Constantino Tsallis

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Nataša Kejžar -- 

Graduated and master's degree in physics from the Balseiro Institute of Bariloche National University of Cuyo (1965), Doctorate d'Etat es Sciences Physiques by the Universite de Paris-Orsay (1974), and post-doctoral students at Oxford University (1983), Cornell University (1983). He is currently a researcher of the Brazilian Center for Physics Research. He has experience in Physics, with emphasis on Condensed Matter Physics and Statistical Mechanics. Member of the Brazilian Academy of Sciences and the Brazilian Academy of Economic Sciences, and Social Policies. Doctor Honoris Causa of the National University of Cordoba (Argentina), the State University of Maringa (Brazil), the Federal University of Rio Grande do Norte (Brazil) and Aristotle University of Tessalonika (Greece). Coordinator of the National Institute of Science and Technology Complex Systems.

Cited 2012

The following cited in Hanel, Rudolf; Thurner, Stefan; Murray Gell-Mann. 2012. [Generalized entropies and logarithms and their duality relations]]. PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA, 109 (47):19151-19154; 10.1073/pnas.1216885109 NOV 20 2012

C. Tsallis, A. M. C. Souza. 2003. Constructing a statistical mechanics for Beck-Cohen superstatistics. Phy Rev E 026106.

C. Tsallis, R. S. Mendez, A. R. Plastino. 1998. Physics A. 261: 534.

New 2014-2015

Constantino Tsallis. 2015. A new entropy based on a group-theoretical structure. Abstract:
A multi-parametric version of the nonadditive entropy Sq is introduced. This new entropic form, denoted by Sa,b,r, possesses many interesting statistical prop- erties, and it reduces to the entropy Sq for b = 0, a = r := 1−q (hence Boltzmann-Gibbs entropy SBG for b = 0, a = r → 0). The construction of the entropy Sa,b,r is based on a general group-theoretical approach recently proposed by one of us [16]. Indeed, essentially all the properties of this new entropy are obtained as a consequence of the existence of a rational group law, which expresses the structure of Sa,b,r with respect to the composition of statis- tically independent subsystems. Depending on the choice of the parameters, the entropy Sa,b,r can be used to cover a wide range of physical situations, in which the measure of the accessible phase space increases say exponentially with the number of particles N of the system, or even stabilizes, by increasing N, to a limiting value. This paves the way to the use of this entropy in contexts where a system ”freezes” some or many of its degrees of freedom by increasing the number of its constituting particles or subsystems.

[16] P. Tempesta, Beyond the Shannon-Khinchin Formulation: The Composability Axiom and the Universal Group Entropy, arXiv: 1407.3807 (2014).

New 2012-2013

Constantino Tsallis. 2009. [ Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World. Springer

Title: Nonadditive entropy Sq and nonextensive statistical mechanics: Applications in geophysics and elsewhere Authors: Tsallis, C Author Full Names: Tsallis, Constantino Source: ACTA GEOPHYSICA, 60 (3):502-525; 10.2478/s11600-012-0005-0 JUN 2012 Language: English SFI Taxonomy: Geophysics Abstract: The celebrated Boltzmann-Gibbs (BG) entropy, S-BG = -k Sigma(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 (1-)Sigma(piq)(i)/q(-1) (S-1 = S-BG). Here we comment some central aspects of this theory, and briefl! y review typical predictions, verifications and applications in geophysics and elsewhere, as illustrated through theoretical, experimental, observational, and computational results.


Constantino Tsallis. 2009. Introduction To Nonextensive Statistical Mechanics: Approaching A Complex World. Springer Physics.

download book

Gell-Mann, Murray, and Constantino Tsallis. Nonextensive Entropy: Interdisciplinary Applications. NY: Oxford University Press.

Front Matter - Table of Contents.

Articles relevant to White, Tsallis, et al

Alexei Vázquez. 2003. Growing network with local rules: Preferential attachment, clustering hierarchy, and degree correlations Phys. Rev. E 67, 056104 Abstract. The linear preferential attachment hypothesis has been shown to be quite successful in explaining the existence of networks with power-law degree distributions. It is then quite important to determine if this mechanism is the consequence of a general principle based on local rules. In this work it is claimed that an effective linear preferential attachment is the natural outcome of growing network models based on local rules. It is also shown that the local models offer an explanation for other properties like the clustering hierarchy and degree correlations recently observed in complex networks. These conclusions are based on both analytical and numerical results for different local rules, including some models already proposed in the literature.

reprinted in yumpu Final paper in pdf as SFI working paper. Reviewed 2005 in Europhysicsnews 36(6):218-220 by Stefan Thurner.

Cited in: Bhatt, R., and Kishor Barman. 2012. [4643-21961-1-PB.pdf Global Dynamics of Online Group Conversations]. And available online at;

Cited in: Constantino Tsallis. 2011. SOME OPEN POINTS IN NONEXTENSIVE STATISTICAL MECHANICS International Journal of Bifurcation and Chaos ⃝c World Scientific Publishing Company. (This paper is for the Special Issue edited by Prof. Gregoire Nicolis, Prof. Marko Robnik, Dr. Vassilis Rothos and Dr. Haris Skokos) "The degree distribution of (asymptotically) scale-free networks [White et al, 2006; Thurner et al, 2007];..."

Google: "Generative Model for Feedback Networks"
Google Scholar: "Generative Model for Feedback Networks" 34 citations


NONEXTENSIVE STATISTICAL MECHANICS AND THERMODYNAMICS: BIBLIOGRAPHY January 2007 - preliminary version - Europhysics News 36 (6) (Nov-Dec 2005)

Generalized entropy and thermostatistics: [1] Connection to thermodynamics, ensembles and Jaynes' information theory: [2{520]

Tsallis Bibliography


Publications: - [1]

Unified model for network dynamics exhibiting nonextensive statistics Stefan Thurner, Fragiskos Kyriakopoulos, Constantino Tsallis published as Phys. Rev. E 76, 036111 (2007) (8 pages)

Preferential attachment growth model and nonextensive statistical mechanics Europhys. Lett., 70 (1), p. 70 (2005) D. J. B. Soares, C. Tsallis, A. M. Mariz, L. R. da Silva.

Nonextensive aspects of self-organized scale-free gas-like networks Stefan Thurner and Constantino Tsallis

Reviewed 2005 in Europhysicsnews 36(6):218-220 by Stefan Thurner.

The Kolmogorov-Smirnov test or KS-test is used in this article to compare actual degree distributions to best-fit distributions using the q-exponental (Tsallis entropy). It is also discussed in


  • C. Tsallis, Entropy, in Encyclopedia of Complexity and Systems Science (Springer, Berlin, 2009);
  • C. Tsallis, Introduction to Nonextensive Statistical Mechanics - Approaching a Complex World (Springer, New York, 2009);
  • S. Umarov, C. Tsallis, M. Gell-Mann and S. Steinberg, J. Math. Phys. 51, 033502 (2010);
  • M. Jauregui and C. Tsallis, J. Math. Phys. 51 (June 2010), in press;
  • CMS Collaboration, J. High Energy Phys. 02, 041 (2010);

On this wiki

  1. The complex network problem
  2. Social-circles network model
  3. Estimating Tsallis q
  4. Tsallis q distribution project
  5. Tsallis q historical cities and city-sizes
  6. Encyclopedia of Complexity and Systems Science can add Tsallis article

At a theoretical level, "the q-exponential is the distribution which, under appropriate constraints, optimizes the nonadditive entropy Sq, in the same manner the exponential is the distribution which optimizes the additive Boltzmann-Gibbs entropy." (personal communication)

See Tsallis website, Nonextensive Statistical Mechanics and Thermodynamics, mini-review by Thurner, Special issue of Europhysics News, European Physical Society.

Comment and new references

Dear Doug,

... you have produced an informational network which might be quite useful to people interested in complexity!

All the best,


PS The q-generalization of the Central Limit Theorem is now published. And a new experimental verification of q-Gaussians in nature has recently appeared, this time in dusty plasma. I am attaching both for your perusal. (below)

Superdiffusion and non-Gaussian statistics in a driven-dissipative 2D dusty plasma. 2008. Bin Liu, J. Goree arXiv:0801.3991

On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics / A generalization of the central limit theorem consistent with nonextensive statistical mechanics. 2007. Sabir Umarov, Constantino Tsallis, Stanly Steinberg arXiv:cond-mat/0603593


Sumiyoshi Abe, Yutaka Nakada. 2006. Temporal extensivity of Tsallis' entropy and the bound on entropy production rate.

Is there a possibility of fitting a q-exponential growth curve?