Universality in edge-source diffusion dynamics

TL;DR

This paper demonstrates that in edge-source diffusion, the integrated concentration follows a universal pattern characterized by a specific time-scale, with distinct short- and long-time behaviors that are largely geometry-independent.

Contribution

The study introduces a universal framework for edge-source diffusion dynamics, revealing a characteristic time-scale and universal asymptotic behaviors.

Findings

Universal dependence of N(t) on domain geometry and diffusion parameters

Short-time behavior follows a universal square-root law

Long-time saturation of N(t) is exponential and weakly geometry-dependent

Abstract

We show that in edge-source diffusion dynamics the integrated concentration N(t) has a universal dependence with a characteristic time-scale tau=(A/P)^2 pi/(4D), where D is the diffusion constant while A and P are the cross-sectional area and perimeter of the domain, respectively. For the short-time dynamics we find a universal square-root asymptotic dependence N(t)=N0 sqrt(t/tau) while in the long-time dynamics N(t) saturates exponentially at N0. The exponential saturation is a general feature while the associated coefficients are weakly geometry dependent.