Diffusion on random hierarchical structures

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Abstract
The authors study diffusion on a statistically self-similar fractal substrate with random transition rates possessing a hierarchical structure. The master equation is solved analytically by means of a recursion technique and different choices of random hierarchies for the transition rates are considered, leading to algebraic, stretched exponential or exponential-logarithmic behaviour for the moments and the correlation decay. Making use of the central limit theorem, analytical expressions for the fluctuation corrections to lowest order in the relative variances are derived for the first two cases where the behaviour is qualitatively the same as in the systems without fluctuations. The sign of the corrections turns out to depend on the temporal and spatial scale at which fluctuations are externally suppressed; they are negative if the diffusion is taking place at smaller scales, positive otherwise. The relevance to diffusion in a turbulent medium (intermittency corrections) is discussed.

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