A two-parameter generalization of Boltzmann-Gibbs-Shannon entropy based on natural logarithm is introduced. The generalization of the Shannon-Khinchin axioms corresponding to the two-parameter entropy is proposed and verified. We present the relative entropy, Jensen-Shannon divergence measure and check their properties. The Fisher information measure, the relative Fisher information, and the Jensen-Fisher information corresponding to this entropy are also derived. Also the Lesche stability and the thermodynamic stability conditions are verified. We propose a generalization of a complexity measure and apply it to a two-level system and a system obeying exponential distribution. Using different distance measures we define the statistical complexity and analyze it for two-level and five-level system. 1. Introduction Entropy is a very important quantity and plays a key role in many aspects of statistical mechanics and information theory. The most widely used form of entropy was given by Boltzmann and Gibbs from the statistical mechanics point of view and by Shannon from information theory point of view. Later certain other generalized measures of entropy like the Rényi entropy [1] and the Sharma-Mittal-Taneja entropy [2, 3] were introduced and their information theoretic aspects were investigated. Recently in [4] a new expression for the entropy was proposed as a generalization of the Boltzmann-Gibbs entropy and the necessary properties like concavity, Lesche stability, and thermodynamic stability were verified. This entropy has been applied to a wide variety of physical systems, in particular to long-range interacting systems [5, 6] and non-Markovian systems [7]. Most of the generalized entropies introduced so far were constructed using a deformed logarithm. But two generalized entropies, one called the fractal entropy and the other known as fractional entropy, were proposed using the natural logarithm. The fractal entropy which was introduced in [8] attempts to describe complex systems which exhibit fractal or chaotic phase space. Similarly, the fractional entropy was put forward in [9] and later applied to study anomalous diffusion [10]. Merging these two entropies in our present work we propose a fractional entropy in a fractal phase space. Thus, there are two parameters, one characterizing the fractional nature of the entropy and the other describing the fractal dimension of the phase space. Thus, the functional form of the entropy depends on the natural logarithm. We give the generalized Shannon-Khinchin axioms corresponding to this two-parameter
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