On the Renyi entropy, Boltzmann Principle, Levy and power-law distributions and Renyi parameter

The Renyi entropy with a free Renyi parameter $q$ is the most justified form of information entropy, and the Tsallis entropy may be regarded as a linear approximation to the Renyi entropy when $q\simeq 1$. When $q\to 1$, both entropies go to the Boltzmann--Shannon entropy. The application of the principle of maximum of information entropy (MEP) to the Renyi entropy gives rise to the microcanonical (homogeneous) distribution for an isolated system. Whatever the value of the Renyi parameter $q$ is, in this case the Renyi entropy becomes the Boltzmann entropy $S_B=k_B\ln W$, that provides support for universality of the Boltzmann's principle of statistical mechanics. For a system being in contact with a heat bath, the application of MEP to the Renyi entropy gives rise to Levy distribution (or, $q$-distribution) accepted as one of the main results of the so-called nonextensive statistics. The same distribution is derived here for a small physical system experiencing temperature fluctuations. The long--range "tail" of the Levy distribution is the power--law (Zipf-Pareto) distribution with the exponent $s$ expressed via $q$. The exponent and free Renyi parameter $q$ can be uniquely determined with the use of a further extension of MEP. Then typical values of $s$ are found within the range $1.3\div 2$ and of $q$ within the range $0.25\div 0.5$, in dependence on parameters of stochastic systems.