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 statistics of a system of diffusing aggregating particles with a steady monomer source. 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 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, $d\leq2$. We calculate non-rigourously the exponents of the higher order correlation functions as an $ε$-expansion where $ε=2-d$. We show that the mass distribution exhibits non-trivial multiscaling. An analogy can be drawn with the case of hydrodynamic turbulence. The origin of this multiscaling is traced to strong correlations between particles participating in large mass aggregation events. These correlations stem from the recurrence of diffusion processes in $d\leq2$.