Aggregation-Fragmentation Processes with Broken Detailed Balance

TL;DR

This paper investigates aggregation-fragmentation processes with broken detailed balance, providing exact solutions for steady states and analyzing the effects of mass-dependent fragmentation rates, including a shattering transition.

Contribution

It offers an exact solution for nonequilibrium steady states in aggregation-fragmentation models and explores the impact of mass-dependent fragmentation rates on system behavior.

Findings

Steady states can be derived from an exact Laplace transform solution despite broken detailed balance.

When $eta eq 1$, the models exhibit steady states similar to the mass-independent case.

A shattering transition with continuous mass loss occurs for $eta<0$.

Abstract

We study aggregation-fragmentation processes in which pairs of clusters can aggregate, and each cluster can break into two fragments. If the rates of aggregation and fragmentation do not depend on the masses, detailed balance does not hold, but nonequilibrium steady states can still be deduced from an exact solution for the Laplace transform. For models in which aggregation rates remain constant but fragmentation rates scale as $(\text{mass})^\beta$, detailed balance holds only when $\beta=1$. Away from this solvable case, we employ asymptotic techniques and show that when $\beta\geq 0$, the steady states share similarities with those from the mass-independent ($\beta=0$) model. An instantaneous shattering transition with continuous mass loss occurs when $\beta<0$.