Beyond Boltzmann–Gibbs–Shannon in Physics and Elsewhere

Abstract

The pillars of contemporary theoretical physics are classical mechanics, Maxwell electromagnetism, relativity, quantum mechanics, and Boltzmann–Gibbs (BG) statistical mechanics –including its connection with thermodynamics. The BG theory describes amazingly well the thermal equilibrium of a plethora of so-called simple systems. However, BG statistical mechanics and its basic additive entropy S B G started, in recent decades, to exhibit failures or inadequacies in an increasing number of complex systems. The emergence of such intriguing features became apparent in quantum systems as well, such as black holes and other area-law-like scenarios for the von Neumann entropy. In a different arena, the efficiency of the Shannon entropy—as the BG functional is currently called in engineering and communication theory—started to be perceived as not necessarily optimal in the processing of images (e.g., medical ones) and time series (e.g., economic ones). Such is the case in the presence of generic long-range space correlations, long memory, sub-exponential sensitivity to the initial conditions (hence vanishing largest Lyapunov exponents), and similar features. Finally, we witnessed, during the last two decades, an explosion of asymptotically scale-free complex networks. This wide range of important systems eventually gave support, since 1988, to the generalization of the BG theory. Nonadditive entropies generalizing the BG one and their consequences have been introduced and intensively studied worldwide. The present review focuses on these concepts and their predictions, verifications, and applications in physics and elsewhere. Some selected examples (in quantum information, high- and low-energy physics, low-dimensional nonlinear dynamical systems, earthquakes, turbulence, long-range interacting systems, and scale-free networks) illustrate successful applications. The grounding thermodynamical framework is briefly described as well.

Mecânica estatística de sistemas complexos

“…Vemos assim que, para q = 1, S qé não aditiva 2 . Outrossim, ela possui propriedades (tais como, por exemplo, concavidade e estabilidade de Lesche para todo q > 0, convexidade para todo q < 0, composabilidade para todo q, unicidade nos axiomas q-generalizados de Shannon [9] e de Khinchin [10], admissibilidade nos axiomas de Shore-Johnson [11], entre outras [2,3,12]), que conduzemà sua essencial utilidade para formular a MENE. Vemos que, para 1−q k → 0 (portanto para 1/k → 0 para q fixo, ou para q → 1 para k fixo), a relação (9) recupera a aditividade de S BG .…”

“…Seguindo as linhas tradicionais de Boltzmann e Gibbs, ilustremos o ensemble canônico baseado na entropia (3). Mais precisamente, otimizando…”

“…Agradeço calorosamente E.P. Borges por numerosos comentários decorrentes de uma leitura crítica do manuscrito, assim como aos diversos autores identificados nas várias figuras aqui reproduzidas, muitas delas fazendo parte do artigo de revisão [3]. Este trabalho foi parcialmente financiado pela Faperj e pelo CNPq.…”

“…Mencionemos entretanto, por oportuno, que até aparecerem a teoria da relatividade restrita e a mecânica quântica no inicio do século XX, usava-se simplesmente a expressão mecânica, pois só existia a de Newton. Analogamente, até 1988, quando foi proposta a mecânica estatística não extensiva (MENE) [1] (ver detalhes em [2,3]), usava-se simplesmente a expressão mecânica estatística, pois só existia a de BG. A situação tem hoje consideravelmente evoluído, como focalizaremos precisamente neste artigo.…”

Canonical Divergence for Flat α-Connections: Classical and Quantum

“…From a physical viewpoint, it is worth remarking that the Boltzmann-Gibbs distribution in statistical physics is an exponential family such that an invariant flat structure is given to the underlying manifold in terms of the -connection [ 6 ]. Tsallis generalized the concept of the Boltzmann-Gibbs distribution by introducing a generalized entropy, called the -entropy, for studying various phenomena not included in the conventional Boltzmann-Gibbs framework [ 17 , 18 ]. Actually, the -geometry, which is induced by the -divergence, covers the geometry of -entropy physics [ 19 ].…”