Constant Flux Relation for diffusion limited cluster-cluster aggregation

TL;DR

This paper establishes a universal constant flux relation for diffusion-limited cluster aggregation, deriving a unique scaling exponent for correlation functions that is independent of system dimension and transport details.

Contribution

It derives the CFR for binary aggregation with scale-invariant kernels and analyzes the conditions for its realization, including locality criteria and applicability across dimensions.

Findings

The CFR exponent is unique and dimension-independent.

Locality criterion determines the realizability of CFR scaling.

Numerical simulations support the theoretical predictions in one dimension.

Abstract

In a non-equilibrium system, a Constant Flux Relation (CFR) expresses the fact that a constant flux of a conserved quantity exactly determines the scaling of the particular correlation function linked to the flux of that conserved quantity. This is true regardless of whether mean--field theory is applicable or not. We focus on cluster--cluster aggregation and discuss the consequences of mass conservation for the steady state of aggregation models with a monomer source in the diffusion-limited regime. We derive the CFR for the flux-carrying correlation function for binary aggregation with a general scale-invariant kernel and show that this exponent is unique. It is independent of both the dimension and of the details of the spatial transport mechanism, a property which is very atypical in the diffusion-limited regime. We then discuss in detail the ``locality criterion'' which must be satisfied in order for the CFR scaling to be realisable. Locality may be checked explicitly for the mean-field Smoluchowski equation. We show that if it is satisfied at the mean-field level, it remains true over some finite range as one perturbatively decreases the dimension of the system below the critical dimension, $d_c=2$, entering the fluctuation-dominated regime. We turn to numerical simulations to verify locality for a range of systems in one dimension which are, presumably, beyond the perturbative regime. Finally, we illustrate how the CFR scaling may break down as a result of a violation of locality or as a result of finite size effects and discuss the extent to which the results apply to higher order aggregation processes.