Anomalous Transport In Heterogeneous Materials

Abstract

Mixing is an essential process in practically all fields of natural sciences and in industrial operations. But it is also a basic component of the field of non linear science where it is at the origin of experimental tools for diagnosis of reversibility. The notions of diffusion and mixing [1] are closely associated with the random walk of a Brownian particle which is such that the square of the distance R to the origin taken at a time t=0, averaged over many independent walks varies linearly with time where D_m is the molecular diffusion coefficient. This is a robust definition as it applies if the mean free paths, 1, are distributed randomly, provided the statistical law P(l) is limited enough (we will come back to this point later). The expression “black hole of statistical physics ” has been coined by Bouchaud & Georges to express the fact that a broad range of distributions of mean free paths all lead to a gaussian spreading. Applications of brownian statistical laws are too numerous to even be listed here. Two extreme cases deal with continuous geometries or regular lattices where the randomness comes from thermal noise and with geometrically disordered systems with deterministic flows ( hydrodynamic dispersion).