Cluster–cluster aggregation as an analogue of a turbulent cascade: Kolmogorov phenomenology, scaling laws and the breakdown of self-similarity

Abstract

We present a detailed study of the statistical properties of a system of diffusing aggregating particles in the presence of a steady source of monomers. We emphasise the case of low spatial dimensions where strong diffusive fluctuations invalidate the mean-field description provided by standard Smoluchowski kinetic theory. The presence of a source of monomers allows the system to reach a statistically stationary state at large times. This state is characterised by a constant flux of mass directed from small to large masses. It therefore admits a phenomenological description based on the assumption of self-similarity and constant mass flux analogous to the Kolmogorov's 1941 theory of turbulence. Unlike turbulence, the aggregation problem is analytically tractable using powerful methods of statistical field theory. We explain in detail how these methods should be adapted to study the far-from-equilibrium, flux-dominated states characteristic of turbulent systems. We consider multipoint correlation functions of the mass density. By an exact evaluation of the scaling exponents for the one and two-point correlation functions, we show that the assumption of self-similiarity breaks down at large masses for spatial dimensions, d ≤ 2. We calculate nonrigourously the exponents of the higher order correlation functions as an ǫ-expansion where ǫ = 2−d. We show that the mass distribution exhibits non-trivial multiscaling. An analogy can be drawn with the case of hydrodynamic turbulence. The physical origin of this multiscaling is traced to the presence of strong correlations between particles participating in large mass aggregation events. These correlations stem from the recurrence of diffusion processes in d ≤ 2. The analytic methods developed here will have more general applicability beyond the study of this specific problem.

Constant Flux Relation for Driven Dissipative Systems

“…For 0, the scaling h ÿ3 ÿ is already established. It is trivially true for d > 2 where mean field holds, in d 2 due to cancellation of logarithmic corrections [6] and in d < 2 by an exact solution [21]. The scaling of the flux correlation function mCm m ÿ3ÿ agrees with general CFR, Eq.…”

Numerical solutions of the isotropic 3-wave kinetic equation

“…The analogy turns out to be useful in both directions however. For a discussion of stochastic aggregation from the point of view of turbulence theory, see [12,13].…”

Constant flux relation for diffusion-limited cluster-cluster aggregation

“…A huge amount is known about the average mass density in the mean-field case [9] from exact analyses [10] and extensive numerical simulations of the Smoluchowski equation. Relatively less is known about the mass density in the diffusion limited regime but several models have been solved exactly or treated approximately by field-theoretic methods [11,12]. Almost nothing is known about higher order correlation functions in the diffusion-limited regime, despite the fact that they encode the details of the fluctuations which dominate the dynamics.…”