S. Martı́nez

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.

General thermostatistical formalisms, invariance under uniform spectrum translations, and Tsallis q-additivity

The recent proposal by Tsallis of a generalized thermostatistics devised to treat systems endowed with long-range interactions, long-range memory effects, or a fractal-like relevant phase space, has raised interesting and profound issues concerning the properties of general thermostatistical formalisms. In the present paper we identify families of thermostatistical formalisms that share some fundamental characteristics. The canonical ensembles invariance under uniform translations of the Hamiltonians energy spectrum is shown to be a universal property verified by any thermostatistical formalism based upon linear mean energy constraints. We also provide multiparametric families of entropies exhibiting Tsallis q-additivity law.

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.

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.

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.

Dark magnetism revisited

Bacry [Phys. Lett. B 317 (1993) 523] showed that, on the basis of the deformed Poincare group, special relativity yields a non-additive energy for large systems, i.e., a total energy (of the Universe) which would not be proportional to the number of particles. He consistently argued that this effect could explain (part of) the so-called dark matter. By considering non-interacting spins in the presence of an external magnetic field, it was shown in Portesi et al. [Phys. Rev. E 52 (1995) R3317] that Tsallis’ non-extensive thermostatistics could account for a possible “dark” magnetism (the apparent number of particles being different from the actual one). The work of Pennini et al. [Physica A 258 (1998) 446]; Tsallis et al. [Physica A 261 (1998) 534] uses the so-called “generalized” expectation values, that were for some time considered indispensable in dealing with Tsallis’ formalism. Lately, a different sort of expectation values has been regarded as being superior to the old generalized ones [Pennini et al., Physica A 258 (1998) 446; Tsallis et al., Physica A 261 (1998) 534]. We revisit the dark magnetism question in the light of this new way of computing mean values.

A chemical-physics test for nonextensivity in thermodynamics

Abstract An ideal mixture of parahydrogen (with nuclear spin K=0) and orthohydrogen (with K=1), in statistical weights 1/4 and 3/4, respectively, is used as a test ground for the existence of nonextensivity in chemical physics. We report on a new bound on the nonextensivity parameter q−1 that characterizes generalized thermostatistics a la Tsallis. This bound is obtained on the basis of laboratory measurements of the specific heat of hydrogen. Suggestions are advanced for the performance of improved measurements.An ideal mixture of parahydrogen (with nuclear spin K=0) and orthohydrogen (with K=1), in statistical weights 1/4 and 3/4, respectively, is used as a test ground for the existence of non-extensivity in chemical physics. We report on a new bound on the non extensivity parameter q - 1 that characterizes generalized thermostatistics a la Tsallis. This bound is obtained on the basis of laboratory measurements of the specific heat of hydrogen. Suggestions are advanced for the performance of improved measurements.