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International Symposium on Non-Additive Entropies:
Fundamentals, Frontiers, and Applications
A tribute to
Constantino Tsallis
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A little over three decades ago, in 1988, a Brazilian theoretical physicist proposed a promising generalization of Boltzmann-Gibbs entropy. At that moment, few would have anticipated that this physical-mathematical formulation – expressed as Sq = k(1 - Σi piᵍ)/(q-1), where q quantifies the degree of non-extensivity in physical systems – would become the foundation of a robust research program capable of describing and predicting the behavior of dynamical systems across a wide variety of scientific domains. The fundamental elegance of this formulation lies in its property of recovering classical entropy in the limit q→1, preserving compatibility with established theory, while providing an adequate description of nonlinear systems with non-additive entropies that traditional statistical mechanics cannot capture: those with long-range correlations, fractal structures, non-ergodic dynamics, and prolonged temporal memory (TSALLIS, 1988; TSALLIS, 2009).
Three decades of research have shown that Tsallis's approach was indeed visionary. More than 7,000 scientific publications have been produced, research groups on every continent have adopted his approach, and accumulated empirical evidence demonstrates the applicability of non-additive entropies in very diverse domains: fluid turbulence, high-energy particle collisions, self-gravitating systems in astrophysics, economic dynamics and, crucially, complex biological systems (TSALLIS, 2023). Despite this success across multiple areas, the opportunity for broader integration of the fields remains open for its consolidation, considering the fundamentality of non-additive entropies, as proposed by Tsallis. Researchers in astrophysics study power-law distributions in gravitational phenomena without in-depth communication with neuroscientists analyzing epileptic seizures, even though they use essentially the same physical-mathematical framework. Fundamental debates about the physical interpretation of the parameter q and the criteria for identifying non-extensive systems remain partially open, generating a productive but unsynthesized tension between rigorous axiomatic arguments and robust empirical evidence in the consolidation of non-additive entropies.
Read the article published in the newspaper O Globo
The honorary chairmanship of Constantino Tsallis confers unique and profound historical legitimacy. The confirmed participation of recognized leaders – Evaldo Curado (CBPF), Markus Müller (ICTP), Christian Beck (Queen Mary University, United Kingdom), Didier Sornette (ETH Zurich, Switzerland), William Bialek (Princeton University, USA), Joseph Loscalzo (Harvard Medical School, USA) – ensures that the synthesis will be truly representative. Holding the event in Rio de Janeiro, with the central involvement of the Brazilian Center for Physics Research and Fiocruz, signals Brazil's commitment to leadership and positions the country as an undisputed hub for research in complex systems. The expected products – a seminal synthesis document and educational materials for graduate programs, including recordings of the lectures and other materials – will help foster global teaching and scientific research over the next decade.
The field of non-additive entropies and non-extensive complex systems, despite its growing recognition, has not yet reached full scientific maturity. When compared to consolidated areas, such as general relativity or quantum mechanics, it lacks integrated theoretical cohesion and strategic maps that would allow the beginning researcher to understand the complete landscape, the practitioner to identify with confidence where to use the tools, and the funder to set priorities based on demonstrated soundness. By promoting a structured synthesis, the Symposium will explicitly identify which foundations are consolidated versus speculative, in which domains applicability has been rigorously demonstrated versus where extrapolation remains conjectural, and which fundamental questions have satisfactory answers versus which remain open.
This unprecedented clarification is absolutely essential for continued maturation. It will identify priority research frontiers that may influence the strategies of agencies such as CNPq, CAPES, FINEP, and international institutions in the coming years. Fundamental unresolved questions, such as the dynamical origin of q, the a priori selection of q, and its role in quantum gravity, will receive coordinated momentum. Similarly, applications identified as promising but incipient may receive structured prioritization in future funding calls.
The symposium's explicit focus on health applications responds to the critical contemporary demand for precision and for big data. Biomedical data produced in clinical practice, multi-omics, multimodal imaging, and continuous physiological monitoring require sophisticated analytical tools capable of capturing multiscale structures and long-range correlations. Non-extensive statistical mechanics offers a rigorous and computationally tractable framework for problems on which conventional approaches have been failing. In computer-aided diagnosis, segmentation algorithms using Tsallis entropy have already demonstrated superiority in identifying brain tumor lesions (WACHS-LOPES et al., 2020). The new synthesis to be carried out will allow for the systematic documentation of best practices, the availability of optimized open-source code, and guidelines for developers. Conservative estimates indicate that CAD systems incorporating these techniques could increase diagnostic sensitivity by 10-15%, with a direct impact on early detection.
In clinical neurology, epileptic seizures can be interpreted as critical phenomena describable by non-extensive statistics, providing a basis for prediction systems (EFTAXIAS et al., 2011). Implantable devices could monitor entropic brain signals to detect pre-ictal states, enabling preventive interventions. For heart rate variability, entropic biomarkers show incremental prognostic value for cardiovascular events, allowing for refinement of risk stratification (RIBEIRO et al., 2017).
In personalized medicine, the entropic characterization of biological networks can guide therapy selection and optimized dosing. Studies in oncology reveal a correlation between tumor entropic indices and response to immunotherapy. In public health, the dynamics of epidemics can be adequately modeled by non-extensive approaches that capture rare high-impact events and heavy-tailed distributions.
Brazil has contributed disproportionately to the development of non-extensive statistical mechanics and non-additive entropies. Constantino Tsallis, founder of the field and one of the most cited scientists in Brazil, is a permanent Brazilian researcher at CBPF; in addition, strong groups have been created at CBPF, USP, UFRGS, UFPE, and Unicamp.
Holding the symposium and producing its outcomes in Rio de Janeiro positions Brazil as a world reference hub. The concrete outcomes are manifold. First, in human resources training: 15 young researchers, competitively selected, will have direct exposure to international leaders, creating collaborative networks that will help structure their careers. Educational materials will be incorporated into graduate programs, raising the quality of training. Second, increased international visibility attracts private investment: health-tech startups may emerge organically; algorithms may be licensed; software may be developed as a digital public good with global impact.
Third, in science diplomacy, established collaborations translate into joint research projects with multilateral funding through H2020, NIH, and NSFC. Brazil will position itself in global science policy discussions at UNESCO, WHO, and OECD, particularly in digital health. The alignment with the National Strategy for Science, Technology and Innovation (ENCTI 2021-2031), the Rio de Janeiro State Plan, and the Sustainable Development Goals is unequivocal and strategic.
EFTAXIAS, K. et al. Are epileptic seizures quakes of the brain? An approach by means of nonextensive Tsallis statistics. Physica A: Statistical Mechanics and its Applications, v. 390, n. 23-24, p. 4253-4266, 2011.
RIBEIRO, H. V. et al. Characterizing time series via complexity-entropy curves. Physical Review E, v. 95, n. 6, 062106, 2017.
TSALLIS, C. Possible generalization of Boltzmann-Gibbs statistics. Journal of Statistical Physics, v. 52, n. 1-2, p. 479-487, 1988.
TSALLIS, C. Introduction to nonextensive statistical mechanics: approaching a complex world. New York: Springer, 2009.
TSALLIS, C. When may a system be referred to as complex? An entropic perspective. Frontiers in Complex Systems, v. 1, 1284458, 2023.
WACHS-LOPES, G. A. et al. Recent nature-inspired algorithms for medical image segmentation based on Tsallis statistics. Communications in Nonlinear Science and Numerical Simulation, v. 88, 105256, 2020.