Geometry, Information Growth and Thermodynamics of Some Nonextensive Systems

This is a study of the information evolution of complex systems through a 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 information calculation in fractal support at any scale, the incomplete normaliza-tion i p q i = 1 is applied throughout the paper. It is shown that the information growth is nonadditive and is proportional to the trace-form i p i − i p q i 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. This trace-form information can also be used for the study of the thermodynamics of nonadditive systems each having its own q. It is argued that, within this thermodynamics, the Stefan-Boltzmann law of blackbody radiation can be preserved.