utilization of fractal geometry into the extended the self organization principle of nature. unification of physical theories, since the fractal geom- In the following, we summarized and discussed the etry includes properties of stochastic multiscale and self- highlights of complexity theory as the road of the global similar character. Scientists like Ord [76], El Naschie [27], unification of the physical theory from microscopic to Nottale [13] and others, introduced the idea of fractality macroscopic level of physical reality. into the geometry of space-time itself negating the notion of differentiability of physical variables. Also the fractional geometry is connected to non-commutative geometry since 7.1 The Thermodynamics point of view for fractional objects the principle of self similarity negates the Eycleidian notion of the simple geometrical point (as Complexity theory becomes effective when dynamical that which has no-parts), just like it negates the idea of dif- system exists far from equilibrium, where the entropy ferentiability and determinism. follows the equation: Therefore, the fractional geometry of space-time leads 2 SS=+δδSS+ (181) to the fractal extension of dynamics exploiting the fractal 0 172 G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one Far from equilibrium, according to Tsallis theory, the The Langevin equation (184) is equivalent to a func- Boltzmann-Gibbs entropy: tional Fokker-Planck equation: SP=−κκlnPP=− ()xPln ()xdx (182) ∂P δδΗδ BG ∑ ii ∫ =+PD()P (185) ∂t δφδφ δφ is the limit of a more general and non-extensive entropy. The non-extensive Tsallis entropy Sq given by the equation: where P[(ϕ xt, ), t] is the probability density functional [21]. 1 − Pq 1[− Px()]q dx S ==∑ i ∫ (183) The probability density functional can be calculated q qq−−11 by the method of math integral formalism. According to Chang [102], for nonlinear stochastic systems near critical- For q = 1 the new and non-extensive statistical mechanics ity the correlations among the fluctuations of the random is transformed to the known Boltzmann-Gibbs statistical dynamical fields are extremely long-ranged and there mechanics [2]. exist many correlations states. The far from equilibrium Every thermodynamical system is also a dynamical generalization of the renormalization group transforma- system with internal dynamical states and this point of tion includes the existence of fixed points (singular point) view exists as the propabilistic character of the internal for the Lagrangian scaling flow on the parameter stochas- dynamics. Thus, the entropy corresponds to the informa- tic Lagrangian affine space. As the stochastic nonlinear- tion that can be obtained by knowing the internal micro- complex system is perturbed from a particular fixed point scopic state of the system. (meta-equilibrium state) it may be attached to another Near equilibrium δS = 0 and δ2S is a Lyapunov func- critical state revealing symmetry breaking and phase tran- tion of positive time derivative. The time derivative corre- sition processes. sponds to entropy productions. Anomalous diffusion and non-Gaussian distribu- However far from equilibrium the time derivative of tions, low dimensional chaos, self organized critically, δ2S is may be negative corresponding to decreasing of spatiotemporal chaos, spatiotemporal patterns, directed entropy and development of order [2]. Significant con- percolation processes, scaling and other macroscopic cepts of Thermodynamics developed for the near equi- complex phenomena can be described by the application librium phenomena can be used for the description of far of the generalized renormalization group theory. In the from equilibrium phenomena. Such concepts are: thermo- above description the Langevin equations can be of more dynamical potentials, phase-transition, order parameters, general type corresponding to a general type of master nucleation phenomena. Also the Ginzburg-Landau theory equations and discrete spatio-temporal stochastic or can be extended for the description of the far from equilib- deterministic chaotic processes. According to the relation: rium phase transition processes and pattern formation in ∂〉||PL=〉P (186) many different dynamical systems [31]. tt t where |Pt〉 is the probability distribution vector and L the so-called Liouville operator which generates the temporal 7.2 The dynamical point of view evolution [116]. It is a truth that from the microscopic to the macroscopic level, the modern physical theory includes Complex systems can be at the same time thermodynami- non-linear dynamical processes in finite-dimensional or cal and dynamical systems. The far from equilibrium infinite-dimensional state spaces. The dynamical pro- dynamics can be described by the generalized free energy cesses can be of different and independent form at every H function which depends generally upon the macro- level of dynamical description of nature. Superstrings, scopic field ϕ(,xt ) (order parameter field). The gen- elementary particles or other physical forms and patterns eralized free energy function can be estimated by using observed at the macroscopic or cosmological level can be generalized Hamiltonians and Lagrangian, while the produced as solutions of non-linear mathematical equa- dynamics is given by a Langevin type equation: tions including Hamiltonians, Lagrangians, and other mathematical forms corresponding to continuing or dis- ∂ϕδ(,xt )(H ϕ) crete fields, quantum or classical. =− + (184) Df The great dream of theoretical scientists is to develop ∂txδϕ(, t) some kind of organizing physical principles valid at where f is stochastic noise. all levels. The strong simultaneous presence of chaos G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one 173 and fractal self-similarity at the macroscopic and at the and non-differentiable paths with fractal dimension D = 2. quantum level indicates the possibility for the existence of The fractal character of space-time indicates that the such global organizing principles. Some significant con- space-time variables can be described as depending also cepts in this direction are described in the following. on resolutions Φ = ϕ(x, ε). Nottale extended the principle of Relativity theory to scale Relativity, which the laws of nature should apply to all reference systems whatever 7.3 Fractal theory space and renormalization their state if scale. This is a scale covariance principle in group theory the spirit of renormalization group theory corresponding to the typical form: According to Nottale [13] theory space can be endowed ∂Φε(,x ) with certain geometrical symmetries based essentially on = βϕ() (188) ∂lnε the renormalization group (RG) transformations. Fractal lattices, known already in condensed matter physics, are In this theory the probabilistic character of quantum used to construct classes of new theory spaces. Theory theory is derived naturally by the fractal nature of space- space lattices are defined recursively by replacing each time while the microscopic quantum laws and the macro- site with a new simplex of S sites, while this process is scopic stochastic processes are the different manifestation iterated k times. of the same physical law under the scale transformation In the limit K→ ∞ this lattice describes a continuum in the fractal space-time. This led us to consider the mac- theory whose properties are determined be certain scaling roscopical chaos and complexity in natural systems as the laws of critical systems. At finite critical temperature the manifestation of the large scale fractal nature of space- Feynman path integral is invariant under the RG transfor- time. From this point of view, the Feynman path integral mations. Wilson’s approach is based on an effective action formulation of physical theory in a fractal space-time is the FtAA(/ϕϕ)~log(Zt/) unifying tool of the microscopic and macroscopic complex- Z (/tDϕϕ)e= xp tOi ()ϕ (187) ity. In this way, the path probability density, the correlation Ai∫ ∑ Α ( i ) function and the generating functional are the common tools in a unified point of view of microscopic and macro- where ZA is the partition function and the integration scopic stochastic processes according to the relation: extends over a functional space of fields. N δδ According to Morozov and Niemi [117], a RG flow Gx(, xx, ..., ; JJ)l= … ogZ[ ] (189) 12 n δδJx() Jx() could indeed tend towards a nontrivial attractor with 1 n even a fractal structure corresponding to a chaotic flow where, Z = ∫Dϕe−H(J) is the partition function of the system in the space of couplings. This could lead to a Big Mess G and GN the correlation functions. scenario in applications to multiphase systems, from Classical macroscopic distributed far from equilib- spin-glasses and neural networks to fundamentals string rium systems and microscopic quantum field systems are theories. The concept of the fractal theory space can be connected through stochastic and chaotic quantization extended in a general unifying scheme to include Lan- theory. According to the Parisi and Wu [124] approach the gevin processes or directed percolation processes corres classical field Φ described by an action S[ϕ] can be quan- ponding to macroscopic far from equilibrium processes tized by means of a stochastic Langevin equation: [118–120]. ∂Φδ(,xt )[S ϕ] =− ⋅+(,xt )(fx, t) (190) ∂t δϕ 7.4 Scale relativity theory where f denotes spatio-temporal Gaussian white noise. The classical stochastic process corresponds to a According to Nottale [121–123] the Scale Relativity theory non-equilibrium D + 1 dimensional space-time system extends the Einstein Relativity theory to scale transfor- while its corresponding Fokker-Planck distributed equa- mations of resolution [25, 48]. In this theory the axiom of tion at the equilibrium state is identified with the theory space-time differentiability is given up and the space-time of the quantum field Φ in the D-dimensional space-time. corresponds to a fractal set. This is in accordance with Fey- The classical stochastic dynamics in the D + 1 dimen- nman’s path integral approach to quantum mechanics, sional space-time is described by the correlators 〈Φ(x1, which at small length and time scales reveals continuous t1), Φ(x2, t2), …., Φ(xn, tn)〉, where Φ(x, t) are the solutions 174 G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one of equation (190). The classical stochastic correlators in included in the general form of Feynman path integral for- the limit t→ ∞ reproduce the Feynman graphs of the cor- mulation of the probability amplitudes of dynamics: responding D dimensional quantum theory. Φ 2 ΦΦ, tt, ≈NDΦeiS/ , (191) On the other hand, according to Beck [3, 125] there 22 11 ∫Φ 1 are deterministic chaotic dynamical systems that can generate the Langevin dynamics of equation (184). This where Φ is the dynamical state and S the action functional approach corresponds to the chaotic quantization theory of the dynamics. as it simulates quantum fields by chaotic dynamics. More- The unification of micro and macro dynamics passes over, Beck introduced a class of coupled map lattices, by through the fractal theory, and the scale invariance of using coupled third – order Tschebyscheff polynomials, Feynman amplitudes. The Feynman principle must be which can simulate quantum field theories. The stochas- related somehow to the general probabilistic Liouville tic and chaotic quantization theories reveal the mathe- dynamical theory. As we show in the following, the prob- matical similarity of the classical and quantum dynamics abilistic character of modern theory reveals the funda- but more than this it indicates the deep unity of physical mental character of becoming or process in the physical theory at the microscopic and the macroscopic level. This theory. According to the radical conjecture of Prigogine point of view can be supported by modern theoretical con- becoming is prior to being in a fundamental way includ- structions, which can produce the quantum theory as the ing even the physical laws. According to Kovacs [51] many manifestation of a subquantum chaotic dynamics. of the pattern formation processes observed in systems Hooft [5, 6] conjectured that quantum mechanics can of quite different genesis exhibit astonishing analogies logically arise as the low energy limit of a microscopically independently of the composition of the system and of deterministic but dissipative theory. Biro et al. [4] showed the interactions between individual components. As that quantum field theory can emerge in the infrared limit the formation of structures is a geometrical event, the of a high-dimensional quantum field theory as Yang- Mills observed qualitative analogies indicate common topolog- fields can quantize themselves. ical origin. From this point of view, physics is geometry and topology. In this direction bifurcation theory and symmetry 7.5 Becoming before being breaking processes are the common base of the far from equilibrium dynamics in micro and macro cosmos. In Quantum theory and Relativity theory (special and the bifurcation theory, bifurcation phenomena include general) introduced the notion of becoming in the heart topological equivalence while the bifurcations are largely of being showing that being is the production of becom- determined by geometrical constraints as the dimension ing. Complexity theory is a theory of being as Becoming and shape or inherent symmetries. The structural stabil- in its essence. From this point of view we believe that ity of physical patterns at every level of physical reality is complexity theory, as the essential theory of becoming inherent in the fundamental law (laws) of physics derived and pattern formation, shows the route for a global uni- by geometrical invariance principles such as space-time fication of physical theory. Particularly, the quantum symmetries or gauge symmetries. The link between geom- states include becoming and potentiality in an objective etry and physics is due to topological invariance princi- way. Because of this the physical magnitudes are related ples which cause inherent stability properties. Structural probabilistically with the quantum states. Entropy and stability proves includes the interaction and correlation complexity are inherent in the quantum states. Simi- physical phenomenology of physical systems. On the larly, in relativity material particles are condensed other hand, self-organization is a far from equilibrium energy and space and time are continuously. In this thermodynamical process revealing entropy decrease and point of view the great Hellenic philosophers Heracli- order development according to the equation: tus, Plato and Aristotle meet one another. For Heraclitus dS=+dS dS (192) all the cosmos is one. For Aristotle matter is the poten- ei tiality of the being and for Plato the essence of things is where deS is the reversible entropy transfer and diS is the the form. irreversible producedentropy within the system. According to the modern theoretical concepts, Thermodynamics is coupled to dynamic through the the essence of the physical theory is the mathematical general Liouville equation: structure-equations with solutions corresponding to the ∂=ft() Lf ()t (193) observed phenomena. All this state of knowledge can be t G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one 175 where f(t) is the infinite dimensional distribution vector theory of El Naschie introduces randomness and scale f(t) = {f0, f1(x1), f2(x1, x2), ….}, including the Bogolinbov- invariance to the very concept of space-time geometry. Born-Green-Kirkwood-Yvon (BBGKY) [18] correlations Space-time is an infinite-dimensional, hierarchical and hierarchy. random geometrical manifold with infinite numbers of The dynamics of correlations corresponds to a func- equivalent paths. In this space, any so-called point will tional Hilbert space H∞. In order the dynamical theory to always reveal a structure on a close examination. Canto- be in agreement with the thermodynamical theory, the rian space-time E∞ of the El Naschie theory is a form of non- Liouville equation (193) must be solved in an extended commutative geometry and it can be constructed using functional space called rigged Hilbert space or Gelfand an infinite number hierarchy of Cantor fractal sets mixed space according to Petrosky and Prigogine [126, 127]. For together in all possible forms of union and intersections. the integrable classical or quantum dynamical systems Let us remember that in the Euclidean theory of space, the dynamical theory does not lead to thermodynamics. the curved surface is included as a subset of Euclidean Chaotic dynamical systems are non-integrable Poincare points. In the Riemannian theory the Euclidean space is a systems. For chaotic or non-integrable dynamical systems, local characteristic of the general Riemannian manifold. the Liouville equation can be solved in an extended func- For Einstein, the space-time Riemannian manifold is the tional space which includes the thermodynamical states primitive or fundamental physical reality, which includes and the irreversible Markov stochastic processes under- every other physical form as a geometrical characteristic. lying to the far from equilibrium thermodynamics. The Ofcourse the dynamical cosmology and the expanded extended physical space in a Rigged Hilbert space or space included in the equations of general relativity of Gelfald space, in which the infinite dimensional correla- Einstein was for him an undesirable trouble. For modern tion hierarchy and the corresponding extended Liouville complexity theory the Mandelbrot fractal theory, the equation must include as general solutions the observed Euclidean or Riemannian manifolds are the smoothed physical forms or states in the microscopic or the macro- out character of a primitive fractal manifold with scale scopic level. Also, the physical states in the Rigged Hilbert dependent dimensions and other topological invariant space correspond to possibilities or potentialities. In this properties. For El Naschie and other modern scientists way, the extended in the Rigged Hilbert space Liouville every physical form in the universe is a scale depend- equation can be transformed to a Fokker-Planck equa- ent geometric characteristic of the infinite dimensional tion including dynamics and thermodynamics simul- fractal Cantor’s manifold. In El Naschie’s theory, space- taneously with breaking of time reversal symmetry and time is an infinite dimensional fractal that happens to entropy production. According to Prigogine the flow of have D = 4 as the expectation value for the topological correlations in the extended Riggen Hilbert space creates dimension. The topological dimension 3 + 1 means that in the observable physical states and the observed dynam- our low energy resolution the world appears to us as it ics. In this way, the process as becoming is before being were four-dimensional. According to Iovane et al. [128], and correlations as potentialities are the fundamental Iovane [129], Ahmed [130], and Agop et al. [44] the differ- physical reality. ence between micro and macro physics depends only on the resolution in which the observer looks at the world. Nature shows us structures with scaling rules where hier- 7.6 Chaos, fractals and unification of archy clustering properties are revealed from cosmologi- dynamics to thermodynamics cal to microscopical objects. The universe is self-similar from the quantum to the cosmological level according to In this chapter, we conjecture that the above mathemati- the scale invariance law: cal construction must be related somehow to the infi- h a nite dimensional fractal space included in the E-infinite R()NN= (194) Mc theory of El Naschie [57]. Von Neumann understood that the quantum dynamics includes non-commutative geom- where R is the radius of the astrophysical structures, M etry of the quantum space which has no points at all. This is the total Mass of the self-gravitating system, h is the means that the physical reality is not created by geometri- Planck constant, N is the number of nucleons, c the speed cal points or by simple material points without internal of light and a = 3/2. structure. At every level physical reality is complex includ- For N = 1, R is equal to the Compton wavelength. There ing forms, structures and patterns. Non-commutative is no breaking point between microscopic and large scale geometry corresponds also to fuzzy spaces. The E-infinite universe. This is in agreement with El Naschie’s E-infinite 176 G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one theory of cantorian space-time, the Golden Mean and the characteristic of sensitivity to initial conditions (chaotic Fibonacci numbers, as well as to the general theory of the instability) or the older Heisenberg’s uncertainty princi- stochastic self-similar processes in a fractal space. Also it ple do not cause any trouble to the theoretical or dogmatic appears that the universe has a memory of its quantum determinism rooted to the Cartesian metaphysic realism. origin as was suggested by Penrose. Consequently the uni- After all it is significant to search for a unifying route verse with its structures as all scales from the quantum through complexity of the physical theory at the micro- level to the organic cell and human to super clusters or scopical and macroscopical or cosmological level even galaxy and the cosmological level is a self-similar complex of it is too early for such a dream. In the following, we system and includes self-similar processes at every level. present significant evidence for such a route. In this theory the fractal space-time includes in a funda- mental way the probabilistic character of the physical dynamics as well as the symmetry braking of time reversal. 7.8 Complexity as a kind of macroscopic quanticity 7.7 Complexity as a new theory The central point of complexity theory is the possibility for a physical system, which includes a great number of Complexity is considered as a new and independent parts or elements, to develop internal long-range corre- physical theory which was developed after the Relativ- lations leading to macroscopic ordering and coherent ity Theory and Quantum mechanics. It is related to far patterns. These long-range correlations can also appear from equilibrium dynamics and concerns the creation at the quantum level. In particular, according to the and destruction of spatiotemporal patterns, forms and general entanglement character of the quantum theory, structures. According to Prigogine [34], Nicolis [49, 131] the quantum mechanical states of a system with two or and others, complexity theory corresponds to the flow more parts cannot be expressed as the conjunction of and development of space-time correlations instead of quantum states of the separate parts. This situation gen- the fundamental local interactions. Moreover, according erally reflects the existence of non-local interactions and to Sornette [132], systems with a large number of mutually quantum correlations while the measurements bearing interacting parts and open to their environment can self- on either part correspond to random variables which are organize their internal structure and their dynamics with not independent and can be correlated independently of novel and sometimes surprising macroscopic emergent the spatial distance of the parts [133]. This means that properties. These general characteristics make the com- the quantum density operator cannot be factored while plexity theory, a fundamentally probabilistic theory of the the quantum state corresponds to the global and undi- non-equilibrium dynamics. vided system. The macroscopic manifestation of the Until now complexity concerns mainly macroscopic quantum possibility for the development of long-range physical systems. The traditional point of view is to explain correlations is the spontaneous appearance of ordered macroscopic complex dynamics by the basic microscopi- behavior in a macroscopic system examples of which are cal physical theory which is a fundamental Quantum phenomena like superfluidity and superconductivity or Field Theory (QFT) with all its modern manifestations lasers [134]. like supersymmetry and gauge symmetries or super- These quantum phenomena display coherent behav- string theory (ST) including the M-theory (superstring) or ior involving the collective cooperation of a huge number D-branes theories [12]. Also, such a type of fundamental of particles or simple elements and a vast number of field theory belongs to the general class of time reversible degrees of freedom. They correspond also to equilibrium and deterministic physical theory. However, the quantum phase transition processes which constitute the meeting probabilism is not included inherently in the fundamental point of quantum theory and complexity. Here the devel- theory but as an external interpretation of its mathemati- opment of quantum long range correlations leads to a cal formulation. It concerns the observer and the action macroscopic phase transition process and macroscopic measurement (observation), but not the dynamics of the ordering. It is not out of logic or physical reality to extend objective physical system which corresponds to an objec- the unifying possibility of quantum process for develop- tive and deterministic process in a general state space. ment of long-range correlations or the quantum entangle- At the macroscopic level, we are obliged as observers ment character to a macroscopic self-organizing factor or as knowing subjects to use statistical or probabilistic causing also the far-from equilibrium symmetry breaking methods. From this point of view, chaos with its famous and macroscopic pattern formation. From this point of G.P. Pavlos: Complexity theory, time series analysis and Tsallis q-entropy principle part one 177 view we can characterize complexity as a kind of a macro- which can generate Langevin dynamics in an appropriate scopic quanticity [19, 135, 136]. scaling limit. In this approach, quantum field theories can be simulated by chaotic dynamics. Non-extensive statis- tics [2], fractal string and branes, fractal statistics, fractons 7.9 Quantum theory as a kind of microscopic and anyons particles as well as chaotic M(atrix) theory chaoticity or complexity indicate the establishment of chaos and complexity at the microscopic and the quantum level [12]. In this direction, Bohm and Hiley [137] imagined that the quantum theory Gerard Hooft raised the conjecture that quantum theory must be the manifestation of subquantum complex can be derived as the low-energy limit of a microscopically dynamics. During the last years, we observe the produc- deterministic but dissipative theory [6]. According to this tive onset or the impetus invasion of chaos and complex- concept classical Perron-Frobenius operators or deter- ity from macroscopic to the microscopic quantum level ministic automata can produce quantum states in Hilbert [5]. In the following we present some novel concepts in spaces as well as the Schrödinger equation [141–144]. this direction: Analytical continuation and fractal space-time can convert an ordinary diffusion equation into a Schrödinger 7.10 The road of complexity for the physical equation and a telegraph equation into a Dirac’s equation. theory unification From this point of view analytical continuation is a short of cut quantization. In this way, the Feynman rules and diagrams become Positive Lyapunov exponents of non-abelian gauge common tool from the estimation of probabilistic processes fields reveal the significance of chaos for the quantum at the microscopic quantum level or the macroscopic level field theory [4]. of continuous media as they are being described by the Coupled map lattices with spatiotemporal chaotic Ginzburg-Landau model [118, 119]. In this direction, we profile can be used to simulate quantum field theories in could imagine Feynman rules and renormalization group an appropriate scaling limit. theory as the universal characteristics of probabilistic pro- Kaneko coupled map lattices including chaotic strings cesses at the microscopic and the macroscopic level. The provide the background for the Parisi-Wu stochastic quan- renormalization group equations have many common fea- tization of ordinary string and quantum field theories [3, tures with non-linear dynamical systems, so that besides 125]. Chaotic strings can be used also to provide a theo- the existence of isolated fixed points, the coupling in a retical argument why certain standard model parameters renormalizable field theory may flow also towards more are realized in nature reproducing numerical values of the general even fractal attractors. This could lead to Big Mess electroweak and strong coupling constants masses of the scenarios in application to multiphase systems, from spin- known quarks and leptons neutrino, W boson and Higgs glasses and neural networks to fundamental string theory mass. [117]. In this direction Cristopher Hill [145] introduced Renormalization group (RG) flows on the superstring the fractal theory space where the key idea is that the world sheet becomes chaotic and leads to non-Mark- Feynman path integral is invariant under a sequence of ovian Fokker-Planck equation with solutions describ- renormalization group transformations that map the kth ing the transition from order to chaos and revealing the lattice into the k – 1 lattice. In the continuum limit these Feigenbaum universal constant [138]. The appearance models produced quantum field theories in fractal dimen- of this constant reveals the scaling of space-time curva- sions D = 4 + ε. These theories are connected to the scaling tures at the fixed points of the RG flow, which becomes behavior of fractal strings (branes), while the couplings chaotic near singularities where the curvature is very oscillate on a limit cycle. Moreover, the concept of fractal large [117, 139]. space-time can be used for the foundation of an extended The Parisi and Wu [124] stochastic quantization theory Einstein’s Relativity Principle unifying the micro and relates the quantum field theory in D-dimensions to a macro levels [121–123, 136]. classical Langevin equation in D + 1-dimensions where In this direction, Ord [76] showed that fractal trajecto- the Parisi-Wu fictitious time plays the role of an extra ries in space with Hansdorff dimension D = 2 exhibit both dimension. In this picture there exists a short of classical an uncertainty principle and a De Broglie wave – particle stochasticity and quantum theory duality [140]. The sto- duality relations. Furthermore, L. Nottale introduced the chastic quantization can be transform to chaotic quanti- principle of Scaled Relativity according to which the laws zation, similar to chaotic deterministic dynamical systems of physics are scale invariant. 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