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Proceedings, 5th Chaotic Modeling and Simulation International
Conference, 12 – 15 June 2012, Athens Greece
“simplicity”, introducing theories like stochastic quantum field theory or
chaotic field theory. This new perception started to appear already through the
Wilsonian theories of renormalization which showed the multiscale cooperation
of the physical reality [25]. At the same time, the multiscale cooperativity goes
with the self similarity characters of nature that allows the renormalization
process. This leads to the utilization of fractal geometry into the unification of
physical theories, as the fractal geometries are characterized by the scaling
property which includes the multiscale and self similar character. Scientists like
Ord [26], El Naschie [4], Nottale [6] and others, will introduce the idea of
fractal geometry into the geometry of space-time, negating the notion of
differentiability of physical variables. The fractal geometry is connected to non-
commutative geometry since at fractal objects the principle of self similarity
negates the notion of the simple geometrical point just like the idea of
differentiability. Therefore, the fractal geometry of space-time is leading to the
fractal extension of dynamics exploiting the fractal calculus (fractal integrals-
fractal derivatives) [27]. Also, the fractal structure of space-time has
intrinsically a stochastic character since a presupposition for determinism is
differentiability [6, 14]. Thus, in this way, statistics are unified with dynamics
automatically, while the notion of probability obtains a physical substance,
characterized as dynamical probabilism. The ontological character of
probabilism can be the base for the scientific interpretation of self-organization
and ordering principles just as Prigogine [1] had imagined, following
Heisenberg’s concept. From this point of view, we could say that contemporary
physical theory returns to the Aristotetiles point of view as Aristotelianism
includes the Newtonian and Democritian mechanistical determinism as one
component of the organism like behavior of Nature [28].
Modern evolution of physical theory as it was described previously is
highlighted in Tsallis q-statistics generalization of the Boltzmann-Gibbs
statistics which includes the classical (Gaussian) statistics, as the q=1 limit of
thermodynamical equilibrium. Far from equilibrium, the statistics of the
dynamics follows the q-Gaussian generalization of the B-G statistics or other
more generalized statistics. At the same time, Tsallis q-extension of statistics
can be produced by the fractal generalization of dynamics. The traditional
scientific point of view is the priority of dynamics over statistics. That is
dynamics creates statistics. However for complex system their holistic
behaviour does not permit easily such a simplification and division of dynamics
and statistics. Tsallis
q−
statistics and fractal or strange kinetics are two faces
of the same complex and holistic (non-reductionist) reality.
Moreover, the Tsallis statistical theory including the Tsallis extension of
entropy to what is known as q-entropy [29], the fractal generalization of
dynamics [6, 7] and the scale extension of relativity theory C [6, 7] are the
cornerstones of modern physical theory related with nonlinearity and non-
integrability as well as with the non-equilibrium ordering and self organization.
In the following, in section (2) we present the theoretical concepts of q-statistics
and fractal dynamics, while in section (3) we present indicative experimental
