Slow Lévy flights.

Among Markovian processes, the hallmark of Lévy flights is superdiffusion, or faster-than-Brownian dynamics. Here we show that Lévy laws, as well as Gaussian distributions, can also be the limit distributions of processes with long-range memory that exhibit very slow diffusion, logarithmic in time. These processes are path dependent and anomalous motion emerges from frequent relocations to already visited sites. We show how the central limit theorem is modified in this context, keeping the usual distinction between analytic and nonanalytic characteristic functions. A fluctuation-dissipation relation is also derived. Our results may have important applications in the study of animal and human displacements.

Diffusion in narrow channels on curved manifolds.

In this work, we derive a general effective diffusion coefficient to describe the two-dimensional (2D) diffusion in a narrow and smoothly asymmetric channel of varying width, embedded on a curved surface, in the simple diffusion of non-interacting, point-like particles under no external field. To this end, we extend the generalization of the Kalinay-Percus' projection method [J. Chem. Phys. 122, 204701 (2005); Phys. Rev. E 74, 041203 (2006)] for the asymmetric channels introduced in [L. Dagdug and I. Pineda, J. Chem. Phys. 137, 024107 (2012)], to project the anisotropic two-dimensional diffusion equation on a curved manifold, into an effective one-dimensional generalized Fick-Jacobs equation that is modified according to the curvature of the surface. For such purpose we construct the whole expansion, writing the marginal concentration as a perturbation series. The lowest order in the perturbation parameter, which corresponds to the Fick-Jacobs equation, contains an additional term that accounts for the curvature of the surface. We explicitly obtain the first-order correction for the invariant effective concentration, which is defined as the correct marginal concentration in one variable, and we obtain the first approximation to the effective diffusion coefficient analogous to Bradley's coefficient [Phys. Rev. E 80, 061142 (2009)] as a function of the metric elements of the surface. In a straightforward manner, we study the perturbation series up to the nth order, and derive the full effective diffusion coefficient for two-dimensional diffusion in a narrow asymmetric channel, with modifications according to the metric terms. This expression is given as D(ξ)=D(0)/w'(ξ)√(g(1)/g(2)){arctan[√(g(2)/g(1))(y(0)'(ξ)+w'(ξ)/2)]-arctan[√(g(2)/g(1))(y(0)'(ξ)-w'(ξ)/2)]}, which is the main result of our work. Finally, we present two examples of symmetric surfaces, namely, the sphere and the cylinder, and we study certain specific channel configurations on these surfaces.

Diffusion in two-dimensional conical varying width channels: comparison of analytical and numerical results.

This study is devoted to the unbiased motion of a point-like brownian particle in two-dimensional tilted asymmetric channels of varying width formed by straight walls. An effective one-dimensional description in terms of the generalized Fick-Jacobs equation is used to derive formulas that yield the particle's effective diffusion coefficient as a function of the geometric parameters of the channel. To such end, we use the formulas obtained by Bradley [Phys. Rev. E 80, 061142 (2009)] and by Dagdug and Pineda [J. Chem. Phys. 137, 024107 (2012)] to study two-dimensional diffusion in narrow and smoothly asymmetric channels of varying width. Comparison with brownian dynamics simulation results allows us to establish the domain of applicability of both the one-dimensional description and the effective diffusion coefficient formulas.

Projection of two-dimensional diffusion in a curved midline and narrow varying width channel onto the longitudinal dimension.

This study focuses on the derivation of a general effective diffusion coefficient to describe the two-dimensional (2D) diffusion in a narrow and smoothly asymmetric channel of varying width, in the simple diffusional motion of noninteracting pointlike particles under no external field. We present a generalization to the case of an asymmetric channel using the projection method introduced earlier by Kalinay and Percus [J. Chem. Phys. 122, 204701 (2005); and Phys. Rev. E 74, 041203 (2006)] to project the 2D diffusion equation into an effective one-dimensional generalized Fick-Jacobs equation. The expression for the diffusion coefficient given in Eq. (23) is our main result. This expression is a more general effective diffusion coefficient for narrow channels in 2D, which contains the well-known previous results as special cases, namely, those obtained by Bradley [Phys. Rev. E 80, 061142 (2009)], and more recently by Berezhkovskii and Szabo [J. Chem. Phys. 135, 074108 (2011)]. Finally, we study some specific 2D asymmetric channel configurations to test and show the broader applicability of Eq. (23).

Diffusion in periodic two-dimensional channels formed by overlapping circles: comparison of analytical and numerical results.

We study two-dimensional diffusion in a channel formed by periodic overlapping circles. Periodic variation of the channel width leads to the slowdown of diffusion along the channel axis. There are several approximate approaches, which allow one to analyze the slowdown. We use these approaches to derive five expressions for the effective diffusion coefficient of a point Brownian particle in the channel. To check the accuracy of the expressions we compare their predictions with the effective diffusion coefficient obtained from Brownian dynamics simulations.