Nonextensive statistics in viscous fingering

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.