Tsallis Constantino
Centro Brasileiro de Pesquisas Fisicas National Institute of Science and Technology of Complex Systems, Rua Xavier Sigaud 150, 22290- Rio de Janeiro, Brazil.
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, 87501 NM, USA.
Philos Trans A Math Phys Eng Sci. 2023 Oct 2;381(2256):20220293. doi: 10.1098/rsta.2022.0293. Epub 2023 Aug 14.
The Boltzmann-Gibbs (BG) statistical mechanics constitutes one of the pillars of contemporary theoretical physics. It is constructed upon the other pillars-classical, quantum, relativistic mechanics and Maxwell equations for electromagnetism-and its foundations are grounded on the optimization of the BG (additive) entropic functional [Formula: see text]. Its use in the realm of classical mechanics is legitimate for vast classes of nonlinear dynamical systems under the assumption that the maximal Lyapunov exponent is (currently referred to as ), and its validity has been experimentally verified in countless situations. It fails however when the maximal Lyapunov exponent (referred to as ), which is virtually always the case with complex natural, artificial and social systems. To overcome this type of weakness of the BG theory, a generalization was proposed in 1988 grounded on the non-additive entropic functional [Formula: see text]. The index [Formula: see text] and related ones are to be calculated, whenever mathematically tractable, from first principles and reflect the specific class of weak chaos. We review here the basics of this generalization and illustrate its validity with selected examples aiming to bridge natural and social sciences. This article is part of the theme issue 'Thermodynamics 2.0: Bridging the natural and social sciences (Part 2)'.
玻尔兹曼 - 吉布斯（BG）统计力学是当代理论物理学的支柱之一。它建立在其他支柱之上——经典力学、量子力学、相对论力学以及麦克斯韦电磁方程——其基础是BG（加法）熵泛函[公式：见原文]的优化。在最大李雅普诺夫指数为（当前称为）的假设下，它在经典力学领域对于大量非线性动力系统的应用是合理的，并且其有效性已在无数情况下得到实验验证。然而，当最大李雅普诺夫指数（称为）时，它就失效了，而这在复杂的自然、人工和社会系统中几乎总是如此。为了克服BG理论的这种弱点，1988年提出了一种基于非加法熵泛函[公式：见原文]的推广。只要在数学上易于处理，指数[公式：见原文]及相关指数就应从第一原理计算得出，并反映特定类别的弱混沌。我们在此回顾这种推广的基础知识，并用旨在连接自然科学和社会科学的选定示例来说明其有效性。本文是主题为“热力学2.0：连接自然科学和社会科学（第2部分）”的一部分。
