When do generalized entropies apply? How phase space volume determines entropy
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In this article, the authors show that generalized entropies can only exist when the dynamically (statistically) relevant fraction of degrees of freedom in the system vanishes in the thermodynamic limit.Abstract:
We show how the dependence of phase space volume Ω(N) on system size N uniquely determines the extensive entropy of a classical system. We give a concise criterion when this entropy is not of Boltzmann-Gibbs type but has to assume a generalized (non-additive) form. We show that generalized entropies can only exist when the dynamically (statistically) relevant fraction of degrees of freedom in the system vanishes in the thermodynamic limit. These are systems where the bulk of the degrees of freedom is frozen and statistically inactive. Systems governed by generalized entropies are therefore systems whose phase space volume effectively collapses to a lower-dimensional "surface". We illustrate these results in three concrete examples: accelerating random walks, a microcanonical spin system on networks and constrained binomial processes. These examples suggest that a wide class of systems with "surface-dominant" statistics might in fact require generalized entropies, including self-organized critical systems such as sandpiles, anomalous diffusion, and systems with topological defects such as vortices, domains, or instantons.read more
Citations
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Maximum Entropy Principle in Statistical Inference: Case for Non-Shannonian Entropies
Petr Jizba,Jan Korbel,Jan Korbel +2 more
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Co-Evolutionary Mechanisms of Emotional Bursts in Online Social Dynamics and Networks
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References
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Book
Nonextensive Entropy: Interdisciplinary Applications
TL;DR: A great variety of complex phenomena in many scientific fields exhibit power-law behavior, reflecting a hierarchical or fractal structure as mentioned in this paper, and these phenomena seem to be susceptible to description using approaches drawn from thermodynamics or statistical mechanics, particularly approaches involving the maximization of entropy and of Boltzmann-Gibbs statistical mechanics.
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Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive
TL;DR: It is conjecture that these mechanisms are deeply related to the very frequent emergence, in natural and artificial complex systems, of scale-free structures and to their connections with nonextensive statistical mechanics.
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Generalized entropies and the transformation group of superstatistics
TL;DR: In this article, the first three Shannon-Khinchin axioms are assumed to hold, and it is shown that for a given distribution there are two different ways to construct the entropy.