By considering generalized logarithm and exponential functions used in nonextensive statistics, the four usual algebraic operators : addition, subtraction, product and division, are generalized. The properties of the generalized operators are investigated. Some standard properties are preserved, e.g., associativity, commutativity and existence of neutral elements. On the contrary, the distributivity law and the opposite element is no more universal within the generalized algebra.Comment: 11 pages, no figure, TeX. Reports on Mathematical Physics (2003), in pres

A path information is defined in connection with different possible paths of irregular dynamic systems moving in its phase space between two points. On the basis of the assumption that the paths are physically differentiated by their actions, we show that the maximum path information leads to a path probability distribution in exponentials of action. This means that the most probable paths are just the paths of least action. This distribution naturally leads to important laws of normal diffusion. A conclusion of this work is that, for probabilistic mechanics or irregular dynamics, the principle of maximization of path information is equivalent to the least action principle for regular dynamics.We also show that an average path information between the initial phase volume and the final phase volume can be related to the entropy change defined with natural invariant measure of dynamic system. Hence the principles of least action and maximum path information suggest the maximum entropy change. This result is used for some chaotic systems evolving in fractal phase space in order to derive their invariant measures.

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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.

We study the minimal thermodynamically consistent model for an adaptive machine that transfers particles from a higher chemical potential reservoir to a lower one. This model describes essentials of the inhomogeneous catalysis. It is supposed to function with the maximal current under uncertain chemical potentials: if they change, the machine tunes its own structure fitting it to the maximal current under new conditions. This adaptation is possible under two limitations. i) The degree of freedom that controls the machine's structure has to have a stored energy (described via a negative temperature). The origin of this result is traced back to the Le Chatelier principle. ii) The machine has to malfunction at a constant environment due to structural fluctuations, whose relative magnitude is controlled solely by the stored energy. We argue that several features of the adaptive machine are similar to those of living organisms (energy storage, aging).