Tsallis Entropy

As humans explore more complex systems and experience heavier tails, Tsallis entropy will attract much more attention. It is the one-parameter generalization of the Boltzmann, Gibbs, and Shannon entropy; for a discrete distribution with probabilities $p_i$ it reads

$$S_q = k\,\frac{1 - \sum_i p_i^{q}}{q - 1}$$ (1)

At $q = 1$ the classical entropy returns, along with the assumption that a system decomposes into independent parts. Long-range dependence, memory, multifractal geometry, and power-law statistics break that assumption, and there $q$ departs from one by an amount the data decides. Finance, networks, extreme risk, and artificial intelligence and machine learning models all supply such systems.

What organizes the rest is nonadditivity. For subsystems that are independent in the ordinary sense, entropy is additive; the Tsallis form composes as

$$S_q(A \cup B) = S_q(A) + S_q(B) + \frac{1-q}{k}\,S_q(A)\,S_q(B)$$ (2)

with $k$ setting the scale. The sign of that cross term, and so the position of $q$ relative to one, fixes how entropy behaves under joining: additive at $q = 1$, superadditive below, subadditive above. Passed through the associated $q$-distributions, the three regimes describe different degrees of correlation and tail weight.

Heavy tails are usually handled by patching the extremes of a thin-tailed model. Maximum entropy runs the other way: constrain only what is genuinely known, such as a mean or a left-tail, ruin-style risk bound, and the least committal distribution consistent with those constraints is heavy-tailed. Maximizing Tsallis entropy under a second-moment constraint in the escort measure gives the $q$-Gaussian family

$$p_q(x) \propto \Bigl[1 - (1-q)\,\beta x^{2}\Bigr]_{+}^{\,1/(1-q)}$$ (3)

which returns the ordinary Gaussian at $q = 1$ and carries power-law tails above it. Taleb develops this nonadditive route to thick tails, and the $q$-distributions that come with it, in Statistical Consequences of Fat Tails.

Overview

Tsallis entropy, Wikipedia. Article
N. N. Taleb, Statistical Consequences of Fat Tails (2020). arXiv:2001.10488 PDF

Selected papers

C. Tsallis, Possible generalization of Boltzmann–Gibbs statistics, J. Stat. Phys. 52, 479 (1988). DOI
C. Tsallis, Nonextensive statistical mechanics: a brief review of its present status (2002). arXiv:cond-mat/0205571 PDF
C. Tsallis and E. Brigatti, Nonextensive statistical mechanics: a brief introduction (2003). arXiv:cond-mat/0305606 PDF
F. Caruso and C. Tsallis, Nonadditive entropy reconciles the area law in quantum systems with classical thermodynamics (2006). arXiv:cond-mat/0612032 PDF
C. Tsallis, Nonadditive entropy: the concept and its use (2008). arXiv:0812.4370 PDF
C. Tsallis, Introduction to Nonextensive Statistical Mechanics, Springer monograph. Springer

Broader foundations and applications: CBPF bibliography · NONEXTENSIVE.pdf

News

E. Boor, R. da Silva, J. C. S. de Siqueira, and J. R. Iglesias, Does the Power-Law Advantage in the Tails Outweigh the Global q-Gaussian Description? (2026-08-31). Fits power laws and q-Gaussians to stock index and cryptocurrency returns from Brazil and the United States across time scales. Restricted to the extreme tails the power law describes the data better, as the asymptotic behaviour of the q-Gaussian itself predicts, but not by enough to outweigh the q-Gaussian advantage of covering peak, bulk, and tails with one functional form, and the evolution of the fitted $q$ tracks aggregational Gaussianity directly. arXiv:2608.30219 PDF
K. Chen and Q. Wang, Nearly Sample-Optimal Estimators for Quantum Rényi and Tsallis Entropies (2026-08-18). Gives estimators for quantum Rényi and Tsallis entropies whose sample complexity is nearly optimal in the order, the dimension, and the additive error, with an explicit split between the regimes below and above order one. The upper bounds improve earlier quantum Rényi and Tsallis estimators and match lower bounds established recently. arXiv:2608.18070 PDF
K. Yang, M. Asgharian, and C. M. T. Greenwood, Maximum Tsallis Entropy Distributions for Robust and Efficient Sparse Learning from Correlated Data (2026-08-18). Replaces the Gaussian assumption in sparse statistical learning with the q-Gaussian derived from maximising Tsallis entropy, re-deriving the multivariate density for correlated and heterogeneous data and building a numerically stable sparse learning algorithm on the Hager and Zhang conjugate gradient method. The motivating setting is biostatistics, where genetic and longitudinal studies carry both correlation and outliers. arXiv:2608.17244 PDF
I. S. Gomez, Finite nonlinear mathematical structures induced by the Tsallis q-sum (2026-08-11). Builds finite structures on the Tsallis $q$-sum: an axiomatic $q$-cardinality of finite sets obeying nonlinear additivity, unique under the stated axioms and recovering ordinary cardinality as $q \to 1$, plus a nonlinear matrix composition whose noncommutativity is characterised through the ordinary commutator. A worked example over $M_2(\mathbb{Z}_6)$ shifts the centre of a finite matrix algebra when $1-q$ is a zero divisor. arXiv:2608.11455 PDF
K. da Costa and B. Modenesi, Perspectives on Tsallis Statistics for Artificial Intelligence (2026-08-02). Surveys where Tsallis statistics meets machine learning: sparse attention such as sparsemax and $\alpha$-entmax, maximum entropy reinforcement learning with controllable exploration, heavy-tailed probabilistic models, and families of generalised losses and regularisers. The recurring design pattern is an interpolation between dense and sparse behaviour governed by $q$, and the paper argues that the heavy-tailed weight spectra and gradient noise observed in deep networks are themselves nonextensive signatures, so $q$ is best treated as a learnable parameter. arXiv:2608.01223 PDF
C. Tsallis, Reminiscences about Hans Capel (2026-07-27). A personal homage to Hans Willem Capel, collecting recollections and thoughts rather than results. arXiv:2607.24380 PDF
C. Tsallis, SigmaPhi Prize 2026 (2026-07). The SigmaPhi Prize, awarded every three years by the SigmaPhi conference for exceptional achievement in statistical physics, went to Constantino Tsallis (CBPF, Rio de Janeiro) jointly with Euthimios Kaxiras (Harvard) and David McComas (Princeton). The diplomas are presented at the opening session of the SigmaPhi 2026 conference. CBPF SigmaPhi
S. Eroglu, Nonextensive Statistical Signatures of the Bilaterian Transition in Proteome Length Distributions (2026-06-26). Fits truncated discrete q-exponentials by maximum likelihood to the protein length distributions of 22 reference proteomes spanning Archaea, Bacteria, and Eukarya. The entropic index separates three regimes: superextensive for prokaryotes, unicellular and non-animal eukaryotes and basal animals, borderline for the cnidarians and the basal bilaterian Capitella teleta, and subextensive for every higher bilaterian, with $q$ rising monotonically from the sea urchin Strongylocentrotus purpuratus to Homo sapiens. arXiv:2606.27985 PDF