Heterogeneous diffusion in comb and fractal grid structures

Abstract

We give an exact analytical results for diffusion with a power-law position dependent diffusion coefficient along the main channel (backbone) on a comb and grid comb structures. For the mean square displacement along the backbone of the comb we obtain behavior x 2 (t) ∼ t 1/(2−α) , where α is the powerlaw exponent of the position dependent diffusion coefficient D(x) ∼ |x| α . Depending on the value of α we observe different regimes, from anomalous subdiffusion, superdiffusion, and hyperdiffusion. For the case of the fractal grid we observe the mean square displacement, which depends on the fractal dimension of the structure of the backbones, i.e., x 2 (t) ∼ t (1+ν)/(2−α) , where 0 < ν < 1 is the fractal dimension of the backbones structure. The reduced probability distribution functions for both cases are obtained by help of the

Information Geometry of Nonlinear Stochastic Systems

“…Multiplicative noise would also be interesting, but the situation becomes somewhat more complicated as a deterministic force could appear as −x n as here, but the noise ξ could appear as ξx m , with m not necessarily the same as n. Fractional noise would also be worth pursuing, although this would be challenging both analytically and numerically. Finally, it will also be interesting to extend our work to analyse a heterogeneous, linear system [35][36][37], and to explore applications of our work to estimators.…”

Time-averaging and emerging nonergodicity upon resetting of fractional Brownian motion and heterogeneous diffusion processes

“…with > 0, was solved by Pattle 174 (see also some recent "reincarnations" 175,176 ). Contemporary models of diffusion with space-dependent diffusion coefficients 154,[177][178][179][180][181][182][183][184] -with HDPs being a specific example that assumes the functional diffusivity form (17)can be used to describe (•) the non-Brownian diffusion in crowded, porous, and heterogeneous media [185][186][187][188][189][190][191][192][193][194][195][196][197][198][199][200][201][202] (such as densely macromolecularly crowded cell cytoplasm), (•) the reduction of a critical "patch size" required for survival of a population in the case of heterogeneous diffusion of its individuals 181 , (•) diffusion in heterogeneous comb-like and fractal structures 182 , (•) escalated polymerization of RNA nucleotides by a spatially confined thermal (and diffusivity) gradient in thermophoresis setups 203 , (•) motion of active particles with space-dependent friction in potentials [both of power-law forms] 204 , and (•) transient subdiffusion in disordered space-inhomogeneous quantum walks 205,206 . We mention also a class of diffusion models with (•) particle-spreading scenarios with concentration-dependent power-law-like diffusivity (20) 175,207 , (•) concentration-dependent dispersion in the population dynamics, with a nonlinear dependence of mobility on particle density, D(ρ) ∼ ρ κ (yielding a migration from more-to less-populated areas) [208]…”

Getting Acquainted with the Fractional Laplacian

“…A very interesting phenomenon to observe is that the geometry of the diffusion medium can naturally transform classical diffusion into an anomalous one. This feature can be very well understood by an elegant model, introduced in [10] (see also [105] and the references therein for an exhaustive account of the research in this direction) consisting in random walks on a "comb", that we briefly reproduce here for the facility of the reader. Given ε > 0, the comb may be considered as a transmission medium that is the union of a "backbone" B := R × {0} with the "fingers" P k := {εk} × R, namely Figure 6.…”