  In the present follow-up article of a previous one [1] we illustrate the use
of the Unconventional Statistical Mechanics described and discussed in the
latter. This is done via the analysis, resorting to Renyi approach, of
experimental results in the case of so-called "anomalous" luminescence in
nanometric quantum wells in semiconductor heterostructures, and the so-called
"anomalous" cyclic voltammetry in fractal-like electrodes in microbatteries.
Also a purely theoretical analysis is done in the cases of an ideal gas and of
radiation comparing the conventional and unconventional approaches. In all of
these situations it is discussed which is the failure to satisfy the Criteria
of Efficiency and/or Sufficiency thus requiring to resort to the unconventional
approach, and what determines the value of the infoentropic index in each case,
and its dependence on the system characteristics. Moreover, on the basis of the
results we obtain, it is conjectured that the infoentropic index may satisfy
what we call a law defining a "path to sufficiency". PACS: 05.70.Ln; 82.20.Mj;
82.20.Db Keywords: Renyi Statistics; Escort Probability; Fractal Structured
Systems; Power Law Properties.
