Our results help choose reliable diversity measures based on the experimental accuracy of the abundance distributions in particular frequency ranges. We enumerate the various possible behaviours of the Rnyi entropy and its singularities. This allows us to obtain the singularities in the Rnyi entropy from those of the thermodynamic potential, which is directly related to the free energy density of the model. We illustrate our results on simple examples, and emphasize the limitations of the equivalence of ensembles when a thermodynamic limit is not well defined. We calculate analytically the Renyi entropy for the zeta-urn model with a Gibbs measure definition of the micro-state probabilities. This mapping predicts that non-concave regions of the rank-frequency curve should result in kinks in the Rényi entropy as a function of its order. In this limit, we show how the Rényi entropy can be constructed geometrically from rank-frequency plots. Hartley entropy, Shannon entropy, collision entropy, min-entropy. This surprising phenomenon is due to the growth of quantum correlations, or entanglement, among different subsystems. Nevertheless, all the parts of a large quantum-mechanical system out of equilibrium experience an increase in their entropy. The two quantities are related in the thermodynamic limit by a Legendre transform, by virtue of the equivalence between the micro-canonical and canonical ensembles. Distributions of abundances or frequencies play an important role in many fields of science, from biology to sociology, as does the Rnyi entropy. The laws of thermodynamics stipulate that the entropy of a closed system cannot increase. This is in analogy to the same question for the Shannon and von Neumann entropy (alpha1) which are known to satisfy several. The Renyi entropy is essentially the trace of some power of the density matrix of the block. We investigate the universal inequalities relating the alpha-Renyi entropies of the marginals of a multi-partite quantum state. Noah Linden, Milán Mosonyi, Andreas Winter. The abundance distribution is mapped onto the density of states, and the Rényi entropy to the free energy. The structure of Renyi entropic inequalities. We derive a mathematical relation between the abundance distribution and the Rényi entropy, by analogy with the equivalence of ensembles in thermodynamics. Distributions of abundances or frequencies play an important role in many fields of science, from biology to sociology, as does the Rényi entropy, which measures the diversity of a statistical ensemble. We introduce a novel measure for the quantum property of nonstabilizerness - commonly known as 'magic' - by considering the Rnyi entropy of the probability distribution associated to a pure quantum state given by the square of the expectation value of Pauli strings in that state.
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