Multiple logarithmic oscillations for statistical fractals on ultrametric spaces with application to recycle flows in hierarchical porous media

Abstract

The interference of logarithmic oscillations characteristic to statistical fractal and ultrametric structures is analyzed. We introduce an ultrametric model consisting in a hierarchy of branches for which the probability distribution ξ̃(n) of the total number, n, of branches has a long tail of the inverse power law type ξ̃(n) ∼ A(ln n)n -(1 + H) as n → ∞ where H is a fractal exponent characteristic for the ultrametric structure and A(ln n) is a periodic function of ln n. A comparison is made with a statistical fractal derived by means of the Shlesinger-Hughes stochastic renormalization approach. We consider a positive random variable, X, selected from a narrow unimodal probability density with finite moments. A process of stochastic amplification of the random variable is introduced which leads to a probability density of the amplified variable with a long tail: P̃(X) dX ∼ dXX-(1 + ℋ)B[ln (X/Xm)] as X → ∞ where ℋ is another fractal exponent, Xm is a cutoff value of the random variable X and B[ln (X/Xm)] is a periodic function of ln (X/Xm). Finally, the interaction between these two types of logarithmic oscillations is analyzed by assuming that the stochastic amplification of the random variable X takes place on an ultrametric structure. The final probability density of X, P*(X) dX, displays a phenomenon of amplitude modulation P*(X) dX ∼ d[ln (X/Xm)] [ln (X/Xm)](-1 + H)Ξ as X → ∞ where Ξ = Ξ(1) + Ξ(2) is made up of two additive contributions: a periodic function Ξ(1)[ln (X/Xm)] of ln (X/Xm) and a superposition Ξ(2) {ln (X/Xm), ln [ln (X/Xm)]} of periodic functions of ln (X/Xm) modulated by much slower periodic functions in ln [ln (X/Xm)]. The model is applied for the analysis of recycle flows in porous media. We assume that a hierarchical structure of pores exists which corresponds to an ultrametric space and that the recycle flow leads to a stochastic amplification of the residence time of fluid elements in the system. We show that the probability density of the residence time has an asymptotic behavior similar to that of P*(X) dX and investigate the possibilities of measurement of multiple logarithmic oscillations by means of tracer experiments. A physical interpretation of multiple logarithmic oscillations is given in the case of flow systems: they are generated by two delayed feedback processes occurring in two different logarithmic time scales, ln t and ln ln t. The fast delayed feedback process in ln t is given by the recycle flows corresponding to a given level of the porous structure whereas the slow delayed feedback process in ln ln t is due to the exchange of fluid among the different levels of the hierarchical structure of pores.