Nonextensive statistics precursors

– 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 non-linear porous media 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 form 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 to present an analysis of the statistical properties of hydrodynamic field fluctuations in fingering processes where we find non-exponential and power law distributions which are shown to be compatible with the solution of the generalized non-linear diffusion equation. More precisely, 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 approach, the lattice Boltzmann simulation method [4], which is based on a kinetic theoretical analysis where the macroscopic description is not pre-established, and (2) a phenomenological