Extended Entropies and Disorder

Landsberg’s notion of disorder, entropy normalized to maximum entropy, was originally proposed for the Shannon information-theoretic entropy to overcome deficiencies of entropy as a measure of disorder due to extensivity. We generalize Landsberg’s concept to three classes of extended entropies: Rényi, Tsallis and Landsberg-Vedral. Three examples are treated, including one based on the logistic map and another for power law distributions. On the basis of the three examples it is demonstrated that all three classes of extended disorder are required to fully characterize the corresponding properties of a system. We also show an intimate connection between the Rényi disorders and the spectrum of dimensions known as multifractals. It follows that again three spectra – Rényi, Tsallis and Landsberg-Vedral are required to completely determine a dimension spectrum.