A proof of the quantum H-theorem taking into account nonextensive effects on the quantum entropy S Q q is shown. The positiveness of the time variation of S Q q combined with a duality transformation implies that the nonextensive parameter q lies in the interval [0,2]. It is also shown that the equilibrium states are described by quantum q-power law extensions of the Fermi-Dirac and Bose-Einstein distributions. Such results reduce to the standard ones in the extensive limit, thereby showing that the nonextensive entropic framework can be harmonized with the quantum distributions contained in the quantum statistics theory.

We study theoretically the transmission spectra in one-dimensional photonic quasicrystals, made up of SiO 2 (A) and TiO 2 (B) materials, organized following the Octonacci sequence, where the nth-stage of the multilayer S n is given by the rule S n = S n−1 S n−2 S n−1 , for n ≥ 3 and with S 1 = A and S 2 = B. The expression for transmittance was obtained by employing a theoretical calculation based in the transfer-matrix method. To normally incident waves, we observe that, for a same generation, the transmission spectra for TE and TM waves are equal, at least qualitatively, and they present a scaling property where a self-similar behavior is obtained, as an evidence that these spectra are fractals. The spectra show regions where the omnidirectional band gaps emerges for specific generations of Octonacci photonic structure, except to TM waves. For TE waves, we note that all of them have the almost same width, for different generations. We also report the localization of modes as a consequence of the quasiperiodicity of the heterostructure.

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The localization and scaling behaviour of quasiperiodic structures are studied for a geometry where the magnetization is perpendicular to the interfaces of the superlattices. Numerical results for the bulk and surface spin waves in the magnetostatic regime are presented for the Fibonacci, Thue-Morse and period-doubling sequences. The results are obtained for both ferromagnetic and antiferromagnetic ordering by using the transfer-matrix method. Interesting features of the localized modes are shown for Fe, EuS and MnF2 .

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