Random walk in chemical space of Cantor dust as a paradigm of superdiffusion

Abstract

We point out that the chemical space of a totally disconnected Cantor dust K(n) [Symbol: see text E(n) is a compact metric space C(n) with the spectral dimension d(s) = d(ℓ) = n > D, where D and d(ℓ) = n are the fractal and chemical dimensions of K(n), respectively. Hence, we can define a random walk in the chemical space as a Markovian Gaussian process. The mapping of a random walk in C(n) into K(n) [Symbol: see text] E(n) defines the quenched Lévy flight on the Cantor dust with a single step duration independent of the step length. The equations, describing the superdiffusion and diffusion-reaction front propagation ruled by the local quenched Lévy flight on K(n) [Symbol: see text] E(n), are derived. The use of these equations to model superdiffusive phenomena, observed in some physical systems in which propagators decay faster than algebraically, is discussed.

Anomalous diffusion of fluid momentum and Darcy-like law for laminar flow in media with fractal porosity

“…Therefore, the mapping of the problem for fractal with ν = 3 into the corresponding problem in F 3 ⊆ E 3 implies a simple change of spatial variables x i → i , where i is a power law function of x i , while index i = 1, 2.3 denotes the Cartesian axis (see Refs. [47][48][49]). However, if ν < 3, the number of independent mutually orthogonal fractal coordinates in F ν ⊆ E 3 is less then the number of Cartesian coordinates in the embedding space E 3 .…”

“…[43,[47][48][49][50][51][52]). Furthermore, the mapping F ⊂ E 3 → F ν ⊆ E 3 implies the use of metric partial derivatives ∂/∂ i instead of conventional partial derivatives ∂/∂x i [43].…”

“…This mapping implies a change from the Cartesian coordinates x i 1 Notice that in order to explain the increase of k L observed in this work the entrance length for the inverse Menger sponge should be more than 3 orders of magnitude larger than the entrance length for samples with the cubic pore network. in the embedding Euclidean space to the fractal coordinates i in the fractional dimensional space allied to the fractal [43,[47][48][49]. Generally, the number of effective spatial degrees of freedom of a random walker on a fractal ν = 2d − d s is less than or equal to the Euclidean dimension of the embedding Euclidean space E 3 [43].…”

“…(4a). Consequently, the gradient of function f (x j ) in the flow direction i = 1 reads as ∂ f /∂ 1 = (∂ 1 /∂x 1 ) −1 ∂ f /∂x 1 [43,[47][48][49][50][51][52]. Accordingly, for the one-dimensional laminar flow through a fractal the Darcy-like equation can be obtained by changing the spatial variable (x 1 → 1 ) and gradient operator (∂/∂ 1 → ∂/∂x 1 ) in the equation q = (k/μ)∂ P /∂x 1 derived in Ref.…”

Statistics of energy dissipation and stress relaxation in a crumpling network of randomly folded aluminum foils

“…The dynamics of random walk on a fractal network is governed by the network spectral dimension d s = 2D/D W , where D is the fractal dimension (e.g., mass, self-similarity, or Hausdorff dimension) of the network and D W is the random walk dimension [57][58][59]. If d s < 2, then D W > D and the random walk is recurrent, whereas if d s > 2, then D W < D and so the random walk is transient [58].…”