Mass Exchange Processes with Input

TL;DR

This paper explores mass exchange processes with input, analyzing different growth regimes, providing exact solutions for specific cases, and examining the effects of localized input on cluster distributions.

Contribution

It introduces a detailed analysis of mass exchange with input, including exact solutions and the impact of localized input on cluster distributions.

Findings

Identifies conditions for continuous growth, gelation, and instantaneous gelation.

Provides an exact solution for exchange rates proportional to the product of masses.

Shows stationary distributions with algebraic tails under certain conditions.

Abstract

We investigate a system of interacting clusters evolving through mass exchange and supplemented by input of small clusters. Three possibilities depending on the rate of exchange generically occur when input is homogeneous: continuous growth, gelation, and instantaneous gelation. We mostly study the growth regime using scaling methods. An exchange process with reaction rates equal to the product of reactant masses admits an exact solution which allows us to justify the validity of scaling approaches in this special case. We also investigate exchange processes with a localized input. We show that if the diffusion coefficients are mass-independent, the cluster mass distribution becomes stationary and develops an algebraic tail far away from the source.