Constant Flux Relation for diffusion limited cluster-cluster aggregation

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.

10 pages, 6 figures

References in corpus (4)

• Cluster-Cluster Aggregation as an Analogue of a Turbulent Cascade : Kolmogorov Phenomenology, Scaling Laws and the Breakdown of self-similarity
• Breakdown of Kolmogorov scaling in models of cluster aggregation with deposition
• Constant flux relation for driven dissipative systems
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Cited by in corpus (5)

• New relations for correlation functions in Navier-Stokes turbulence
• Stationary mass distribution and nonlocality in models of coalescence and shattering
• On the Non-equilibrium Phase Transition in Evaporation-Deposition Models
• Multi-Avalanche Correlations in Directed Sandpile Models
• Scaling properties of one-dimensional cluster-cluster aggregation with Levy diffusion