  Measurements in turbulent flows have revealed that the velocity field in
nonequilibrium systems exhibits $q$-exponential or power law distributions in
agreement with theoretical arguments based on nonextensive statistical
mechanics. Here we consider Hele-Shaw flow as simulated by the Lattice
Boltzmann method and find similar behavior from the analysis of velocity field
measurements. For the transverse velocity, we obtain a spatial $q$-Gaussian
profile and a power law velocity distribution over all measured decades. To
explain these results, we suggest theoretical arguments based on Darcy's law
combined with the non-linear advection-diffusion equation for the concentration
field. Power law and $q$-exponential distributions are the signature of
nonequilibrium systems with long-range interactions and/or long-time
correlations, and therefore provide insight to the mechanism of the onset of
fingering processes.
