Assortative Exchange Processes

TL;DR

This paper investigates maximally assortative exchange processes where only clusters of equal size exchange monomers, analyzing their behavior through mean-field theory, scaling, and heuristic methods across different spatial dimensions and system sizes.

Contribution

It introduces a detailed analysis of maximally assortative exchange processes, including mean-field solutions and scaling approaches for low-dimensional systems and finite systems with localized input.

Findings

Mean-field theory accurately describes high-dimensional systems.

Scaling and heuristic methods reveal behavior in low-dimensional systems.

Finite systems with localized input exhibit unique dynamics.

Abstract

In exchange processes clusters composed of elementary building blocks, monomers, undergo binary exchange in which a monomer is transferred from one cluster to another. In assortative exchange only clusters with comparable masses participate in exchange events. We study maximally assortative exchange processes in which only clusters of equal masses can exchange monomers. A mean-field framework based on rate equations is appropriate for spatially homogeneous systems in sufficiently high spatial dimension. For diffusion-controlled exchange processes, the mean-field approach is erroneous when the spatial dimension is smaller than critical; we analyze such systems using scaling and heuristic arguments. Apart from infinite-cluster systems we explore the fate of finite systems and study maximally assortative exchange processes driven by a localized input.