Renyi information

The three dimensions introduced above, Dp, Di and Dq, are actually three members of an (uncountably infinite) hierarchy of generalized fractal dimensions introduced by Heiitschel and Procaccia [hent83]. The hierarchy is defined by generalizing the information function /(e) (equation 4.88) used in defining Dj to the i/ -order Renyi information function, Io e) -... 

Renyi (78) generalized the concept of information entropy to measure different aspects of system homogeneity, and Alemanskin et al. (79), and Alemanskin and Manas-Zloczower (80) adopted the Renyi entropy for measuring mixing. Considering only statistically independent partition, Renyi determined that the information entropy can be replaced with the following single-variable function... 

The formalism of the statistical mechanics agrees with the requirements of the equilibrium thermodynamics if the thermodynamic potential, which contains all information about the physical system, in the thermodynamic limit is a homogeneous function of the first order with respect to the extensive variables of state of the system [14, 6-7]. It was proved that for the Tsallis and Boltzmann-Gibbs statistics [6, 7], the Renyi statistics [10], and the incomplete nonextensive statistics [12], this property of thermodynamic potential provides the zeroth law of thermodynamics, the principle of additivity, the Euler theorem, and the Gibbs-Duhem relation if the entropic index z is an extensive variable of state. The scaling properties of the entropic index z and its relation to the thermodynamic limit for the Tsallis statistics were first discussed in the papers [16,17],... 

Renyi, A., Some fundamental questions of information theory, MTA III Oszt. Kdzl., 10, 251, 1960 On measures of information and entropy, in Proceedings of the Fourth Berkeley Symposium on Mathematics, Statistics and Probability, Berkeley University Rress, Berkeley, CA, 1960, 547 Probability Theory, North Holland, Amsterdam, 1970. 

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