Diffusive transport in networks built of containers and tubes

TL;DR

This paper introduces analytical and numerical methods to efficiently model diffusive transport of particles in complex networks of containers and tubes, capturing dynamic behaviors like wave-like concentration variations.

Contribution

The authors developed a set of rate equations reducing the complex diffusion problem to manageable integro-differential equations, enabling efficient analysis of large, intricate networks.

Findings

Networks can exhibit wave-like concentration dynamics.

The method simplifies complex diffusion problems to first-order equations.

Transport behavior varies significantly with network geometry.

Abstract

We developed analytical and numerical methods to study a transport of non-interacting particles in large networks consisting of M d-dimensional containers C_1,...,C_M with radii R_i linked together by tubes of length l_{ij} and radii a_{ij} where i,j=1,2,...,M. Tubes may join directly with each other forming junctions. It is possible that some links are absent. Instead of solving the diffusion equation for the full problem we formulated an approach that is computationally more efficient. We derived a set of rate equations that govern the time dependence of the number of particles in each container N_1(t),N_2(t),...,N_M(t). In such a way the complicated transport problem is reduced to a set of M first order integro-differential equations in time, which can be solved efficiently by the algorithm presented here. The workings of the method have been demonstrated on a couple of examples: networks involving three, four and seven containers, and one network with a three-point junction. Already simple networks with relatively few containers exhibit interesting transport behavior. For example, we showed that it is possible to adjust the geometry of the networks so that the particle concentration varies in time in a wave-like manner. Such behavior deviates from simple exponential growth and decay occurring in the two container system.