The entropy of non-ergodic complex systems - A derivation from first principles

Thurner, S. & Hanel, R. (2012). The entropy of non-ergodic complex systems - A derivation from first principles. International Journal of Modern Physics: Conference Series 16 105-115. 10.1142/S2010194512007817.

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In information theory the 4 Shannon-Khinchin (SK) axioms determine Boltzmann Gibbs entropy, S ~ -Sigma_i p_i log p_i, as the unique entropy. Physics is different from information in the sense that physical systems can be non-ergodic or non-Markovian. To characterize such strongly interacting, statistical systems - complex systems in particular - within a thermodynamical framework it might be necessary to introduce generalized entropies. A series of such entropies have been proposed in the past decades. Until now the understanding of their fundamental origin and their deeper relations to complex systems remains unclear. To clarify the situation we note that non-ergodicity explicitly violates the fourth SK axiom. We show that by relaxing this axiom the entropy generalizes to, S ~Sigma_i Gamma(d+1,1-c log p_i), where Gamma is the incomplete Gamma function, and c and d are scaling exponents. All recently proposed entropies compatible with the first 3 SK axioms appear to be special cases. We prove that each statistical system is uniquely characterized by the pair of the two scaling exponents (c, d), which defines equivalence classes for all systems. The corresponding distribution functions are special forms of Lambert-W exponentials containing, as special cases, Boltzmann, stretched exponential and Tsallis distributions (power-laws) - all widely abundant in nature. This derivation is the first ab initio justification for generalized entropies. We next show how the phasespace volume of a system is related to its generalized entropy, and provide a concise criterion when it is not of Boltzmann-Gibbs type but assumes a generalized form. We show that generalized entropies only become relevant when the dynamically (statistically) relevant fraction of degrees of freedom in a system vanishes in the thermodynamic limit. These are systems where the bulk of the degrees of freedom is frozen. Systems governed by generalized entropies are therefore systems whose phasespace volume effectively collapses to a lower-dimensional "surface". We explicitly illustrate the situation for accelerating random walks, and a spin system on a constant-conectancy network. We argue that generalized entropies should be relevant for self-organized critical systems such as sand piles, for spin systems which form meta-structures such as vortices, domains, instantons, etc., and for problems associated with anomalous diffusion.

Item Type: Article
Uncontrolled Keywords: Shannon-Khinchin axioms; Generalized entropy; Extensivity; Phasespace volume; Super-diffusion; Ising on networks
Research Programs: Advanced Systems Analysis (ASA)
Bibliographic Reference: International Journal of Modern Physics: Conference Series; 16:105-115
Depositing User: IIASA Import
Date Deposited: 15 Jan 2016 08:46
Last Modified: 27 Aug 2021 17:39

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