F. Pennini

Tsallis’ entropy maximization procedure revisited

The proper way of averaging is an important question with regards to Tsallis’ Thermostatistics. Three different procedures have been thus far employed in the pertinent literature. The third one, i.e., the Tsallis–Mendes–Plastino (TMP) (Physica A 261 (1998) 534) 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 Lagrange multipliers involved in the extremization process that leads to Tsallis’ statistical operator. It is seen that the new approach considerably simplifies the pertinent analysis without losing the beautiful properties of the Tsallis–Mendes–Plastino formalism.

Blackbody radiation in a nonextensive scenario

An exact analysis of the N-dimensional blackbody radiation process in a nonextensive a la Tsallis scenario is performed for values of the nonextensives index in the range (0<q<1). The recently advanced “Optimal Lagrange Multipliers” technique has been employed. The results are consistent with those of the extensive, q=1 case. The generalization of the celebrated laws of Planck, Stefan-Boltzmann and Wien is investigated.

How fundamental is the character of thermal uncertainty relations

Abstract We discuss here two different information measures of the Tsallis type, and their associated probability distributions, in order to repeat the Mandelbrot Cramer–Rao steps that lead to a thermal uncertainty relation for exponential distributions. We deal first with the original Tsallis measure and discuss afterwards a second entropic measure associated with the concept of escort distribution. In neither case it is possible to re-obtain a thermal uncertainty relationship. We conclude therefore that the thermal uncertainty, as derived from the Cramer–Rao inequality, cannot be as fundamental as the quantum one.

On the equipartition and virial theorems

We revisit the celebrated equipartition and virial theorems from a non-extensive viewpoint. We show that both theorems still hold in a non-extensive scenario, independently of the value of Tsallis’ index q.

q-Thermostatistics and the black-body radiation problem

We give an exact information-theory treatment of the n-dimensional black-body radiation process in a non-extensive scenario. We develop a q-generalization of the laws of (i) Stefan–Boltzmann, (ii) Planck, and (iii) Wien, and show that conventional, canonical results are obtained at temperatures above 1K. Classical relationships between radiation, pressure, and internal energy are recovered (independently of the q value). Analyzing the particles’ density for q≈1, we see that the non-extensive parameter q introduces a fictitious chemical potential. We apply our results to experimental data on the cosmic microwave background and reproduce it with acceptable accuracy for different temperatures (each one associated to a particular q value).

Thermodynamics and the Tsallis variational problem

An appropriate redefinition of the Lagrange multipliers entering the q-MaxEnt variational treatment neatly reconciles his formalism with classical thermodynamics. The non-extensive approach is seen to reproduce classical results for all q-values.

Power-law distributions and Fisher's information measure

We show that thermodynamic uncertainties (TU) preserve their form in passing from Boltzmann–Gibbs’ statistics to Tsallis’ one provided that we express these TU in terms of the appropriate variable conjugate to the temperature in a nonextensive context.

Entropic uncertainty relation for power-law wave packets

For the power-law quantum wave packet in the configuration space, the variance of the position observable may be divergent. Accordingly, the information-entropic formulation of the uncertainty principle becomes more appropriate than the Heisenberg-type formulation, since it involves only the finite quantities. It is found that the total amount of entropic uncertainty converges to its lower bound in the limit of a large value of the exponent.