Effective one-dimensional description of confined diffusion biased by a transverse gravitational force

Diffusion of point-like non interacting particles in a two-dimensional (2D) channel of varying cross section is considered. The particles are biased by a constant force in the transverse direction. We apply our recurrence mapping procedure, which enables us to derive an effective one-dimensional (1D) evolution equation, governing the 1D density of the particles in the channel. In the limit of stationary flow, we arrive at an extended Fick-Jacobs equation, corrected by an effective diffusion coefficient D(x), depending on the longitudinal coordinate x. Our result is an approximate formula for D(x), involving also influence of the transverse force. Our calculations are verified on the stationary diffusion in a linear cone, which is exactly solvable.Comment: 10 pages, 7 figures, submitted in Phys. Rev.

Phase space reduction of the one-dimensional Fokker-Planck (Kramers) equation

A pointlike particle of finite mass m, moving in a one-dimensional viscous environment and biased by a spatially dependent force, is considered. We present a rigorous mapping of the Fokker-Planck equation, which determines evolution of the particle density in phase space, onto the spatial coordinate x. The result is the Smoluchowski equation, valid in the overdamped limit, m->0, with a series of corrections expanded in powers of m. They are determined unambiguously within the recurrence mapping procedure. The method and the results are interpreted on the simplest model with no field and on the damped harmonic oscillator.Comment: 13 pages, 1 figur

Survival probability (heat content) and the lowest eigenvalue of Dirichlet Laplacian

We study the survival probability of a particle diffusing in a two-dimensional domain, bounded by a smooth absorbing boundary. The short-time expansion of this quantity depends on the geometric characteristics of the boundary, whilst its long-time asymptotics is governed by the lowest eigenvalue of the Dirichlet Laplacian defined on the domain. We present a simple algorithm for calculation of the short-time expansion for an arbitrary "star-shaped" domain. The coefficients are expressed in terms of powers of boundary curvature, integrated around the circumference of the domain. Based on this expansion, we look for a Pad\'e interpolation between the short-time and the long-time behavior of the survival probability, i.e. between geometric characteristics of the boundary and the lowest eigenvalue of the Dirichlet Laplacian.Comment: Accepted in IJMP

UV-Cured Green Polymers for Biosensorics: Correlation of Operational Parameters of Highly Sensitive Biosensors with Nano-Volumes and Adsorption Properties

The investigated polymeric matrixes consisted of epoxidized linseed oil (ELO), acrylated epoxidized soybean oil (AESO), trimethylolpropane triglycidyl ether (RD1), vanillin dimethacrylate (VDM), triarylsulfonium hexafluorophosphate salts (PI), and 2,2-dimethoxy-2-phenylacetophenone (DMPA). Linseed oil-based (ELO/PI, ELO/10RD1/PI) and soybean oil-based (AESO/VDM, AESO/VDM/DMPA) polymers were obtained by cationic and radical photopolymerization reactions, respectively. In order to improve the cross-linking density of the resulting polymers, 10 mol.% of RD1 was used as a reactive diluent in the cationic photopolymerization of ELO. In parallel, VDM was used as a plasticizer in AESO radical photopolymerization reactions. Positron annihilation lifetime spectroscopy (PALS) was used to characterize vegetable oil-based UV-cured polymers regarding their structural stability in a wide range of temperatures (120–320 K) and humidity. The polymers were used as laccase immobilization matrixes for the construction of amperometric biosensors. A direct dependence of the main operational parameters of the biosensors and microscopical characteristics of polymer matrixes (mostly on the size of free volumes and water content) was established. The biosensors are intended for the detection of trace water pollution with xenobiotics, carcinogenic substances with a very negative impact on human health. These findings will allow better predictions for novel polymers as immobilization matrixes for biosensing or biotechnology applications