The Tsallis $q$-Gaussian distribution is a powerful generalization of the standard Gaussian and commonly used in various fields, including non-extensive statistical mechanics, financial markets image processing. It belongs to $q$-distribution family, which characterized by non-additive entropy. Due their versatility practicality, $q$-Gaussians are natural choice for modeling input quantities me...

In a previous paper (1) it was shown how, for a dynamical system, the probability distribution function of the sojourn–times in phase–space, defined in terms of the dynamical orbits (up to a given observation time), induces unambiguously a statistical ensemble in phase–space. In the present paper it is shown which is the p.d.f. of the sojourn–times corresponding to a Tsallis ensemble (this, by ...

The Boltzmann–Gibbs and Tsallis entropies are essential concepts in statistical physics, which have found multiple applications in many engineering and science areas. In particular, we focus our interest on their applications to image processing through information theory. We present in this article a novel numeric method to calculate the Tsallis entropic index q characteristic to a given image...

The proper way of averaging is an important question with regards to Tsal-lis' Thermostatistics. Three different procedures have been thus far employed in the pertinent literature. The third one, i.e., the Tsallis-Mendes-Plastino (TMP) [1] normalization procedure, exhibits clear advantages with respect to earlier ones. In this work, we advance a distinct (from the TMP-one) way of handling the L...

We find the value of constants related to constraints in characterization of some known statistical distributions and then we proceed to use the idea behind maximum entropy principle to derive generalized version of this distributions using the Tsallis’ and Renyi information measures instead of the well-known Bolztmann-Gibbs-Shannon. These generalized distributions will depend on q ∈ R and in t...

We study the microscopic dynamics of the metastable Quasi-Stationary States (QSS) in the Hamiltonian Mean Field (HMF) model, a Hamiltonian system of N classical inertial spins with infinite-range interactions which shows a second order phase transition. In order to understand the origin of metastability, which appears in an energy region below the critical point, we consider two different class...

In this lecture we briefly review the definition, consequences and applications of an entropy, Sq, which generalizes the usual Boltzmann-Gibbs entropy SBG (S1 = SBG), basis of the usual statistical mechanics, well known to be applicable whenever ergodicity is satisfied at the microscopic dynamical level. Such entropy Sq is based on the notion of q-exponential and presents properties not shared ...

We study dynamical phase transitions in systems with long-range interactions, using the Hamiltonian mean field model as a simple example. These systems generically undergo a violent relaxation to a quasistationary state (QSS) before relaxing towards Boltzmann equilibrium. In the collisional regime, the out-of-equilibrium one-particle distribution function (DF) is a quasistationary solution of t...

Using the Feigenbaum renormalization group (RG) transformation we work out exactly the dynamics and the sensitivity to initial conditions for unimodal maps of nonlinearity ζ > 1 at both their pitchfork and tangent bifurcations. These functions have the form of q-exponentials as proposed in Tsallis’ generalization of statistical mechanics. We determine the qindices that characterize these univer...