49 research outputs found
A mathematical structure for the generalization of the conventional algebra
An abstract mathematical framework is presented in this paper as a
unification of several deformed or generalized algebra proposed recently in the
context of generalized statistical theories intended to treat certain complex
thermodynamic or statistical systems. It is shown that, from mathematical point
of view, any bijective function can be used in principle to formulate an
algebra in which the conventional algebraic rules are generalized
Fractal geometry, information growth and nonextensive thermodynamics
This is a study of the information evolution of complex systems by
geometrical consideration. 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
of the state number counting at any scale on fractal support, the incomplete
normalization is applied throughout the paper, where is the
fractal dimension divided by the dimension of the smooth Euclidean space in
which the fractal structure of the phase space is embedded. It is shown that
the information growth is nonadditive and is proportional to the trace-form
which can be connected to several nonadditive
entropies. This information growth can be extremized to give power law
distributions for these non-equilibrium systems. It can also be used for the
study of the thermodynamics derived from Tsallis entropy for nonadditive
systems which contain subsystems each having its own . It is argued that,
within this thermodynamics, the Stefan-Boltzmann law of blackbody radiation can
be preserved.Comment: Final version, 10 pages, no figures, Invited talk at the
international conference NEXT2003, 21-28 september 2003, Villasimius
(Cagliari), Ital
Incomplete normalization of probability on multifractals
This work is an extension of the incomplete probability theory from the
simple case of monofractals previously studied to the more general case of
multifractals which can occur in the phase space without equiprobable
partition.Comment: 10 pages, to appear in Physica A (2007