Singular dynamical renormalization group and biased diffusion on fractals

Abstract

An exact renormalization group describes extremely slow, logarithmic diffusion in the presence of a biasing field on ramified fractal structures. Recursion equations are singular at the fixed point and the standard analysis to extract asymptotic behaviors has to be reconsidered. The model reproduces mechanisms working for biased diffusion on percolation clusters. For 1 ~d structures, logarithmic diffusions generalizing that discussed by Sinai [Theory Probab. Its Appl. 27, 256 (1982)] are obtained by the same methods.PACS numbers: 64.60. Ak, 71.55Jv Slow diffusion, with the average displacement R, growing in time t, as some power of lnf, has been exactly demonstrated by Sinai for a particle hopping on a onedimensional chain and subject at each site to an independent random bias [1]. There is numerical evidence that logarithmic diffusion could replace the anomalous power law one, in fractal structures, like the infinite incipient cluster (IIC) of percolation, when the diffusing particle is subject to the action of an external biasing field [2,3]. In this case the field is not random, but the structure of the fractal conspires with it and determines a localization effect, e.g., by pushing the particle towards dangling ends [4].Interestingly enough, the current understanding of such phenomena relies almost entirely on the picture of a particle diffusing in one dimension. This is the situation of the Sinai model, whose behavior is indeed easily understood once realized that, over a distance R along the chain, a potential barrier oc J~R develops by adding the local random biases with zero average. On the basis of Arrhenius law, overcoming such a barrier requires a time cce , from which R<x(\nt) 2 follows. Another mechanism of slow diffusion under bias is that arising in a comblike structure with teeth of variable length, along which a bias field pushes the particle towards the tips. A suitable power law distribution of teeth lengths is then easily shown to imply waiting times, which make diffusion along the comb basis logarithmic [5]. In this case again the model is basically one dimensional, and still very far from the complexity of fully self-similar structures, like the IIC of percolation. For example, the linear character of the teeth is essential in order to derive the waiting times associated with them.While it is intuitive to ascribe the bias induced logarithmic diffusion in a fractal to mechanisms of trapping into dangling ends, so far, a proper mathematical description of such phenomena within a fully self-similar geometry has never been accomplished. In the present Letter we address this issue on deterministic fractal models by a suitable dynamical renormalization group (RG) approach.While the RG is the most natural method to treat dynamics on self-similar structures [6], one should realize that logarithmic long time behavior must somehow imply a peculiar, singular structure for an RG transformation. Indeed, standard dynamical RG yields the time rescaling factor, / ,/v , under length rescaling / [7]. This ...

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