Fractals finitely ramified

It is interesting to compare eq. (15) with the results obtained on finitely ramified fractals by means of Green function renormalization [9-10]. It has been shown that the fractional uptake curve for a structure possessing fractal dimension dj, walk dimension d, and adsorbing from a reservoir at constant concentration c through an exchange manifold B (which represents the permeable boundary for treuisfer) possessing fractal dimension d scales as... 

This article also discusses a general scaling theory for sorption on bulk fractals and across fractal interfaces based on the results obtained by applying Green function renormalization to finitely ramified structures. Numerical simulations of batch sorption kinetics on infinitely ramified structures confirm the validity of the scaling expression eq. (17). 

In the case of fractal substrates, one has to distinguish between two main subclasses of structures, namely deterministic and random fractals. Within the class of deterministic fractals, one additionally has a subdivision in finitely and infinitely ramified fractals. Here, (either finite or infinite) ramification refers to the number of cut operations which are required to disconnect any given subset of the structure, the upper limit of which is independent of the chosen subset [7,8]. An example of a finitely ramified structure is the Sierpinski triangular lattice, whereas the Sierpinski square lattice is an example of an infinitely ramified structure. See Figs. 2(a) and 6 in Section 4 for the respective sketches of these structures in d = 2. 

Renormalisation group (RG) techniques have been applied to several finitely ramified structures, so that results are available for some deterministic fractals including Sierpinski triangular lattices [40-47] (for a comprehensive discussion see Ref. [48]). For infinitely ramified structures, there is no RG result available and one has to rely on numerically evaluating SAWs on these fractals (note, however, the study of Taguchi [49] of SAWs on Sierpinski square lattices). Nonetheless, even in the former case when RG results are available, it is instructive to apply munerical schemes as mentioned in the Introduction. 

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