Bridging the gap between ecological diversity indices and measures of biodiversity with Shannon's entropy: Comment to Izsák and Papp

Most ecological diversity indices summarize the information about the relative abundances of community species without reflecting taxonomic differences between species. Nevertheless, in the environmental conservation practice, data on species abundances are generally unknown. In such cases, to summarize the conservation value of a given site, so-called ‘biological diversity’ measures need to be used. Most of these measures are based on taxonomic relations among species and ignore species relative abundances. In a recent paper, Izsa´k and Papp suggest that the quadratic entropy index (Q) is the only diversity index used to date in the ecological practice that incorporates both speciesrelative abundances and a measure of the pairwise taxonomic differences between species in the analyzed data set. I show here that a number of traditional ecological diversity measures can be generalized to take into account a taxonomic weighting factor. Since these new indices violate part of the mathematical properties that an index should meet to be termed an ecological diversity index, I defined this new family of indices ‘weak diversity indices’

Phase transitions and quantum effects in anharmonic crystals

The most important recent results in the theory of phase transitions and quantum effects in quantum anharmonic crystals are presented and discussed. In particular, necessary and sufficient conditions for a phase transition to occur at some temperature are given in the form of simple inequalities involving the interaction strength and the parameters describing a single oscillator. The main characteristic feature of the theory is that both mentioned phenomena are described in one and the same setting, in which thermodynamic phases of the model appear as probability measures on path spaces. Then the possibility of a phase transition to occur is related to the existence of multiple phases at the same values of the relevant parameters. Other definitions of phase transitions, based on the non-differentiability of the free energy density and on the appearance of ordering, are also discussed

The Fundamental Concepts of Classical Equilibrium Statistical Mechanics

A critical examination of some basic conceptual issues in classical statistical mechanics is attempted, with a view to understanding the origins, structure and statuts of that discipline. Due attention is given to the interplay between physical and mathematical aspects, particularly regarding the role of probability theory. The focus is on the equilibrium case, which is currently better understood, serving also as a prelude for a further discussion of non-equilibrium statistical mechanics.Comment: 33 pages, overview, conceptual discussio