Fractal geometry, information growth and nonextensive thermodynamics

Details Fractal geometry, information growth and nonextensive thermodynamics Fractal geometry, information growth and nonextensive thermodynamics

2003-11-07

arxiv.org/abs/cond-mat/0311175v3

Physica A, 340(2004)117

Physics

Condensed Matter

Statistical Mechanics

Final version, 10 pages, no figures, Invited talk at the international conference NEXT2003, 21-28 september 2003, Villasimius

Scientific paper

10.1016/j.physa.2004.03.086

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 $\sum_ip_i^q=1$ is applied throughout the paper, where $q$ 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 $\sum_ip_i-\sum_ip_i^q$ 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 $q$. It is argued that, within this thermodynamics, the Stefan-Boltzmann law of blackbody radiation can be preserved.

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