Pseudo-scaling and multifractality

Random systems with stretched (and fluctuating) exponential form of probability density functions (PDFs), such as turbulence (distribution of velocity differences) and percolation (visiting frequency in percolative diffusion), are studied. Dimensionless moments are defined as Fnp(r) = ƒp/ƒnp/n, where ƒp(r) is the standard moments of order p (p > n). Pseudo-scaling (PS) is defined as existence of power relationships Fnp Fnqρ, where exponent ρnpq depends on p, q and n. It is shown that for large enough n, p, q, the PS takes place even if the ordinary scaling is broken. It is shown that there are two kinds of the asymptotical pseudo-scaling: with ρnpq = (p−n)/(q−n) and with ρnpq = (p/q) [ln(p/n)/(ln(q/n)]. If ordinary scaling also takes place in the systems then these two kinds of the pseudo-scaling lead to two kinds of corresponding ordinary scaling laws. A speculation concerning genesis of the stretched exponential PDFs is briefly discussed. Phase transition from random fractal structures to homogeneity is studied in the framework of the pseudo-scaling approach and it is shown that multifractality of passive scalar concentration has the pseudo-scaling form of the second kind near the transition (whereas for systems far from the transition the multifractality has the pseudo-scaling form of the first kind). The condition of compatibility of the pseudo-scaling and of the generalized dimension approaches is also investigated. Agreement between the theoretical results and experimental data of different authors is established both for the situations where ordinary scaling takes place and for situations where ordinary scaling does not take place.