Generalized entropies, density of states, and non-extensivity

Abstract

The concept of entropy connects the number of possible configurations with the number of variables in large stochastic systems. Independent or weakly interacting variables render the number of configurations scale exponentially with the number of variables, making the Boltzmann–Gibbs–Shannon entropy extensive. In systems with strongly interacting variables, or with variables driven by history-dependent dynamics, this is no longer true. Here we show that contrary to the generally held belief, not only strong correlations or history-dependence, but skewed-enough distribution of visiting probabilities, that is, first-order statistics, also play a role in determining the relation between configuration space size and system size, or, equivalently, the extensive form of generalized entropy. We present a macroscopic formalism describing this interplay between first-order statistics, higher-order statistics, and configuration space growth. We demonstrate that knowing any two strongly restricts the possibilities of the third. We believe that this unified macroscopic picture of emergent degrees of freedom constraining mechanisms provides a step towards finding order in the zoo of strongly interacting complex systems.

Learn your entropy from informative data: an axiom ensuring the consistent identification of generalized entropies

“…One approach requires a priori knowledge of the system (e.g. how certain properties of the entropy or of the system change with the number of accessible configurations [8,24,25]) as in Eqs. ( 20) and ( 21), implying that, in absence of such knowledge, the entropic parameters cannot be consistently derived purely from data as the other parameters.…”

Numerical studies for an ab initio investigation into the Boltzmann prescription in statistical mechanics of large systems

“…The statistical mechanics of non-extensive systems is still considered an open problem [1,2]. On the one hand, many studies in the past three decades have addressed the issue of generalizations of Boltzmann-Gibbs statistical mechanics [3,4,5,6,7,8,9,10].…”

An approach to the quasi-equilibrium state of a self-gravitating system

“…In order to deal with non-extensivity in systems with long-range interactions, the so-called generalized entropies, such as the Tsallis entropy, are suggested. They are generalizations of the well-known Boltzmann-Gibbs entropy and contain it as a special limit [12,13]. These entropies are not extensive in general and so it seems good to make use of them in such systems; but there are reasons which prevent us to do so.…”