Diffusion in the Presence of Scale-Free Absorbing Boundaries

TL;DR

This paper investigates the behavior of diffusing particles near scale-free absorbing surfaces, revealing power-law dependencies characterized by a single exponent, which also relates to polymer conformations and entropic forces.

Contribution

It provides a detailed analysis of diffusion near scale-free surfaces and determines the key exponent governing the behavior, linking it to polymer physics and entropic forces.

Findings

Derived the exponent $$ for diffusion near scale-free surfaces.

Linked the diffusion exponent to polymer conformations and entropic forces.

Provided numerical values and insights into polymer behavior near such surfaces.

Abstract

Scale-free surfaces, such as cones, remain unchanged under a simultaneous expansion of all coordinates by the same factor. Probability density of a particle diffusing near such absorbing surface at large time approaches a simple form that incorporates power-law dependencies on time and distance from a special point, such as apex of the cone, which are characterized by a single exponent $\eta$. The same exponent is used to describe the number of spatial conformations of long ideal polymer attached to the special point of a repulsive surface of the same geometry and can be used in calculation of entropic forces between such polymers and surfaces. We use the solution of diffusion equation near such surfaces to find the numerical values of $\eta$, as well as to provide some insight into the behavior of ideal polymers near such surfaces.