Constant flux relation for diffusion-limited cluster-cluster aggregation

Abstract

In a nonequilibrium 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 realizable. 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.

New relations for correlation functions in Navier–Stokes turbulence

“…Similar relations can be derived for other systems with fluxes of conserved quantities, particularly in wave turbulence (see e.g. Zakharov, L'vov & Falkovich 1992) and in stochastic coagulation (see Connaughton, Rajesh & Zaboronski 2008).…”

Stationary mass distribution and nonlocality in models of coalescence and shattering

“…(1) into the analytic integral expression for the flux J and checking that it remains finite in the limit of small shattering probability p (see Refs. [29,31] for details of the derivation). It turns out that the realizability of constant flux solution depends on the locality of the collision mechanism.…”

Scaling properties of one-dimensional cluster–cluster aggregation with Lévy diffusion

“…Upon substitution into Eq. (25) we arrive at the following suggestion for the asymptotic behaviour, now in mass, of the multi-point correlation functions in stationary CCA:…”