Statistics of precursors to fingering processes

– We present an analysis of the statistical properties of hydrodynamic field fluctuations which reveal the existence of precursors to fingering processes. These precursors are found to exhibit power law distributions, and these power laws are shown to follow from spatial q-Gaussian structures which are solutions to the generalized non-linear diffusion equation. Over the past fifteen years, a nonextensive variant to the Boltzmann-Gibbs formulation of Statistical Mechanics has been developed for the analysis of systems away from the equilibrium state [1] characterized by the appearance of non-exponential distributions and power laws. For instance, for diffusion type processes, non-exponential distributions are obtained from generalized Fokker-Planck type equations, and it has been shown, in a generic manner from a generalization of classical linear response theory, that such distributions may arise from first-principle considerations [2]. The goal of this letter is (i) to show that in fingering phenomena, patterns are preceded by precursor processes, and (ii) to present an analysis of the statistical properties of hydrodynamic field fluctuations in these precursors; we find power law distributions which are shown to follow from the solution of the generalized non-linear diffusion equation. Although we do not use nonextensive statistical mechanics to explain our results, we find mathematical similarities which suggest a possible connection as discussed in the concluding paragraph. The phenomena investigated here arise before the onset of fingering, a generic phenomenon that results from the destabilization of the interface between two fluids with different mobilities in systems such as a shallow layer or a porous medium, when the fluid with highest mobility is forced through the medium filled with the other fluid. Here before means that the constrained fluid is in a state where no fingering pattern is as yet visible, but where hydrodynamic field fluctuations are enhanced as precursors to the onset of fingering. The analytical form of the statistical properties of these precursors are compatible with the solution of the generalized diffusion equation [2] which has formally the same structure as the “porous media equation” [3], but where the diffusion coefficient depends on the solution of the equation. This leads to the fact that the diffusion process is classical in the sense that there is linear scaling with time, but the solutions are not Gaussian: they have the canonical q-exponential form [1]. To date at least to the best of our knowledge such precursor properties have not yet been obtained from laboratory measurements. Here we use two methods: (1) a mesoscopic