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Complexity Theory, Time Series Analysis and Tsallis Q-Entropy

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Complexity Theory, Time Series Analysis and Tsallis Q-Entropy 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
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