  Competing styles in Statistical Mechanics have been introduced to investigate
physico-chemical systems displaying complex structures, when one faces
difficulties to handle the standard formalism in the well established
Boltzmann-Gibbs statistics. After a brief description of the question, we
consider the particular case of Renyi statistics whose use is illustrated in a
study of the question of the ''anomalous'' (non-Fickian) diffusion that it is
involved in experiments of cyclic voltammetry in electro-physical chemistry. In
them one is dealing with the fractal-like structure of the thin film morphology
present in eletrodes in microbatteries which leads to fractional-power laws for
describing voltammetry measurements and in the determination of the interphase
width derived using atomic force microscopy. The fractional-powers associated
to these quantities are related to each other and to the statistical fractal
dimension, and can be expressed in terms of a power index, on which depends
Renyi's statistical mechanics. It is clarified the important fact that this
index, which is limited to a given interval, provides a measure of the
microroughness of the electrode surface, and is related to the dynamics
involved, the non-equilibrium thermodynamic state of the system, and to the
experimental protocol. \\
