Fractal geometry, information growth and nonextensive thermodynamics

Abstract

This is a study of the information evolution of complex systems by 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 state number counting at any scale on fractal support, the incomplete normalization i p q i = 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 traceform i p i − i p q i 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 lead to a nonextensive thermodynamics for heterogeneous systems which contain subsystems each having its own q. It is shown that, within this thermodynamics, the Stefan-Boltzmann law of blackbody radiation can be preserved.

Isotropic anomalous filtering in Diffusion-Weighted Magnetic Resonance Imaging

“…Another method used to analyze the images was the structural similarity index (SSIM) [19]. This index is widely used in image processing to estimate the visual quality of digital images.…”

Composition of Fractals

“…The main aim is to find the relationship between the dimensions of the component fractals and the dimension of the composite fractal, and to see whether or not this relationship is dependent on the mixing manner. This question has been raised several years ago within a statistical theory called incomplete statistics (IS) which has been proposed by physical consideration in order to make statistics in, amount others, fractal phase spaces of dynamic systems [8,9,10,11,12,13]. A problem in this framework was to find the characteristic parameter (associated to the fractal dimension, see section 5.2 below) of a composite system from the same parameters of the component systems with the condition of the thermodynamic equilibrium [13,15,16].…”

Incomplete nonextensive statistics and the zeroth law of thermodynamics

“…Introduction. Incomplete statistics (IS) proposed by Wang [1] and Tsallis' statistics [2,3] have been two important branches of nonextensive statistical mechanics [4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Recently, IS has been used to research the thermostatistic properties of a variety of physical systems with long-range interacting and/or long-duration memory and many significant results have been obtained [6][7][8][9][10][11][12][13][14][15][16].…”

“…Incomplete statistics (IS) proposed by Wang [1] and Tsallis' statistics [2,3] have been two important branches of nonextensive statistical mechanics [4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Recently, IS has been used to research the thermostatistic properties of a variety of physical systems with long-range interacting and/or long-duration memory and many significant results have been obtained [6][7][8][9][10][11][12][13][14][15][16]. For example, it has been found that for some chaotic systems evolving in fractal phase space [7,8], the entropy change in time due to the fractal geometry is assimilated to the information growth through the scale refinement; and that the generalized fermion distributions based on incomplete information hypothesis can be useful for describing correlated electron systems [9,10].…”

“…This T definition is different from original one of IS in Refs. [1,4,6,8,11,13,15] where T β is defined as the physical (measurable) temperature through the zeroth law. It shows that the Lagrange multiplier β in Refs.…”