Constantino Tsallis introduced what is presently known as Tsallis entropy and Tsallis statistics, also known as q-statistics, in his 1988 paper "Possible generalization of Boltzmann–Gibbs statistics" published in the Journal of Statistical Physics. Tsallis-statistics is based on the idea of generalizing the Boltzmann-Gibbs entropy by using probabilities q; with the index q introducing a bias in the weight of the probability of microscopic events. Given the increasing importance of the Tsallis distribution in the domain of heavy-ion physics, Constantino Tsallis recently visited CERN and delivered a lecture during a Heavy-Ion Forum. We met him and discussed about the importance of q-statistics and the possible interpretations.

When did you decide to pursue research in the field of statistical mechanics?

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 immunology, finance, populational genetics, quantum information and others. During the last 25 years I focus on the foundations of statistical mechanics.

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.

Which are the fields in which the Tsallis distribution is applied? 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. 

Is this your first time at CERN?

I visited CERN invited by the organizers of the Heavy-Ion Forum and I must say that it has been a very interesting visit. I would like to take this opportunity and thank Yiota Foka and Urs Wiedemann for organizing this wonderful workshop. During my stay I had the chance to visit the ALICE cavern and see the detectors and the electronics that were built for these experiments.

Is there a growing interest over the last few years in the q distribution?

Over the recent years there has been an increasing use by the LHC experiments of the q-statistics and particularly the distribution associated with a stationary state within q-statistics. Q-statistics seems to describe very well the transverse momentum distributions of all different types of hadrons. All four LHC experiments have published results of these distributions that are well fitted by the q-exponential function. The resulting value of q is around 1.15; a neat departure from q=1 that corresponds to the Boltzmann-Gibbs distribution.

This result means that the stationary states of the particles before the hadronization are not in thermal equilibrium. Nevertheless the distribution is very robust and practically the same for different hadrons spanning a range of different energies.

Perhaps, one of the most impressive results, published a few months ago, is the measurement of the p_T distribution over a logarithmic range of 14 decades. It was found that the same expression of a q exponential q=1.15, fits the data over the full range of these fourteen decades. A theory that fits a range of couple of decades is already very interesting but fitting such a large range of decades, with the same distribution, is surprisingly rare.

Think for example Einstein’s and Newton’s equations of energy. When you compare them for protons with the highest cosmic ray energies using measurements from the AUGER project, the expression of Einstein is verified along eleven decades. This gives you an idea of how difficult is to measure experimentally a distribution over fourteen decades as it is the case for ALICE, ATLAS and CMS that all fit well with a value of q=1.15

What is the reason for that?

There is not a detailed explanation but the fact itself gives an idea of what is the physical scenario within which one should try to establish a specific model. The robustness of this distribution –that is not in thermal equilibrium- shows that it is a long-lasting stationary state or a quasi-stationary state. In turn this indicates that between the elements of the system there are very strong correlations.

If only weak correlations were taking place, like in the molecules of the air in this room, then you would expect these states to be in thermal equilibrium and hence the value of q would equal one. However, this is not the case and points to strong correlations with a very strong hierarchical structure. This could be either long memory effects at very elementary level or important long-distance correlations or both. This provides a framework of thinking about the possible detailed mechanism.

What can we learn from the new data?

I think that it would be interesting to check whether this q statistical frame could also describe not only the transverse momentum distribution but also the rapidity distribution. There are a few indications in the literature that the rapidity distribution is also described by q statistics but it’s interesting to explore whether this is the case or not and what would be the value of q. Personally, I think that q will have a different value but still related to the value of the transverse component.

Was it expected that the value of q would be the same for different hadrons?

In the beginning it was a big surprise. An interesting effect is how q changes with the collision energy. Below TeV energies, the value of q increases very slowly and we can hypothtically assume that when the energy is increased to infinity the value of q will reach an asymptotic value close to 1.21. That value of q coincides with the value that q has when we fit the cosmic ray fluxes with q-statistics. In these observations q has a value that is very close to the very high-energy limit of what is currently observed at the LHC. These connections between what is observed at the LHC and what is measured in observations of cosmic rays is fascinating.

Do you have a scenario for this?

There is a mathematical feature that is very suggestive. If you sum random variables, that are independent or nearly independent under certain general conditions, the attractor of the sum is a Gaussian. This is the so-called Central Limit Theorem (CLT), one of the most important theorems in the theory of probabilities.

As a consequence of this theorem, in nature we find many Gaussians. Under some circumstances when the variance associated of those random variables is not finite the attractor is not Gaussian but a Levy distribution. Both Gaussian and Levy distributions are related to independent random variables. However, if the variables are strongly correlated the attractor is a q-gaussian that also has a tail described with a power law but the central and intermediate regions of the distribution are very different from those corresponding to the Levy distribution.

Q-Gaussian and Levi distributions have different bodies although they both end with a power law. The CLT does not tell you that the attractor ends like a Gaussian but that it is a Gaussian everywhere. The generalized central limit theorems show that the attractors are Q-gaussians and that they are easily found in nature like it happens with the gaussians. However, the difference is that gaussians are found when the random variables are independent while q-gaussians are found when random variables are strongly correlated in a specific way related to some kind of hierarchical structure.

Q–gaussians and q-exponentials can be found in the solar wind, cold atoms, granular matter and also in high-energy collisions at CERN. The ubiquity of this distribution found in so many different systems is consistent with the CLT and the observations from the LHC experiments fit quite well with this scenario.

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.