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Nonadditive entropy Sq and nonextensive statistical mechanics: Applications in geophysics and elsewhere

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Abstract

The celebrated Boltzmann-Gibbs (BG) entropy, S BG = −kΣ i p i ln p i , and associated statistical mechanics are essentially based on hypotheses such as ergodicity, i.e., when ensemble averages coincide with time averages. This dynamical simplification occurs in classical systems (and quantum counterparts) whose microscopic evolution is governed by a positive largest Lyapunov exponent (LLE). Under such circumstances, relevant microscopic variables behave, from the probabilistic viewpoint, as (nearly) independent. Many phenomena exist, however, in natural, artificial and social systems (geophysics, astrophysics, biophysics, economics, and others) that violate ergodicity. To cover a (possibly) wide class of such systems, a generalization (nonextensive statistical mechanics) of the BG theory was proposed in 1988. This theory is based on nonadditive entropies such as \(S_q = k\frac{{1 - \sum\nolimits_i {p_i^q } }} {{q - 1}}\left( {S_1 = S_{BG} } \right)\). Here we comment some central aspects of this theory, and briefly review typical predictions, verifications and applications in geophysics and elsewhere, as illustrated through theoretical, experimental, observational, and computational results.

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  1. Centro Brasileiro de Pesquisas Fisicas and National Institute of Science and Technology for Complex Systems, Rio de Janeiro, Brazil

    Constantino Tsallis

  2. Santa Fe Institute, Santa Fe, NM, USA

    Constantino Tsallis

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Tsallis, C. Nonadditive entropy Sq and nonextensive statistical mechanics: Applications in geophysics and elsewhere. Acta Geophys. 60, 502–525 (2012). https://doi.org/10.2478/s11600-012-0005-0

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