Constantino Tsallis works in Rio de Janeiro at CBPF, Brazil. He was born in Greece, and grew up in Argentina, where he studied physics at Instituto Balseiro, in Bariloche. In 1974 he received a Doctorat d'Etat et Sciences Physiques degree from the University of Paris-Orsay. Tsallis introduced what is known as Tsallis entropy and Tsallis statistics in his 1988 paper "Possible generalization of Boltzmann–Gibbs statistics" published in the Journal of Statistical Physics. The generalization is considered to be one of the most viable and applicable candidates for formulating a theory of non-extensive thermodynamics and has many implications in heavy-ion physics.
1) First of all I would like to ask you about your background and your previous studies
Born in Athens, I grew up in Argentina. I studied two and a half years engineering, but I then moved to physics. I concluded a "doctorat d'etat es sciences physiques" at the University of Paris-Orsay in 1974. I simultaneously taught physics at the University of Paris and at the ecole speciale de physique et chimie de Paris during 7 years. Then I moved to Brazil with my family in 1975, where I live permanently until now (except for several long visiting stages in Europe, Usa, Japan).
2) When did you decide to pursue research in the field of statistical mechanics/physics?
From early times I was fascinated by statistical mechanics and thermodynamics, its ubiquitous and wide fields of applications, its remarkably practical aspects, but, over all, its mathematical and conceptual beauty and subtleties. I started basically with the theory of phase transitions and critical phenomena, renormalization group, then I became also interested in chaos and nonlinear dynamical systems. I became occasionally interested in theoretical biogenesis, immunology, finance, populational genetics, quantum information and others. During the last 25 years I focus on the foundations of statistical mechanics.
3) How did you come up with the idea of generalizing the Boltzmann-Gibbs entropy and statistical mechanics back in 1988? What was the motivation but also what inspired you?
The inspiration came from multifractals (as I mention in my 1988 paper). This is why I use the notation q, although the meaning is definitively different from the one this letter has in multifractal theory. I was participating in a French-Mexican-Brazilian workshop in 1985 in Mexico city, and during a coffee-break I had the idea of generalizing the Boltzmann-Gibbs entropy by using probabilities^q. The index q would introduce a bias in the weight of the probability of microscopic events. Like this, events with say low probability could control a macroscopic phenomenon. For example, a vortex (i.e. zillions of molecules moving one following the other in circles) is a highly improbable event unless very strong correlations exist. But if these correlations are indeed there, the vortex can be the most important aspect of a large-scale phenomenon and eventually control it. Back to Rio de Janeiro from that short trip to Mexico, I just wrote down the non additive entropy sq and had a first look on its properties. However I could not really understand what its meaning was and what it could be useful for. And this is why I did not publish until 3 years later, in 1988. In particular, the frequently used q-generalization of the Boltzmann actor was calculated in August 1987 during a domestic flight between Maceio and Rio de Janeiro. My main motivation was the feeling of beauty and curiosity that the theory inspired to me. In the beginning I certainly had no idea of the physical predictions, verifications and applications that, along the years, would emerge in theoretical, experimental, observational and computational aspects of natural, artificial and social systems.
4) Which are the fields in which the Tsallis distribution is applied? Do you have the time to follow all of them? How do the different problems in each of these fields motivate your current research - are you "revisiting" your original ideas?
The non additive entropies and their related functions and probability distributions have so amazingly many applications that for sure no human being can follow all of them. It has been applied in cosmology, gravitation, high energy physics, condensed matter physics, astronomy, astrophysics, plasmas of various kinds (including the solar wind), earthquakes, heart and brain signals, recognition algorithms, medicine (detection of cancer and guided surgery among others), linguistics, biology, turbulence, granular matter, glasses, spin-glasses, scale-invariant networks, computational science, chemistry, engineering problems, quantum information, urban description, train and airlines statistics, nonlinear dynamical systems, financial theory, hydrology, ecology, ergodic theory, optimization techniques, polymers, cognitive psychology, paleontology, the list is virtually endless.
The bibliography is regularly updated at http://tsallis.cat.cbpf.br/biblio.htm It contains over 4000 papers from nearly 6000 authors from 75 countries. Easy therefore to understand why no human being can follow all that! The unbelievable ubiquity of the concept of entropy and its related functions has gradually become, along the years, one of the deepest perplexities of my life.
5) Now I would like to come particularly to the field of heay-ion physics and the application of the Tsallis function. Are you familiar with these applications and what are the physical pictures (particularly I would like to ask you about the q parameter and it physical interpretation).
High energy physics has gradually become during recent years an important domain of application of this theory. For example, the flux of cosmic rays appears to follow the q-exponential function along 27 decades in flux as function of the energy. Transverse momenta of hadronic jets in electron-positron experiments such as the DELPHI experiment at LEP were also amazingly well fitted using the same function. Nowadays, at the LHC (ALICE, CMS, ATLAS collaborations, among others) and the RHIC at Brookhaven (PHENIX collaboration) the same function fits very satisfactorily (in some cases within 15 decades of the probability distribution!) results in proton-proton as well as in heavy-ion experiments. A strongly remarkable fact is that the corresponding value of q [q = 1+ 1/n, as sometimes noted] appears to be nearly universal, namely close to 1.1 - 1.2, in all these experiments. This reflects some fundamental lack of ergodicity at the most microscopic level, and constitutes up to now a sort of physical mystery (let me remind that q=1 recovers the usual Boltzmann-Gibbs statistics). As already illustrated in the literature in some simple cases, the calculation of the index q from first principles is to be done within dynamics (Newtonian, quantum, relativistic, or say quantum chromodynamics for high-energy physics). By no means do I state that it is easily feasible but it belongs to the level of dynamics. However, the entire function characterized by that specific value of q demands the methods of statistical mechanics. The situation is analogous to the calculation of the orbit of, say, Mars. Within Newtonian mechanics we can easily calculate that it indeed has an elliptic form (Keplerian orbit). The calculation of its specific eccentricity would nevertheless demand the knowledge of the initial conditions of all masses of the planetary system, as well as the access to a formidable computer in order to handle Newton's law within such a very complex system. Similarly, within nonextensive statistical mechanics we can easily obtain the q-exponential form for the energy distribution (which recovers the celebrated Boltzmann factor for q=1). The specific value of q (and of the temperature) remains to be done at the microscopic mechanical level. Let me by the way point out a regrettable error commonly done in some papers. The q-exponential distributions differ from Levy distributions and from infinitely many other distributions which asymptotically behave as power-laws. They all (relevantly) differ however in the non-asymptotic regions. The distribution which systematically fits the data for the hadronic transverse momenta in the LHC experiments is the q-exponential and not at all the Levy distribution. Therefore expressions such as "Levy-Tsallis function" are deplorably misleading. The quantitative difference between Levy and q-exponential distributions is so neat that it can be checked even within Google!
6) Which are the open directions for future research?
Many open problems constitute presently interesting challenges within this research area. Among those one might mention (i) what is the analytic dependence of q on the dimensionality d of the system and on the range of the interactions (characterized by the index alpha) in many-body Hamiltonian systems?; (ii) what is the general algebra of relations involving the so called q-triplet? (Murray Gell-Mann, Yuzuru Sato and myself have found such algebra for a relatively simple case, related to the magnetic field within the solar wind plasma, but the general connections still are elusive); (iii) the entire set of interconnected fundamental theorems within theory of probabilities, involving q-generalized central limit theorems and large-deviation theory; (iv) the generic fundamental connections between q and the sensitivity to the initial conditions (e.g., q-generalized Lyapunov exponents) within the theory of classical and quantum nonlinear dynamical systems; (v) the first-principle calculation of the value of q for high-energy physics, within say quantum chromodynamics; (vi) the relevance of these ideas for the entropy of black holes. I have only focused here on theoretical aspects because there already exists a plethora of experimental and observational verifications of the theory. Nevertheless, many experimental challenges also do exist, essentially concerning the identification of the relevant physical ingredients connecting the real systems with the theory.
7) Finally, I would like to ask you about the role that science has in the Brazilian public sphere. Do you think that science communication is important and in what sense?
Since several years science and technology occupy central concerns of successive governments in Brazil. This is very important and obviously welcome. The penetration into the entire society, with all its diversified instruction levels, of the understanding of the corresponding needs, advantages and peculiarities can only be achieved with intelligent and efficient efforts from talented science communicators.