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We examine the relationship between two different types of ranked data, frequencies and magnitudes. We consider data that can be sorted out either way, through numbers of occurrences or size of the measures, as it is the case, say, of moon craters, earthquakes, billionaires, etc. We indicate that these two types of distributions are functional inverses of each other, and specify this link, first in terms of the assumed parent probability distribution that generates the data samples, and then in terms of an analog (deterministic) nonlinear iterated map that reproduces them. For the particular case of hyperbolic decay with rank the distributions are identical, that is, the classical Zipf plot, a pure power law. But their difference is largest when one displays logarithmic decay and its counterpart shows the inverse exponential decay, as it is the case of Benford law, or viceversa. For all intermediate decay rates generic differences appear not only between the power-law exponents for the midway rank decline but also for small and large rank. We extend the theoretical framework to include thermodynamic and statistical-mechanical concepts, such as entropies and configuration.
We show that size-rank distributions with power-law decay (often only over a limited extent) observed in a vast number of instances in a widespread family of systems obey Tsallis statistics. The theoretical framework for these distributions is analog
We propose a new physically-based ``multifractal stress activation model of earthquake interaction and triggering based on two simple ingredients: (i) a seismic rupture results from activated processes giving an exponential dependence on the local st
Path-dependent stochastic processes are often non-ergodic and observables can no longer be computed within the ensemble picture. The resulting mathematical difficulties pose severe limits to the analytical understanding of path-dependent processes. T
In this paper we review some general properties of probability distributions which exibit a singular behavior. After introducing the matter with several examples based on various models of statistical mechanics, we discuss, with the help of such para
Biomolecular folding, at least in simple systems, can be described as a two state transition in a free energy landscape with two deep wells separated by a high barrier. Transition paths are the short part of the trajectories that cross the barrier. A