Anomalous diffusion with absorbing boundary

Abstract

In a very long Gaussian polymer on time scales shorter that the maximal relaxation time, the mean squared distance travelled by a tagged monomer grows as ∼ t1/2 . We analyze such sub-diffusive behavior in the presence of one or two absorbing boundaries and demonstrate the differences between this process and the sub-diffusion described by the fractional Fokker-Planck equation. In particular, we show that the mean absorption time of diffuser between two absorbing boundaries is finite. Our results restrict the form of the effective dispersion equation that may describe such sub-diffusive processes.

Fractional Brownian motion approach to polymer translocation: The governing equation of motion

“…While this latter property may readily be verified in numeric experiments, the unambiguous determination of the precise functional shape of W (s,t) is rather difficult as a result of progressively deteriorating statistics of the distribution at late times. Meanwhile, several recent publications [12][13][14], devoted to the unbiased translocation dynamics of a Gaussian onedimensional (1D) chain [12] as well as to that of a twodimensional (2D) self-avoiding chain [13], have validated the subdiffusive behavior of s 2 (t) with an exponent α 0.8. Nonetheless, these new findings cast serious doubts as to whether the FFPE indeed provides an adequate description of nondriven translocation dynamics:…”

Eavesdropper Localization in Random Walk Channels

“…The solutions are also well known and some of which can be found in [9]. As before, the PDF of the free molecule can be expressed as a superposition of the original boundary-free RW process P (x, t) and mirrored RW processes.…”

“…Most existing applied physics and statistics research has focused on singular or symmetrical boundaries [9], and asymmetric boundaries are more complex and received less attention in literature. In this paper, we consider the general case of two absorbing walls placed at x=-K and x=+L, where K = L and the parameters can take on any values.…”

“…By exploiting the linear property of the RW process, the PDF of the free molecule can be expressed as a superposition of the original boundary-free RW process P (x, t) and another RW process starting at a negative mirror location P (x−2L, t) (see Fig. 2a), such that the PDF of the free molecules is [9] P 1 (x, L, t) = P (x, t) + P (x − 2L, t) reflect,…”

“…In this paper, we consider the general case of two absorbing walls placed at x=-K and x=+L, where K = L and the parameters can take on any values. The geometric reasoning (similar to the symmetric case [9]) is based on cancelling out the molecule residue at the walls, so that the boundary conditions of zero flux and zero concentration are maintained. This is achieved with the aid of negative and positive mirror pulses, which are transmitted from a cascade of mirrors, which extend in distance from 0 to +∞ on the positive axis (wall L side) and from 0 to -∞ on the negative axis (wall K side).…”