  In this letter, we address the relationship between the statistical
fluctuations of grain displacements for a full quasistatic plane shear
experiment, and the corresponding anomalous diffusion exponent, $\alpha$. We
experimentally validate a particular case of the so-called Tsallis-Bukman
scaling law, $\alpha = 2 / (3 - q)$, where $q$ is obtained by fitting the
probability density function (PDF) of the measured fluctuations with a
$q$-Gaussian distribution, and the diffusion exponent is measured independently
during the experiment. Applying an original technique, we are able to evince a
transition from an anomalous diffusion regime to a Brownian behavior as a
function of the length of the strain-window used to calculate the displacements
of grains in experiments. The outstanding conformity of fitting curves to a
massive amount of experimental data shows a clear broadening of the fluctuation
PDFs as the length of the strain-window decreases, and an increment in the
value of the diffusion exponent - anomalous diffusion. Regardless of the size
of the strain-window considered in the measurements, we show that the
Tsallis-Bukman scaling law remains valid, which is the first experimental
verification of this relationship for a classical system at different diffusion
regimes. We also note that the spatial correlations show marked similarities to
the turbulence in fluids, a promising indication that this type of analysis can
be used to explore the origins of the macroscopic friction in confined granular
materials.
