On Non-Extensive Entropies: With Applications in Stochastic Dynamics and Information Theory

Abstract

Among the generalised measures of entropy, there is a special class of measures whose functional dependence dismisses all free parameters, but instead relies exclusively on probability. For this class, we will pay attention to the full-stable measures of entropy having a well defined thermodynamic limit, provided these attributes are necessary for physical observables to be recovered from entropy. To our knowledge, there are only two generalised entropies fulfilling these requirements. Then we investigate their basic mathematical aspects as well as their impact on physics, information and computer sciences. We will prove formally such entropies converge asymptotically to the Boltzmann-Gibbs measure, whereas they induce a generalised classification of entropies. We study the consequences these entropies convey in diffusion and transport phenomena, which leads us to derive master equations out of equilibrium. Interestingly, our master equations adopt a similar structure to some chemotaxis-aggregation models studied in biology. Further, given that entropy is at the interface between statistical mechanics and information theory, we propose a non-extensive information theory, where data compression and channel capacities are improved, in relation to Shannon's formulation, in a scenario of high probabilities. Finally, we bring this non-extensive information theory in its algorithmic counterpart to obtain generalisations to Kolmogorov's statistical complexity.