M. C. Rocca

Inversion of Umarov–Tsallis–Steinberg’s q-Fourier transform and the complex-plane generalization

We introduce a complex q-Fourier transform as a generalization of the (real) one analyzed in [S. Umarov, C. Tsallis, S. Steinberg, Milan J. Math. 307 (2008)]. By recourse to tempered ultradistributions we show that this complex-plane generalization overcomes all the troubles that afflict its real counterpart.

Reflections on the q-Fourier transform and the q-Gaussian function

The standard q-Fourier Transform (qFT) of a constant diverges, which begs for a better treatment. In addition, Hilhorst has conclusively proved that the ordinary qFT is not of a one-to-one character for an infinite set of functions [H.J. Hilhorst, J. Stat. Mech. (2010) P10023]. Generalizing the ordinary qFT analyzed in [S. Umarov, C. Tsallis, S. Steinberg, Milan J. Math. 76 (2008) 307], we appeal here to a complex q-Fourier transform, and show that the problems above mentioned are overcome.

New Solution of Diffusion-Advection Equation for Cosmic-Ray Transport Using Ultradistributions

In this paper we exactly solve the diffusion–advection equation (DAE) for cosmic-ray transport. For such a purpose we use the Theory of Ultradistributions of J. Sebastiao e Silva, to give a general solution for the DAE. From the ensuing solution, we obtain several approximations as limiting cases of various situations of physical and astrophysical interest. One of them involves Solar cosmic-rays’ diffusion.

The fractionary Schrödinger equation, Green functions and ultradistributions

In this work, we generalize previous results about the Fractionary Schrodinger Equation within the formalism of the theory of Tempered Ultradistributions. Several examples of the use of this theory are given. In particular we evaluate the Green function for a free particle in the general case, for an arbitrary order of the derivative index.

Dimensionally regularized Tsallis’ statistical mechanics and two-body Newton’s gravitation

Abstract Typical Tsallis’ statistical mechanics’ quantifiers like the partition function and the mean energy exhibit poles. We are speaking of the partition function Z and the mean energy 〈 U 〉 . The poles appear for distinctive values of Tsallis’ characteristic real parameter q , at a numerable set of rational numbers of the q -line. These poles are dealt with dimensional regularization resources. The physical effects of these poles on the specific heats are studied here for the two-body classical gravitation potential.

On the Nature of the Tsallis–Fourier Transform

By recourse to tempered ultradistributions, we show here that the effect of a q-Fourier transform (qFT) is to map equivalence classes of functions into other classes in a one-to-one fashion. This suggests that Tsallis’ q-statistics may revolve around equivalence classes of distributions and not individual ones, as orthodox statistics does. We solve here the qFT’s non-invertibility issue, but discover a problem that remains open.

The Tsallis–Laplace transform

We introduce here the q-Laplace transform as a new weapon in Tsallis’ arsenal, discussing its main properties and analyzing some examples. The q-Gaussian instance receives special consideration. Also, we derive the q-partition function from the q-Laplace transform.

Dimensional regularization of Renyi’s statistical mechanics

Abstract We show that typical Renyi’s statistical mechanics’ quantifiers exhibit poles. We are referring to the partition function Z and the mean energy 〈 U 〉 . Renyi’s entropy is characterized by a real parameter α . The poles emerge in a numerable set of rational numbers belonging to the α -line. Physical effects of these poles are studied by appeal to dimensional regularization, as usual. Interesting effects are found, as for instance, gravitational ones. In particular, negative specific heats.

Rescuing the MaxEnt treatment for q-generalized entropies

Abstract It has been recently argued that the MaxEnt variational problem would not adequately work for Renyi’s and Tsallis’ entropies. We constructively show here that this is not so if one formulates the associated variational problem in a more orthodox functional fashion.

Reciprocity relations and generalized, classic entropic quantifiers that lack trace-form

Abstract In this effort we show that the Legendre reciprocity relations, thermodynamics’ essential formal feature, are respected by general classic entropic functionals, even if they are NOT of trace-form nature, in contrast with Shannon’s or Tsallis’ cases. Further, with reference to the Maximum Entropy (MaxEnt) variational process, we encounter important cases, relevant to physical applications currently discussed in the research literature, in which the associated reciprocity relations exhibit anomalies. We show that these anomalies can be cured by carefully discriminating between apparently equivalent entropic forms.