Anomalous Diffusion in Systems with Concentration-Dependent Diffusivity: Exact Solutions and Particle Simulations

Abstract

We explore the anomalous diffusion that may arise as a result of a concentration dependent diffusivity. The diffusivity is taken to be a power law in the concentration, and from exact analytical solutions we show that the diffusion may be anomalous, or not, depending on the nature of the initial condition. The diffusion exponent has the value of normal diffusion when the initial condition is a step profile, but takes on anomalous values when the initial condition is a spike. Depending on the sign of the exponent in the diffusivity the diffusive behavior will then be either sub-diffusive or super-diffusive. We introduce a particle model that behaves according to the non-linear diffusion equation in the macroscopic limit. This correspondence is demonstrated via kinetic theory, i.e. by means of Chapman-Kolmogorov equation, as well as by direct simulations.

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 .…”

“…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][209][210] , as well as (•) similar nonlinear equations 10 for porous-media dynamics 211 , no...…”

CO2 solubility and diffusivity in 1-ethyl-3-methylimidazolium cation-based ionic liquids; isochoric pressure drop approach

“…This may address the ambiguity in the literature regarding the dependency of the diffusivity coefficient on the concentration. Based on our finding, D could exhibit both behaviors; it is proportional to the concentration at the early stages of the diffusion, such as what could be seen within 0 < t < 2/3 h in Figure 5(a), and is inversely proportional to the concentration after that, such as what could be seen at t > 2/3 h. Therefore, the anomaly in the diffusivity coefficient, which has been the subject of some papers in the literature and is explained by Hansen et al [71] can be confirmed and may be addressed if it is taken to be a variable rather than a constant value. Here, we emphasize that the variable diffusivity coefficients in Figure 5 are not taken as a function of concentration.…”

Hyperballistic Superdiffusion and Explosive Solutions to the Non-Linear Diffusion Equation

“…For negative c this will always lead to sub-diffusion. We have recently shown that in d 1 there are exact solutions with positive c as well [40], which still satisfy Eq. 2, thus yielding superdiffusion with 1/2 < τ < 1 as c < 1 always.…”

“…This defines a Wiener process with η as a random variable with 〈η〉 0 and 〈η 2 〉 1. Now, following the same steps as in [40,43] we use the standard Chapman-Kolmogorov, or master equation, to derive the following Fokker-Planck equation for the particle concentration C(r, t)…”