Generalized Non-extensive Statistical Distributions

We find the value of constants related to constraints in characterization of some known statistical distributions and then we proceed to use the idea behind maximum entropy principle to derive generalized version of this distributions using the Tsallis’ and Renyi information measures instead of the well-known Bolztmann-Gibbs-Shannon. These generalized distributions will depend on q ∈ R and in the limit q → 1 we obtain the “classical” ones. We found that apart from a constant, generalized versions of statistical distributions following Tsallis’ or Renyi are undistinguishable. Introduction The development of statistical mechanics and its application to a wide variety of physical phenomena take a leap forward with the works of Shannon [17] and Jaynes [8, 9]. Before that, the usual line of reasoning was to construct the theory based on the equations of motion, supplemented by additional hypotheses of ergodicity, metric transitivity, or equal a priori probabilities and the identification of entropy was made at the end by comparison of the resulting equations with the laws of thermodynamics. The point of view developed by Shannon and Jaynes took the concept of entropy to play a central role. In this modern approach, the fact that a probability distribution maximizes the entropy subject to certain constraints becomes essential to the justification of the use of that distribution for statistical inference. By freeing the theory from apparent dependence of physical hypotheses they made its principles and mathematical methods to become available for the treatment of many new physical problems. The constraints used to maximize the entropy are interpreted as the real unbiased knowledge about the system (as the mean energy, mean number of particles, etc) with the additional restriction of normalization of the probability distribution over the entire value range of the state variable. This is sometimes referred as the Maximum Entropy Principle (MEP)[8, 9].