Discrete growth models on deterministic fractal substrate

Tang, Gang; Xun, Zhipeng; Wen, Rongji; Han, Kui; Xia, Hui; Hao, Dapeng; Zhou, Wei; Yang, Xiao‐Jun; Chen, Yuling

doi:10.1016/j.physa.2010.06.041

Cited by 29 publications

(35 citation statements)

References 20 publications

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“…In other words, Eqs. (28) and (29) which are obtained for the fractional Langevin equation are showing the same behavior as in the case of the EW and MH equations which were studied prior to this work [10,25].…”

Section: Limiting Time Cases For the Mean Square Surface Widthsupporting

confidence: 59%

“…[26,27] for an analytical study and Ref. [28] for a numerical approach; where the power-counting analysis was implemented to obtain the roughness and growth exponents for the equilibriumrestricted SOS model on a rough surface [27] and for the equilibrium-restricted curvature model on a Sierpinski gasket surface [26]. It is also interesting to see the contribution of the fractional derivative order towards the critical exponents for linear and nonlinear regimes, where it was shown by Leith [29] that in fractional growth of diffusion, the order of fractional derivatives plays the most important role in identifying the critical exponents, see Ref.…”

Section: Mean Square Widthmentioning

confidence: 99%

“…(28) and (29) show the dependence of the mean square surface width on the logarithm of time and the system size L, respectively. In other words, Eqs.…”

Section: Limiting Time Cases For the Mean Square Surface Widthmentioning

confidence: 99%

See 2 more Smart Citations

Scaling properties of fractional continuous growth equations

Masoudi¹,

Zahedifar²,

Farahani³

2013

Physica B: Condensed Matter

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Section: Limiting Time Cases For the Mean Square Surface Widthsupporting

confidence: 59%

Section: Mean Square Widthmentioning

confidence: 99%

See 1 more Smart Citation

Scaling properties of fractional continuous growth equations

Masoudi¹,

Zahedifar²,

Farahani³

2013

Physica B: Condensed Matter

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“…Other growth models have also been studied in substrates with the geometry of SCs [44][45][46][47]. The roughness oscillations were also observed in the simulations of ballistic deposition in SCs [44].…”

Section: Roughening Of the Infiltration Frontsmentioning

confidence: 97%

Scaling relations in the diffusive infiltration in fractals

Reis

2016

Phys. Rev. E

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In a recent work on fluid infiltration in a Hele-Shaw cell with the pore-block geometry of Sierpinski carpets (SCs), the area filled by the invading fluid was shown to scale as F ∼ t n , with n < 1/2, thus providing a macroscopic realization of anomalous diffusion [Filipovitch et al, Water Resour. Res. 52 5167 (2016)]. The results agree with simulations of a diffusion equation with constant pressure at one of the borders of those fractals, but the exponent n is very different from the anomalous exponent ν = 1/D W of single particle diffusion in the same fractals (D W is the random walk dimension). Here we use a scaling approach to show that those exponents are related as n = ν (D F − D B ), where D F and D B are the fractal dimensions of the bulk and of the border from which diffusing particles come, respectively. This relation is supported by accurate numerical estimates in two SCs and in two generalized Menger sponges (MSs), in which we performed simulations of single particle random walks (RWs) with a rigid impermeable border and of a diffusive infiltration model in which that border is permanently filled with diffusing particles. This study includes one MS whose external border is also fractal. The exponent relation is also consistent with the recent simulational and experimental results on fluid infiltration in SCs, and explains the approximate quadratic dependence of n on D F in these fractals. We also show that the mean-square displacement of single particle RWs has log-periodic oscillations, whose periods are similar for fractals with the same scaling factor in the generator (even with different embedding dimensions), which is consistent with the discrete scale invariance scenario. The roughness of a diffusion front defined in the infiltration problem also shows this type of oscillation, which is enhanced in fractals with narrow channels between large lacunas.

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“…Among previous theoretical investigations of continuum equations, as well as numerical simulations of discrete atomistic models, much more were performed on regular or Euclidean substrates with integer dimension, however, less were devoted to fractal substrates. As a result, there is no simple and clear understanding about the interplay between the dynamical growth rules of the system and the self-similarity of fractal structures until the recent works [12][13][14][15][16][17][18][19][20].…”

mentioning

confidence: 99%