Measuring information growth in fractal phase space

We look at chaotic systems evolving in fractal phase space. The entropy change in time due to the fractal geometry is assimilated to the information growth through the scale refinement. Due to the incompleteness, at any scale, of the information calculation in fractal support, the incomplete normalization $\sum_ip_i^q=1$ is applied throughout the paper. It is shown that the information growth is nonadditive and is proportional to the trace-form $\sum_ip_i-\sum_ip_i^q$ so that it can be connected to several nonadditive entropies. This information growth can be extremized to give, for non-equilibrium systems, power law distributions of evolving stationary state which may be called ``maximum entropic evolution''.Comment: 10 pages, 1 eps figure, TeX. Chaos, Solitons & Fractals (2004), in pres

On the generalized entropy pseudoadditivity for complex systems

We show that Abe's general pseudoadditivity for entropy prescribed by thermal equilibrium in nonextensive systems holds not only for entropy, but also for energy. The application of this general pseudoadditivity to Tsallis entropy tells us that the factorization of the probability of a composite system into product of the probabilities of the subsystems is just a consequence of the existence of thermal equilibrium and not due to the independence of the subsystems.Comment: 8 pages, no figure, RevTe

The heuristic power of the non integer differential operator in physics: From chaos to emergence, auto-organisations and holistic rules

© 2015 by Nova Science Publishers, Inc. All rights reserved. The use of fractional differential equations raises a paradox due to the non-respect of the space time noetherian axioms. In environments characterized by scaling laws (hyperbolicgeometry associated with fractional diff integral) energy is no more the invariant of the dynamics. Nevertheless the experimental action requiringthe use of energy, the relevant representation of the fractional process, must be extended. The extension is carried out usingthe canonical transfer functions in Fourier space and explained by their links with the Riemann zeta function. Category theory informs the extension problem.Ultimately the extension can be expressed by asimple change of referential. It leads to embed the time in the complex space. This change unveils the presenceof a time singularity at infinity.The paradox of the energy in the fractality illuminates the heuristic power of thefractional differential equations. In this mathematical frame, it is shown that the dual requirement of the frequency response to differential equations of non-integer order and of the notherian constraints make gushing out a source of negentropique likely to formalize the emergence of macroscopic correlations into self-organized structures as well as holistic rules of behaviour

Electromagnetic field of fractal distribution of charged particles

Electric and magnetic fields of fractal distribution of charged particles are considered. The fractional integrals are used to describe fractal distribution. The fractional integrals are considered as approximations of integrals on fractals. Using the fractional generalization of integral Maxwell equation, the simple examples of the fields of homogeneous fractal distribution are considered. The electric dipole and quadrupole moments for fractal distribution are derived.Comment: RevTex, 21 pages, 2 picture

Maximum Path Information and Fokker-Planck Equation

We present in this paper a rigorous method to derive the nonlinear Fokker-Planck (FP) equation of anomalous diffusion directly from a generalization of the principle of least action of Maupertuis proposed by Wang for smooth or quasi-smooth irregular dynamics evolving in Markovian process. The FP equation obtained may take two different but equivalent forms. It was also found that the diffusion constant may depend on both q (the index of Tsallis entropy) and the time t.Comment: 7 page